 this session, last class we derived the speed of sound or the speed of propagation for an infinite small disturbance or a pressure wave through a compressible medium flowing with a velocity and we have seen that the velocity of the disturbance wave or the pressure wave relative to a sound medium can be written as if I write it as a that is the relative velocity of the disturbance wave or the pressure wave is equal to root over d p by d rho or with the definition of bulk modulus of elasticity of the medium we can so this is the velocity of propagation of the disturbance wave or the pressure wave which is the acoustic velocity the sound velocity acoustic velocity relative to the speed of the fluid medium or relative to the flow of the flow velocity of the fluid medium now we have also recognized or we discussed in the last class that this value of d p by rho or e by rho if you express in terms of the bulk modulus of elasticity of the medium can be evaluated as the can be evaluated explicitly in terms of the state variables that is pressure volume temperature or density provided we know the process constraint that means how the changes take place that means the value of d p by d rho under what process constraint and also the equation of state now for this we have to depend on certain physical factors now let us see that this is the condition that the velocity of flow was like that if we consider this a frame of reference for our deduction we took it attached to this pressure wave that means the pressure wave is fixed we see that this a is the speed of sound that is the acoustic velocity with which it is moving that means v plus c that means it is this equal to a that means with which the fluid flow to this pressure this is the pressure wave well and ultimately here it comes with a velocity is velocity is reduced v plus c by an amount d v that is a minus d v as a result its pressure is increased by p plus d p the pressure is p and density is rho and density is change. So, this acts as a compression wave that means the fluid which is flowing in this direction that means in the upstream fluid has in a higher velocity and a lower pressure and density where the fluid at downstream after the shock wave sorry after the pressure wave sorry after the pressure wave its velocity is reduced and pressure and density is changed now if we consider this pressure wave of infinite small thin that means when we have described this control volume this control volume is very small. So, that the frictional effects are neglected and at the same time if we consider this changes to be very fast. So, that heat transfer across the shock wave that means for this change that means if we consider this control volume we can consider the control volume is in adiabatic condition that means the heat transfer does not take place in the short time then we can tell the changes that occur due to the flow of the fluid through this control volume or across the pressure wave to be adiabatic and also reversible. This is because when it is adiabatic there is no irreversibility due to heat transfer heat transfer is 0 usually the irreversibility takes place due to heat transfer across a finite temperature difference which is absent there and more over if we consider this control volume to be very small considering this pressure wave to be very thin the dissipative effects due to friction can be considered as negligible. So, therefore, we can consider the changes to be both adiabatic and reversible the consequence of which together is the constant entropy that means isentropic that means the changes are isentropic. So, therefore, we can write the speed of this as a change in pressure with respect to the change in density we can write we should write it in partial differential nomenclature because one of the variable state variable this constant that means that constant entropy. So, this should be the correct expression for this speed of sound or this speed of propagation of the disturbance or pressure wave with respect to the velocity of the medium. So, now this value we can evaluate for a particular system which if we take as a perfect gas. Now, what is the definition of a perfect gas a perfect gas as you know is defined as from microscopic point of view as a gas or as a system where the intermolecular forces are assumed to be 0 there is no intermolecular forces and molecules are moving in a rectilinear path, but from the classical thermodynamics point of view we define the perfect gas at those gases whose equation of state bears a functional relationship like this p v is equal to r t equation of state means the equation of state of any substance or any system relates this three variable in terms of a functional relation. So, perfect gases are those systems which behaves whose equation of state is given by p v is equal to r t where r is the characteristic gas constant characteristic characteristic gas constant which is a constant for a particular perfect gas and it varies from gas to gas where p is the pressure v is the specific volume and t is the temperature in absolute thermodynamic temperature scale. So, p v is equal to r t or we can write p is equal to rho r t this is the equation of state for a perfect gas. So, if you consider this equation of state it can be proved or you have seen it earlier from other relationships for perfect gas that for an isentropic change of a perfect gas that means for a change with the entropic constant the pressure and density can be equated or related like this p by rho to the power gamma is equal to constant where gamma is the ratio of specific heats ratio of specific heats well ratio of specific heats. That means it is the ratio of c p that specific heat at constant pressure divided by specific heat at constant volume we know this relation that p by rho to the power gamma is constant for an isentropic change or for an isentropic process executed by a perfect gas whose equation of state is given by p v is equal to r t or p is equal to rho r t. So, with the help of these two equations one can derive that del p by del rho the value of this derivative partial derivative at constant is comes to be gamma p by rho or gamma r t. So, therefore, it is a very simplified expression that a ultimately can be written as gamma r t or simply gamma p by rho. So, this is the final expression for the speed of sound relative to the velocity of the flowing medium as is equal to root over gamma r t it is directly proportional to the square root of the absolute temperature. Now after defining the speed of sound we define mainly three categories of flow one is subsonic flow subsonic flow subsonic flow is that subsonic flow is that where the flow velocity is less than the velocity of speed in that medium with respect to the flow medium that is a we have already recognized a dimensionless number known as mac number it was named after the scientist mac that is mac number which is equal to the ratio of the flow velocity to the acoustic velocity or speed of the sound or velocity of sound in that flow or in that fluid at that particular condition. So, we can write in terms of the non dimensional number mac number. So, mac number less than one the flow is known as subsonic flow then another condition is the sonic flow sonic flow where the flow velocity sorry is equal to exactly equal to the acoustic speed m a is equal to one and another category of flow is supersonic flow supersonic flow where v is greater than a that is the flow velocity is more than the acoustic velocity and m a is greater than one. So, it is not only the mathematical demarcations you will see afterwards there is a great change in the physical say physical processes of the system and in the behavior of the hydrodynamic parameters in three different regimes of flow mainly the fluids are the flow are divided into these three regimes for internal flows for all internal flows that means flow through a duct, but for external aerodynamics for external flows or external aerodynamics for external you write aerodynamics aerodynamics means dynamics of compressible flows usually air is taken to aerodynamics which means for external flows of compressible fluid more stringent definitions or divisions of flow are given from these three distinctions one is the subsonic flow one is the subsonic one is the incompressible flow first of all you here write one is incompressible this is always there incompressible flow one is incompressible flow when v very very less than equal to a or in terms of mach number it is equal to less than point three three now for external aerodynamics flow the incompressible flow remains as it is incompressible flow incompressible flow where mach number is now I am writing only in terms of mach number less than point three three there we call a flow as a subsonic flow subsonic flow when the mach number remain within this regime a regime of mach number is point point three three to point eight in this range of mach number the flow density changes appreciably but what happens is that no shock wave appears no shock wave appears here and flow becomes almost subsonic always subsonic almost no always subsonic but there is a region where the flow is told to be transonic transonic flow when the mach number is within the region of one point two all these divisions have been made corresponding to certain classes of flow depending upon the certain physical differences in the physical changes or change in the trends of hydrodynamic parameters with the pertinent input variables so the different divisions have come like that so it has been found for external aerodynamics within this range of Reynolds mach number the flow behaves in a very mixed way that means there is a mixed region of locally subsonic and supersonic velocity supersonic flow and the shock wave appears here what is a shock wave I will tell afterwards so shock wave you can consider that is a wave which creates a sharp discontinuity in the flow field there is a sharp discontinuity in the direction of the streamlines and in other hydrodynamic properties of the fluid so this is the region where the flow is known as transonic flow another region is the supersonic flow supersonic flow where the mach number lies between one point two less three here what happens the oblique shock wave takes place and the density pressure temperature all varies sharply and flow is totally supersonic that means the flow velocity is always more than the velocity of the sound another region is hypersonic which is definitely a supersonic flow but mach number is greater than three and in this regime of flow density pressure temperature all changes explosively so these are basically the different regimes of flow based on the relative values of the flow velocity with the relative values of flow velocity from the acoustic velocity or the velocity of sound in that flowing medium that means depending upon the relative value of the non dimensional or dimensionless mach number mach number we can divide the flow regimes so after this we will see how a disturbance created is how a pressure field created by a disturbance moves through a compressible medium first of all you consider though it is written pressure field due to moving source first of all let us consider the pressure field due to a stationary source let us consider a stationary source here which propagates pressure fields at different times we have considered a time interval of three delta t after which we see how the pressure field moves and what are the pressure locations so you see it is moves in a spherical way that means the spherical field of propagation is generated that means a pressure wave which was sent here at the initial time t is equal to zero when we start our observation after a time of three delta t it has reached a point which I can be found out by a circle or a sphere is a sphere two dimensional plane it shows at a circle it is shown as a circle whose radius is three a delta t similarly which was emanated at the time of after delta t from the observation it has come here two a delta t then again at after two delta t this is at a delta t that means this is propagating spherically like that when the source is at rest now if we consider when the source is moving that means a source emanating disturbances pressure disturbances also move in some direction let the direction of the movement in this this is the direction of the motion of the source then what happens if the moving source is such that is velocity is less than the acoustic velocity that is the velocity of the disturbance wave of the pressure wave that is emanating from the source then what will happen this disturbance wave which is generated and advanced spherically these are always ahead of the source the source is at one so when after delta t it moves at two it emanates another wave at three emanates another wave so you see this way this moves from one to four this is the four this is wrong this is the four so it moves from one to two this is you delta t its movement it is one to two two delta t it is one to three this point three delta t so therefore you see one to two two one to three is two u delta t one to four is three u delta t when it moves this delta t then it emanates a wave then its two to three another disturbance wave so when it comes at four after a time interval of three delta t the pressure wave which has emanated from one it reaches three a delta t this is the outer sphere similarly when it comes at two when it came at two after delta t time then the pressure wave which was given or which was coming which came from this point source at two this has come here that means two a delta t sphere with this as center similarly the pressure wave emanated from the point three at two delta t time when it came from one to two this has come here with a radius of a delta t when the source has come here so the movement of the source are encompassed by the moving that the disturbance wave from so it cannot go ahead of that all right so therefore one can say that an any observer here will in this direction downstream for that source will receive the disturbance before he receives the source so he will receive the disturbance first now here you see that what happens when the situation is that the moving source velocity is greater than the velocity of sound that is u is greater than a or mach number is greater than one in this case situation is like that let this is at one initially now at a time after a time t it has come to the point two which is u delta t after another delta t time it has come to a point three which is two u delta t and u is greater than a and after another delta t time it has come to point four we are always seeing the picture after a time interval of three delta t now you see the pressure wave which was emanated at the point one this has reached after three delta t like this the special location which is this has been covered by this spherical zone so which radius is three a delta t similarly the wave which has emanated from two has encompassed this zone two a delta t similarly at three it is a delta t and here always you see that when it has moved from one two three four so after a time of three delta t when it has just reached here just reached here just after three delta t all the disturbances which it emanated continuously discretize do you have seen at one two three these are like these so the point source has come ahead of the disturbance wave so at from point four we can draw a common tangent to all these spheres and this angle alpha is known as the mach angle alpha the half of this angle and this can be written as sin alpha you can see from here if you draw a perpendicular this is same for all that is the principle for which you can draw a common tangent that this becomes this divided by this this will be three a delta t and this will be three u delta t or here this will be a delta t or u delta t that means sin of this angle will be a by u a by u which is equal to one by m e tan no sin perpendicular by hypotenuse this is the perpendicular this is the tangent this is the perpendicular so alpha the mach angle is given by sin inverse one by m e so one interesting feature here is that you see at any point if a person stands here at any not at three delta t time he will not be aware of the disturbances created by the source that means the disturbance field will be within this mach cone so therefore it is known as zone of action and this cone is known as mach cone mach cone is the cone form like this with this point as the vertex at any instant the special location of the moving source as the vertex if we draw the common tangent this cone is known as mach cone and this zone within the mach cone is known as zone of action and zone of silence so observer will only have a feeling or will be aware of the disturbances created by the moving source when this point or the observer will be engulfed by the mach cone so this will depend upon the time then it will move in this direction and the disturbances will grow on this disturbance will be like this it will go for a delta t so that the mach cone will be expanded so until and unless the point or an observer is taken within the mach cone the zone of action he will not be aware of that here a person or an observer standing here he will see the man first or he will see he will receive the moving source first before the disturbances are reached there that is the reason for which sometimes we see a supersonic moving object we will seen first after we can receive first after the sound comes to this point that is the disturbance so disturbance field reaches this point okay next I will discuss the stagnation properties next I will discuss the stagnation properties what is stagnation what are stagnation properties it is very very important stagnation properties now we have already heard this word stagnation in your basic fluid mechanics classes what are first of all known as stagnation properties what is that stagnation pressure first is stagnation enthalpy stagnation enthalpy all stagnation properties stagnation pressure stagnation pressure stagnation temperature stagnation density you write density all with an adjective stagnation stagnation temperature stagnation temperature that means all with the adjective stagnation we know the word enthalpy pressure density what are or what is meant by the word stagnation stagnation means it is rest now how do you define the stagnation properties in a fluid flow for example if a fluid flowing with a uniform velocity across a section v its pressure is p density b rho now you see if its fluid is brought to rest that means it is coming through a long duct then it is allowed to come to a closed chamber then the fluid will be at rest so let its velocity will be 0 let this is the v 1 this is the p 1 rho 1 final velocity is v 2 is 0 so there will be p 2 there will be rho 2 definitely if we measure the p 2 by a gauge we will see p 2 will be greater than p 1 is very simple from simple physics we can tell because this velocity will be converted into pressure because of the fact that the fluid has brought to rest but it is true that all the kinetic energy for this velocity v 1 will not be converted into pressure energy or will not be converted into pressure this is because of the friction fluid friction or viscosity of the fluid that some of this mechanical energy corresponding to this velocity v will be dissipated or converted into intermolecular energy which will raise the temperature that means if we write the Bernoulli's equation for example the Bernoulli's equation considering a point here and considering a point here when the fluid has come to rest then we can write p 1 by rho 1 plus v 1 square by 2 is equal to p 2 by rho we know that Bernoulli's equation can be written for a viscous fluid with a consideration as this which is the loss of energy that means which is not appearing in the form of mechanical energy which is lost that some part of the mechanical energy is converted into intermolecular energy so in this case v 2 is 0 so therefore simply we can write p 2 by rho is p 1 by rho plus v 1 square by 2 plus minus h m but this we can write provided we consider the entire duct with this closed end is adiabatic definitely from the general energy point of view there is no energy coming in from outside or going from inside to outside simply the Bernoulli's equation will give us like this so if I start with this stagnation pressure now therefore we see the pressure which is being built up rho 2 rho 1 rho 2 so the pressure energy it is not the sum of the total energy that means the sum of the total mechanical energy is this if there could not be any loss the sum of the total mechanical energy could be same so we could have told the entire kinetic energy is now converted into pressure energy the difference is accounted for this but there is a loss so this is less than the total energy but if we consider the process to be frictionless that means reversible reversible reversible that is frictionless that means if we consider the fluid to be inviscid consider the fluid to be inviscid then reversible adiabatic process means isentropic process isentropic process that means if we consider the flow to be isentropic then what happens h f is 0 in that case the pressure p 2 if it is denoted at p 0 this is known as the stagnation pressure p 0 by rho 2 is equal to p 1 by rho 1 plus p 1 square by rho 2 and this is the definition of the stagnation pressure stagnation. That means physically you will define stagnation pressure corresponding to any pressure and velocity in a flow field is the pressure which could be generated or would be generated if the fluid is brought to rest isentropically that means if we imagine the fluid is brought to rest isentropically which cannot be done because isentropic process is a hypothetical process then the pressure which would be generated is the stagnation pressure. So, therefore, this is known as pressure head that is the pressure energy per unit weight in a flowing fluid this is known as dynamic head that is the kinetic energy dynamic head per unit weight in the flowing fluid. So, the dynamic head is also converted into the pressure head. So, the entire dynamic head is converted into pressure head provided the flow is adiabatic and reversible there is no conversion of the kinetic energy in the intermolecular energy because the agent which converts it that is the friction that is the fluid viscosity is absent. So, therefore, in this case this pressure is known as. So, do not tell that a stagnation pressure is the pressure when the fluid is brought to rest where the fluid is flowing and it is brought to rest in practice the pressure which will be generated not refers to a stagnation pressure though colloquially or verbally fluid is made to be stagnant, but pressure when the fluid is made to be stagnant in practice is not the stagnation pressure stagnation pressure by definition refers to a theoretical situation when fluid is brought to rest isentropically then that is the maximum limit that a pressure we can get from the conservation of energy the entire kinetic energy be converted into pressure energy. So, therefore, this is the definition of this stagnation pressure, but the definition of stagnation enthalpy does not have a restriction of the for example, how this stagnation enthalpy you define does not have the restriction of this isentropicness. Now, let the flow is adiabatic let the flow is adiabatic what is you see that here the section velocity is v pressure p density rho let the enthalpy is h here 0 velocity. So, v is equal to 0. So, let this is the p 2 v 2 is 0 rho 2 and let this is h 2. So, now we can write from the general energy equation that u 1 let this is u 1 that is u 2 u 1 plus p 1 by rho 1 plus v 1 square by 2 if we neglect the changes in potential energy between these two sections one is at here another is at here then we can write and without any other energy interactions with the surrounding because we have already considered adiabatic that is no heat interactions we consider the work interactions to be 0 then u 2 plus p 2 by rho 2 plus v 2 square by 2 which is 0 in this case. So, this is h 1 is the plus v 1 square by 2 is equal to h 2 now one beautiful thing is that. So, long it is adiabatic the enthalpy at this stage corresponds to the enthalpy plus the kinetic energy that means this kinetic energy part is completely converted into enthalpy whether friction is there or not difference is that if friction is there then friction is there or not the difference is that that will make a distribution between u and p by rho quantities you understand if the friction is not there then u quantity will remain same for an a perfect gas because you know for a perfect gas the internal energies are functions of temperature only there will be no rise in temperature because of frictional dissipation from kinetic energy to intermolecular energy is not there. So, entirely it will be converted into pressure energy. So, how much will be converted into pressure energy and how much will be converted to intermolecular energy from this part that is the business of the friction that means whether friction is present or not, but irrespective of this condition if the flow is adiabatic then h 1 plus v 1 square by 2 is h 2. So, therefore, this is the stagnation enthalpy and stag written as h 0 stagnation. So, while defining the stagnation enthalpy we can tell corresponding to a particular corresponding to a particular condition of the flow at a particular location characterized by the flow velocity v pressure density enthalpy h we can tell this is the definition of stagnation enthalpy physically. That means if the fluid is brought to rest adiabatically then the enthalpy which would result from this is known as the stagnation enthalpy h 0. So, well so now we will derive certain important relation h 0 is h plus v square by 2. Now, we know for an ideal gas what is the expression for h 0 if you recall back in your from into your thermodynamics you will see the specific heat at constant pressure is defined as del h del t at constant pressure del h del t at constant pressure. We know for an perfect gas whose equation of state is given by p v is equal to r t or p is equal to rho r t enthalpy is a function of temperature only. So, therefore, we can write this instead of h we can write this d h d t because enthalpy is a function of temperature only. So, it is not a function of pressure. So, therefore, at constant pressure differentiation with temperature does not occur at all when we differentiate with temperature it is a ordinary differential total differential because it is a function of temperature only pressure dependent is not there. So, one can write d h is c p d t. Another assumption is that if the specific heat at constant pressure is constant that is known as a calorically perfect gas calorically. What is a calorically perfect gas that even ideal gas is calorically perfect which is not reacting and whose specific heat at constant pressure and specific heat at constant volumes are constants is not a function of any of the state variables these are known as calorically perfect gases. So, for calorically perfect gases d h we can integrate up to this this assumption is not required only the assumption was taken each as a function of temperature after that we can integrate it with this assumption of a calorically perfect gas where I take c p outside and I can tell that plus some constant arbitrary constant of integration this constant is usually not taken this is because we are not interested in h we are interested in its change. So, therefore, this arbitrary constant does not come into the picture. However, one can also defend that it is a reference datum that arbitrary constant is made forcefully 0 if we consider the absolute enthalpy specific enthalpy or enthalpy whatever you call is 0 when temperature approaches 0 absolute temperature. So, this way also forcefully we can get rid of this. So, simply that is why we can write h c p t we do not bother with the constant because it will appear in the terms of the difference. So, therefore, if we look this equation h 0 is h plus v square by 2 we can simply write then c p t 0 for a perfect and ideal calorically perfect gas c p t plus v square by 2. Now, we are interested in deriving the ratios between this stagnation properties and local properties local properties means the value t which a fluid has at a particular point at a particular instant. So, this t 0 by t therefore, comes out to be one plus v square by 2 c p 1 plus v square by 2 c p t 0 by t 1 plus v square by c p t very good c p t. Now, what is c p we can find out that you know that the difference between the heat capacities or specific heat at constant pressure and volume for an perfect gas is equal to its characteristic gas constant that you can find out from if you recall the here I can write the t d s equation which we develop last classes t d s is d u plus p d v a general property relations and t d s is equal to d h d h minus v d p. Now, d u for a perfect gas we can write as c v d t plus p d v just recapitulating the whole thing this we have already learnt at your school also c p d t minus v d t. So, if you subtract this then you get this is 0 then you get c p minus c v is equal to what is that c p minus c v that means v d p plus p d v p d v plus v d p divided by d t and using this equation of state p v is equal to r t this can be expressed as c p minus c v is equal to r. So, therefore, if we use this then we get t 0 by t is equal to 1 plus v square by now c p minus c v is r and if we define define c p by c v the ratio as gamma then we can with the help of this equation and with the help of this equation we can express c p is equal to gamma by gamma minus 1 into r. So, we write this 2 gamma by gamma minus 1 into r. So, v square that is gamma minus 1 2 gamma r t c p gamma by gamma minus 1 r. So, this can be written as 1 plus gamma minus 1 by gamma into v square what is this 2 gamma r t this is a square. So, 1 plus gamma minus 1 by 2 gamma into m a square. So, this is a very important relation that is the ratio of this stagnation temperature to the local temperature we have this ratio 1 plus gamma minus 1 by 2 gamma m a square all right gamma by gamma minus 1 no well it is gamma c p is gamma by gamma minus 1 this gamma will not be there. So, gamma minus 1 I am sorry because gamma r t contains the gamma minus 1 by 2 m a square this is all right. So, we get t 0 by t as 1 plus gamma minus 1 by 2 m a square then using the relationship that p 0 by p for a perfect gas is t 0 by t to the power gamma by gamma minus 1 that is the p t relationship p by t to the power gamma by gamma minus 1 is constant for a perfect gas. So, we can write that p 0 by p 1 plus gamma minus 1 by 2 m a square to the power gamma by gamma minus 1 again using the relationship of row verses t that t 0 by t to the power 1 by gamma minus 1 we can write 1 plus gamma minus 1 by 2 m a square to the power 1 by gamma minus 1. So, therefore, this is the relationship between the ratios of the stagnation pressure to stagnate local pressure stagnation temperature to local temperature and stagnation density to local density t 0 by t in terms of the mach number p 0 by p and row 0 by row well when mach number equals to 0 that means the fluid is brought to rest then t 0 by t 1 these are all derived considering the process to be isentropic. So, isentropicness is inherent to the definition. So, automatically t 0 becomes t p 0 become p and row 0 sorry sorry row 0 becomes row alright question. Thank you sonic boom yes sonic boom is