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good morning. so, in the last class, we discussedthe flow through a conical nozzle. we have shown that what are the losses, how they comein because of the 3d effect, and then we said that if we kind of reduce the half cone angle,we can reduce those losses. the ultimate reduction is if you can get the flow to be parallelto the axis. and then, we discuss that why a shaped nozzle which goal is to get a parallelflow at the exit will be more at one touches so far a conical nozzle. so, today now weare going to discuss about design of the shaped nozzles, there are two ways of doing it. one is, so we are talking about shaped nozzles.first is if we have given a specified nozzle geometry, this actually is a roundabout way.first we choose nozzle geometry, and then
we can set up a numerical solution of equationsof motion for the whole nozzle. we can use a finite different scheme, a volume scheme, a finiteelement scheme whatever. so, the primary thing is that the shape must be specified with thisspecified shape, then we solve the full navies strokes. we can give proper boundary conditionsthrough the wall, even thermal boundary conditions we can have a full structure interaction codealso. so, we can solve at present, we can solvefor the full flow field, but the problem is that this is very time consuming and expensive.it will give us the full velocity field, full pressure field etcetera. first of all thesolution itself is going to be very time consuming and expensive because the temperature is quitehigh. the flow is going to be turbulent and
it is quite high temperature. it has to becompressible. so, it is a compressible turbulent flow modelling with the wall conditions whichare neither isothermal nor adiabatic. there is heat transfer through the wall.so, all this needs to be model, right. so, therefore, it is quite time consuming andexpensive, and it will give only one solution for one geometry. our goal is to design theshape. so, then what we have to do is, we have to try different shapes and then, optimize,right. so, therefore, then try different shapes and optimize. so, essentially the entire programis coming up with a design of a curved nozzle or shaped nozzle which will give us the parallelfor the exit and required pressure and required velocity is going to be quite expensive. ifyou do it this way, the easier method, then
it is something that gives us the shape asthe solution which specifies the exit conditions or the required conditions. something thatgives the shape as the solution will be an easier method, much more cheaper and thatis where method of characteristics comes in. the point here is the method of characteristicsis applicable only to supersonic flows. it is not applicable to subsonic flows. so, therefore,the method of characteristics of course for supersonic flows is widely used because first of all, it is simple and secondly, supersonic portion isof importance. we have discussed that in a previous class that the subsonic portion,the converging portion or subsonic portion is not that important because since we arefavourable pressure gradient, we can take any simple shape and that will give us a properflow. supersonic portion is the important
part where we need to have the design.so, method of characteristic then will give us initial shape. at present, the practiceis the initial shaped we get from method of characteristics. then, we go to this to dothe full analysis and check the validity of the initial design. that is a much improvedapproach and gives us much better solution. so, now, we will focus on this method of characteristics.method of characteristics depend on the fact that in supersonic flow, the influence ofa small pressure difference is limited in a specific region.so, if you have a small pressure perturbation, it does not influence the entire flow field.it is limited to a particular region and the method of characteristic is built upon that.so, let us understand what i mean. let us
consider that i have a small source, a pressuredisturbance sitting here and we have a flow coming. this is a supersonic flow at a velocityu, this is source of pressure disturbance. it can be a loud speaker; it can be a smallpulse generator or something. now, the flow is coming at supersonic speed. what happensis that we know that the disturbance propagates sound, particularly the compression wavesare, particularly the pressure waves propagates through the air or the medium through soundwaves, right. so, initially if the flow is subsonic, thenlet me put it this way. first, let us look at a subsonic. first day is, let us look asthere is no flow. we have a source here. when this source propagates, it will propagateuniformly all around the disturbance, right.
so, we get a spherical disturbance going allaround. it is like a spherical disturbance going all around. this is unbiased stationaryflow, the stationary condition. if we give a flow here that is a biased flow, biasedcondition, this will change this pattern. so, now, what happens is that more will bepropagating on this side; less will be propagating on this side. so, there is change in periodmay not be look like this. there is a change in pattern that is called biasing.as we keep on increasing this speed, still the flow is subsonic. some information willalways propagate. there will be some effect of bias, but it will always propagate upstream.also, we keep on increasing the flow speed when it becomes sonic. now, this informationwill just be able to real because what happens
is that is trying to go this side and theflow is coming on this side with the same speed. it will stop the information from propagatingwhen it becomes supersonic. then, it will not propagate upstream at all. it will goonly in the downstream region. so, then if i draw the schematic of propagationof this information, it will look like this, like a cone and this is how the informationis going to propagate. we call point b, this is point a. so, at point a, we have a sourceof disturbance. this disturbance propagates with like a spherical wave at a particularwhich is the speed of sound, right. now, the centre of the spherical wave is moving downstreamwith a velocity u. this is a centre of the spherical wave; this is moving downstreamwith a velocity u. so, at time t it has moved
from this location to this location and thedistance between these two will be equal to u t because the flow is moving this sourceaway. now, at any time if i look from time t equalto 0, it was here and time t equal to t plus 0 plus delta t, it was here. slowly it ismoving away and at every time, then depending on the time t, it is moving in a certain locationand is going also. so, the influence of this that is if i put a microphone here, it willnot be here. only if the microphone is put here, it will hear this, right. so, if thedomain of influence is only limited to this pole, then this cone is given by this anglealpha, right. so, q alpha, this is the limit of influence. so, the zone of influence islimited to a cone of half angle alpha here.
now, first of all how do we get this halfangle? at time t, the disturbance is emitting from here is propagating with this speed ofsound a, right. at time t, this source has reached this point. this distance is u t.the disturbance that emitted from here will reach this point. the distance will be a timest, where a speed of sound, so this is a t. so, the radius of this sphere is a t. so,at every location that is how we get this radius. so, therefore, now this angle alphai have not drawn it properly. it will be something like this. so, angle alpha will be sine inversea t by u t, right. this is by angle alpha. this is a t by u t. actually this will be,yeah h e by h, ok. so, i can cancel this thing. what is u bya? this mass number, right. so, therefore,
this is equal to sine inverse 1 by m, rightor alpha equal to tan inverse a t upon square root of u t square minus a t square. thiscan also be written as tan inverse 1 upon n square minus 1. so, alpha is equal to taninverse m square minus 1. remember that i said at the beginning that is applicable onlyto supersonic flow and that is why it is coming from here. if mach number is less than 1,this becomes imaginary. you cannot define alpha, right.so, therefore, alpha is defined only if mach number is greater than 1. even at mach numberequal to 1, this is 0 is infinite, right. so, that is something that is the limitingcase. so, alpha is now the domain of influence or at the mach alpha is called the mach angle,and this is the domain of influence for this
sound source. so, this is how now the measureof characteristics i will come to. first, i have defined the mach angle alpha. now,i will come to the method of characteristics. let us consider now and these lines by theway are called mach lines. so, the limit of influence is essentially bounded by the machlines. so, what do we see here? for this point here, what will be the angle of the mach linesgiven by alpha? it depends on the incoming flow mach number, right.so, essentially how much the mach lines will in turn depend on the incoming flow mach number?so, that is what the mach line is. so, if i look at these two mach lines, this is theupstream of this mach line. if the flow upstream of a mach line is uniform like in this case,then what we can seen here that the mach line
is going to be straight line that is whatin this case is, but if this flow is not uniform, then at different locations, we have differentmach numbers. the mach line will be curved. so, another point here is that on this domainof influence everywhere, the flow properties are uniform, right.so, within the limit of influence, the flow properties are uniform. here, the flow isuniform coming in. therefore, the mach line is straight. so, if we have a straight machline, the flow properties are uniform downstream of the mach line. so, these are the few thingsthat we actually use when we employ a measure of characteristics that if we have uniformflow, the mach lines are straight lines and within the limit of influence, the flow propertiesare uniform. these are the things that we
are going to use. now, let us look at a nozzleand see that how we use this information in the case of a nozzle. so, let me now go to a proper nozzle, propershaped nozzle. let me consider a shaped nozzle like this. i am not still going to the fullnozzle. i am just going, just showing a part of it. this is our central line. we have letus say supersonic flow coming here into this nozzle. now, we have a point here. from thispoint here, we have uniform flow. so, from this point, a mach line will emerge rightgo like this from since we are talking about a symmetric nozzle. just opposite side, thereis another mach line, another point a dash from where another mach line will emerge andgo like this.
so, if nothing happens in between here andhere, these mach lines will be straight and travel like this, but now if this was a straightsection, this is what is going to happen, but here it is curved, right. as it is curved,if i look at a flow, supersonic flow over a curvature, what happens is the flow comeshere because of the curvature. there is an expansion fan about which it turns and then,goes like this. so, it accelerates, right. so, the mach number here is greater than themach number here. the flow is accelerated. so, if i now come to this point, at this pointthe mach number is more than at this point. so, now another mach line will emit from here,right and for that mach number, this is higher. this mach number is higher. so, what happensto alpha? so, this is reducing. so, alpha
is increasing, right because sine alpha is0, sine sine 0, 0 is 0, sine 90 is 1. so, alpha is mach number is increasing. alphais reducing, sorry because sine inverse alpha is reducing, so alpha is reducing. so, alphadecreases which mean that something that was at this angle, now may be at this angle, salaciousangle. so, if i drop another mach line from here, it will go like this, slight decreasein mach angle. now, if i look at these two, this mach linewas straight with sudden property, uniform property. this is straight with some otherproperty, but these two mach numbers are different. so, when they intersect, they will bring ina change, right which will be dependent on the mach number here and the mach number here.so, now, this mach lines will start to curve,
right. it is no longer straight line becausethey are known uniform properties merging. so, the mach lines will start to change. letme draw it on the one side only c d a dash. similarly, from this side. so, we get patternlike this. we get a pattern like this, like this, we get a pattern. now, as you can seethis looks like a grid pattern where the mach lines are now intersecting at different points.so, we got a point d here, a point e here like that at different points, this mach linesare intersecting, ok. now, in between these two mach lines, theflow is still uniform, but now since this and this mach line are different, this attwo different mach lines. so, the uniform flow will neither be this nor be this. itwill be different condition. although it is
uniform, but it will be different conditionand that is how the flow will propagate. when it goes along, there is going to be changein mach number. so, now, this mach number change will depend on this curvature and theincoming flow mach number. so, the initial curvature that is provided, that will dictatewhat kind of flow will be emerging. so, let me just summarize. suppose we havea uniform flow which enters the diverging portion of the nozzle here, the wall curvatureinitially like we have shown here will establish pressure gradients that are going to turnthe stream lines. so, the stream lines are going to be turned because the fluid willbe flowing along the wall like we have shown here. the stream lines are turned like this.since, we are neglecting friction in this
case, the fluid can be assumed to be slidingfreely,. so, the flow is like in this case, in theexpansion fan which is flowing, sliding smoothly over the surface. so, at this point, we havea small variation in angle d theta. let us say a small variation in angle d theta whichcauses a small pressure disturbance d p to be produced and now, this disturbance willpropagate along this line from here to here and at what angle, it will more depend onthe incoming flow mach number. so, it creates a mach line there. similar disturbance willbe propagating from this point, right. so, this small curvature here now is our sourceof disturbance. the flow was coming smoothly here. there is a small curvature that createsa disturbance.
so, that is the initial point which createsthe initial mach lines. so, these are the two mach lines that are created. now, we havea continuous curvature at this point beyond this. so, because of this curvature from everypoint, there will be mach line that will be coming up, right and by the way if i lookat the mach lines here, the mach lines which are emitting from here are moving in thisdirection. the mach lines which are emitting from here are moving in this direction. themach lines emitting from the lower duct valve are called left running mach lines. so, theseare called left running mach lines, and emitting from upper valve are called right runningmach lines. this is the nomenclature that is used.so, we have left running mach lines and right
running mach lines, and they are crossingeach other in a zigzag manner as we have drawn here. now, this designation of left runningand right running refer to the direction in which the lines approximate to propagate,approximately propagates or appears to propagate downstream to an observer looking downstream.so, if i look, if say if we have an observer sitting here. observer is standing let ussay at this point and looking downstream, and that observer looks at this line. so,this line with respect to that observer is left; this line with respect to that observeris right. that is why these are called left running and these are called right running.so, observer is sitting here and looking downstream. that is why this nomenclature is used.now, if i look at this region a, b, c or this
region a prime, b prime, c prime etcetera.if i look at this region, at that region where if i look at this region, this only one typeof mach line here, right. of course, this is bounded by the other type. similarly, onthis region, there is only one type of mach line bounded by other type here. so, it isonly same type of mach line present in both of this, where as we have already discussedthat if we have single type of mach line, then the fluid properties are going to beconstant. just upstream or downstream of each line.so, if i look at this region, the fluid properties are going to be constants. similarly, in thisregion, the fluid properties are constant. of course, it is going to change from hereto here because the two mach lines that are
present, but within a particular region, thefluid properties are going to be constant, same type of mach number. so, specifying theproperties along any stream line within this, the stream lines are moving like this. letus say choose this stream line. we specify the property along any stream line since specifyin that entire region, right. so, specifying the properties along any stream line willbe sufficient to describe everything in that region. now, what is our limiting? streamline a wall, right. so, if we can specify the property along thewall, this is our limiting stream line. then, the properties all along this as specifiedwithin this zone, of course within this zone and this zone. so, specifying the wall propertieswill specify the properties in between everywhere.
now, let us see that what is happening inthis region more closely. this region let us say that this point there is a curvatureof d theta. the flow is turning by amount d theta which is given here. let us see thathow do we get the property variations. now, when the flow is turning by this amount dtheta, so for that i will draw this diagram, bigger diagram to represent the flow. let us consider this that we have a velocitycoming in this direction u, and the flow is turning by a small angle d theta. becauseof that there is a mach line that is emitting. so, upstream mach number here is m. this isthe upstream mach number m, the disturbance propagate at an angle alpha. so, this is myangle alpha, this is a mach angle that we
have already discussed, and we can estimatealpha as tan inverse 1 upon m square minus 1. now, any fluid crossing this mach linewill have a change in its direction, right. so, the stream line, any stream line crossingthe mach line must have its direction change, so that it continues to be parallel to thiswall, right. so, there is only change in the direction.so, let me consider this. this is the stream line. it is coming here and its directionget's changed. so, now, it moves in a manner parallel to this, right. so, this is whatis the changed direction stream line is going to move and now, just draw an analogy. thisis like a pendal mayer flow expansion fan. so, we know that when the supersonic flowchanges direction because of the increase
in angle, it accelerates. so, its velocityis going to increase. so, let say the velocity now is u plus d u, where as this flow velocitywas u, this is the same flow i am just drawing it on this side. velocity is u, then for thisi can have a, there is a change in direction now to flow and this angle is alpha.let me draw it properly. this is better. so, this is u, this is u plus d u. this term isu which is the increasing actual velocity u plus d u and d v velocity is now changing.we have a velocity in the v direction as well this angle is d theta. so, if i draw thistriangle just for the increased d u d v, this is the overall increase. this is the vectordiagram only for the changed part. so, the actual velocity increases by the amount du, there is a d v time. so, the total velocity
increase is d capital d u.so, there is if we look at this, there is no change in momentum component parallel tomach line. parallel to the mach line, it does not change. so, the change in velocity isparticularly let me list this diagram as yeah. so, the change in velocity is primarily inthe v direction, not in the u direction. from this diagram, we can get d v is u d theta.this is my u, right. this is u d theta, d v is u d theta and d u from here is equalto this is the change in the x component. let me write a different notation capitalu for the actual velocity, otherwise because i am representing the change also with thissmall u. so, i can write a different notation here. so, d v is equal to capital u d thetaand d u is equal to d u which is the change
in velocity. therefore, from there we canget tan alpha is equal to d u by d v, right and then is equal to. therefore, now fromhere what do we see tan alpha in place of d u, i write this here. i write u d theta.so, this is equal to d u by u d theta, right and tan alpha i know is equal to this value.so, tan alpha is equal to 1 upon square root of m square minus 1, right.so, this is equal to 1 upon square root of m square minus 1. so, i can get now an expressionfor d theta, that is the change in angle which is equal to square root of rather let me writeit like this. therefore, d u by u equal to d theta upon 1 square root of m square minus1. so, this is the change in velocity. because of this change in angle or else, we can seethe change in velocity is function of the
change in angle as well as the incoming flowmach number. so, based on that, we get this expression. now, from the definition of machnumber, u square is equal to m square gamma r t, right and from isentropic flow assumption,this is actually applicable to any adiabatic flow we called the isentropic. we get theisentropic relationship in word by t given like this. i can get t coming here.so, let me first write it here. this gives me u square equal to gamma r t naught m squaredivided by 1 plus gamma minus 1 by 2 m square. so, the velocity now can be written as a functionof t naught which is the stagnation temperature mach number. that is more important. thiscan be written in terms of mach number. so, we have represented velocity in terms of themach number. now, what we can do is, we can
differentiate this. so, if we differentiate this, differentiatingthat expression, we get d u by u is equal to d m square upon 2 m square 1 plus gammaminus 1 by 2 m square. let me call this equation 1. what actually what we are trying to dois to change, get an expression for change in mach number. when there is a curvature,what we got is a change in velocity. we want to represent it as change in mach number.so, we get this expression from this then. now, if i combine this, this and that expressionif i combine, then i will get d m square is equal to 2 m square 1 plus gamma minus 1 by2 m square by m square minus 1 d theta. let me call this equation 2.so, let me again see what we have done. we
have derived this expression here which isgetting the velocity change to the angle change and of course, it is a function of mach number.the velocity we have expressed in terms of the mach number, then we differentiated thatand got an expression for d u by u in terms of mach number, and here we have an expressionfor d u by u in terms theta and mach number. when we combine these two, we get this expression.so, what is this expression representing the change in mach number as a function of incomingflow mach number and change in angle theta d theta, right.so, this is what we wanted to get because we have the curvature. we want to know whenthere is a curvature, how much change in mach number we can expect or we can estimate fromthat from this equation? so, the change in
mach number here relate to a change in directionof stream line for isentropic flow because d theta is the change in direction of streamline, and we are considering isentropic flow completely here. this is something that youhave to keep in mind that all this derivation isentropic flow is inherent. so, what we aredoing now is completely for isentropic flow, it is applicable only to isentropic flow,ok. so, therefore, because this was isentropic,this description is isentropic. so, we are doing it for an isentropic flow. so, thisis the variation in mach number because of change in stream line which is brought aboutby changing this wall angle. so, now, once we have this mach number change, we can estimatethe change in other properties. also, d t
by t will be equal to just differentiatingthe isentropic relationships, we can get this. let me call this equation 3. this is a changein temperature. so, once again differentiating that isentropic relationship, we get this.since, we are talking about an isentropic flow without any work done, there is no workdone here an isentropic flow. therefore, stagnation temperature remains constant. so, t naughtis constant in this case. therefore, when we differentiate this t naught goes away.similarly, stagnation pressure is constant. so, we can take the isentropic relationshipfor pressure, differentiate it, we can get the relationship for change in pressure, sothat will be given by change in pressure relationship will be given as d t by t is equal to minusgamma by 2 t m square 1 plus gamma minus 1
by 2 m square. let me call this equation 4.similarly, we can get the change in density d rho by rho is minus d m square upon 2, 1plus gamma minus 1 by 2 m square. let me call this equation 5.so, we have 5 equations listed here 1, 2, 3, 4, 5. all these equations actually representthe change in flow properties when the angle theta or the wall curvature is changed fora supersonic flow with a mach number m, right. we are now getting a change in velocity, changein mach number, change in temperature, pressure and density and this is what we wanted toestimate how the properties change when we have the variation in angle. so, now, thewall curvature then is giving us all the required changes that we are looking for.so, the wall cum curvature now can be replaced
here. we have in this case, what we have doneis you have considered a small curvature, right. so, using this small curvature, wecalculate these changes, right. so, then what we can do is, this is our finite curvature.we can break it into many small curvatures, right. so, for this we know how much d thetais. we guess get all this, then come to this. when we come to this point, now this machnumber has come from that solution and then, we have another change in theta. so, anotherchange will occur, we can get the conditions here.so, we move downstream, we keep on changing the wall curvature and we keep on changing,getting the new flow properties because of change in wall curvature till we come to thispoint at the end. so, the wall curvature can
be replaced by finite number of straight linesegments, right. so, this is the curved wall. what we have done is, we have removed, replacedit by many straight line segments and thus, the flow properties in the flow field correspondingto each infinite small turned can be calculated now.so, if we break this curvature into many small straight line curvatures, we can calculatethe entire property variation here during this curved portion now. so, what we havedone so far once again go back to our original try. we are now focusing on this portion only.this is the straight line c a c b a dash b dash. remember what i said is in this portioneither in this region or in this region; there is only one type of mach line. on this side,we have all right running mach lines here;
we have all left running mach lines. so, therefore,the property variation essentially if i get the variation in the wall, that is good enough.so, from here to here, then knowing the wall curvature here, i can get all the propertyvariations and that is what we are doing. so, what you are doing is, from here to here,we are breaking this wall into many small segments and calculating the variation hereand since, for each of them we have the same property, nothing changes. so, we can getthe full property variation within this zone. so, this is the first part of our problem.our problem actually has multiple parts. first part is when we have only single type of machlines that we have now explained how to get the property variation. one thing for thisportion we need to know theta variation, right.
so, this curvature for this portion must bespecified. so, for this portion, it is not designed. it’s initial curvature is specified.so, we get all this from there. so, this theta variation is specified on this wall, we getall this. now, coming back to this. now, when we cometo this zone, so here like this. so, we have seen here, this portion and this portion wehave only single type of mach lines, when we come to this portion. now, we see thatthe mach lines are crossing left running, and right running are crossing, right andnot only that, if i look at this domain, it has come from here and here, but if i lookat this one, it has come from some other location, right. so, therefore, this mach line up tothis is when it crosses this, it has also
crossed this mach line. so, the propertieshave changed here. so, at this point, there is a different property.when it comes here, again is a different property where as this has seen only one change, right.so, there is a continuous change in the properties. so, now, we have mach lines crossing eachother or the flow crossing each other with different properties and now, that makes itlittle more complex. so, now we have is intersection of two mach lines. first of all, it can bean intersection of one left running essentially always a left running and right running machline will intersect, but the point is that these two mach lines may not have emittedfrom the same disturbance. so, therefore, the mach number and the flow properties correspondingto these two mach lines may not be same, they
can be same also.for example, if i look at this point, this mach line and this mach line are same. so,when we come to this domain, it has the same property, but this and this are going across.they may not be coming from the same mach line or emitted from the same property. inthat case, one mach line as say emitted from a flow with mach m 1. other has emitted froma flow with mach m 2 and then, the mach angles are going to be different for these two. so,that needs to be accounted for. then, the next thing what we are going to see is thatwhen the two mach line intersects, how do the properties change and this is one part.finally, what is our goal is to make the flow straight. our design is not complete. so,next thing what we will do is, we will look
at how the mach lines intersect, what willbe the property variation and then, so far here the wall itself was enough to give usthe conditions. now, we will design the wall curvature that what kind of d theta shouldbe given here, d theta was specified. now, we will change the theta in such a way thatwe get finally a uniform flow at the exit. let us say this is our nozzle. we will geta flow which is uniform like this. this is what we want to do for that. first, we haveto understand how the properties will change when the mach lines intersect and the nextis to complete the design. so, i think we have spent all, most used upthe time today. so, what i will do is, i will stop here today. in the next class, we firsttalk about the intersection of mach lines
and then, continue with the design processusing method of characteristics and finish this discussion. after that we talk littlebit about plug nozzles which are other type of shaped nozzles and then, the effect offriction, will talk about the effect of heat transfer. we will talk little bit there willbe just small descriptions. therefore, main focus in the next class will be completingthis measure of characteristics, particularly when the mach lines cross each other whathappens. so, i will stop here now. in the next class, we will continue from here.thank you.