 Welcome to capsule number 5, in this capsule we are going to look at some terms related to wings. So now we move from aerofoils to wings and today we still look at aerofoils but talk about critical mark number wave drag and swept wings. In the next lecture we are going to look at finite wings and I think drag effects. So let us have a quick overview of what we have in store today. So there are a series of concepts to be covered mostly to do with the critical mark number and the drag divergence mark number and then we look at some types of wings which are available to try and give us a better control over wave drag. So what is M critical? So M critical there must be something critical because of the name itself. So to see what is M critical we will watch a short video. What actually happens to the air and the aircraft when approaching the speed of soft? The air flows speeds up as it passes over the wing and reaches the maximum speed at a certain point on the wing. So the mark number of the air at this point is greater than that of the aircraft as a whole. As the aircraft speed increases so does the local mark number at this point on the wing. Eventually just at this point on the wing the air reaches the speed of sound although the aircraft as a whole is still flying slower than sound. The aircraft's mark number when this happens is called its critical mark number usually written M crit. So this is very clear because of the angle of attack or curvature or camber there is going to be acceleration of the fluid flow both above and below the airfoil. So the local mark number is going to increase from the free stream mark number. There will be some free stream mark number at which sonic conditions are first reached anywhere on the airfoil remember it could be upper surface, it could be lower surface, it could be front, it could be behind. The lowest free stream mark number at which sonic conditions are first reached anywhere on the airfoil is called as the critical mark number but the question is why is it critical? It is critical because it tells you that from now onwards there is going to be some portion of the airfoil which will have sonic flow mark number more than one locally. Sonic flow is no problem except the fact that it results in a shock wave and across shock wave there are serious problems. So that mark number is the critical mark number and the pressure coefficient value so the pressure coefficient where you reach the sonic conditions first is called as the critical pressure coefficient or CP critical the local V is 1 mark number 1 and V infinity is critical mark number and the place where it hits or where the place where the sonic conditions start and the shock wave is presented that particular place is where we have the most negative the pressure coefficient. So this free stream mark number 0.8 is the critical mark number and this mark number we would like it to be as high as possible so that we can fly faster and faster without encountering the problems of sonic flow. So if there are two airfoils airfoil A, airfoil B and airfoil A has a higher critical mark number it is a better one because it allows you to travel faster without encountering. Now when you have m equal to 1 you get a weak shock wave. So the actual value of m critical it varies from wing to wing depending on the wing profile wing geometry. The main the two main parameters that affect the critical mark number of the airfoil are its camber and thickness basically camber and thickness are the ones that create acceleration at a given angle of attack if you have a thicker airfoil there will be larger acceleration if you have more camber there will be higher acceleration. So the thick airfoil because it deflects more it is going to have a lower critical mark number that means at a lower value of m you will have sonic conditions first reached. So with this we come to the concept of something called as a sound barrier. So sound barrier is a barrier but it can be easily broken. Now this was one of the very interesting misconceptions in early days of aviation. So what used to happen is as aircraft began flying faster and faster earlier the maximum speed was limited by the power available of the power plant they were using mostly turboprops piston props. So they were not able to fly faster than say mark number 0.6, 0.5 etc. But when we started getting more powerful power plants and eventually the jet engine the thrust available became much larger. So aircraft could fly faster so they were able to fly mark 0.8, 0.9, 0.92 and then the pilots reported that as soon as you cross a particular value of mark number below 1 the whole plane starts shaking there are lots of vibrations and you just cannot fly faster than mark number 1. So many people thought that there is some kind of a physical barrier speed of sound there is a barrier which cannot be surmounted but then some people said hey we have bullets which fly faster than speed of sound and bullets are also flying objects. So if bullets can fly that means given sufficient force you should be able to fly. So we knew that we can fly from anecdotal evidence of bullets but we were just not able to fly. So for many many years there was this so called sound barrier and then it was finally broken in 1947 when Chuck Yeager flew an aircraft called Bell X-1 for the first time at mark number more than 1. So yes it can be broken and let us see what it is and why we call it as a barrier. So let us revisit the pressure coefficient to get an understanding. So at very low speeds and when I say low speeds I talk about mark numbers up to approximately 0.3. At these low speeds you can assume Cp to be constant ok. So Cp will be equal to some Cp called Cp naught does not change with mark number. Now when you go beyond mark 0.3 or so and up to around 0.7 remember I am not using absolute values I am saying approximately. During these conditions the Cp starts changing it starts actually reducing and the formula which is called as the Prandtl-Glauert rule is a very simple formula that correlates the Cp at any mark number with Cp at very low mark numbers and this is because of the compressibility effects. This change in Cp occurs because of compressibility effects and actually if you plot the value of Cp versus mark number you will see that it is almost constant Cp 0 and then it starts increasing and there is one mark number called M critical at which Cp is Cp critical and it keeps on increasing further. But actually the rate of change or the rate of increase in Cp will not be like this it will actually be little bit more non-linear after little bit after Cp critical. Remember you cannot apply this formula in supersonic flow where M infinity is going to be more than 1. In fact interestingly you just reverse and call it as root of M square minus 1. So let us see the critical pressure coefficient. So if it is a thin aerofoil you can fly longer higher mark numbers in the free stream to achieve critical mark number if it is medium it is lesser if it is thick then it will start increasing. So therefore that is a line which will give you the locus of the location of the critical mark number and this line is a universal curve it does not depend upon it does not change with whether it is a thick aerofoil or a thin aerofoil in the sense that there is a curve which can be applied to almost any geometry. So this line is available if you have the geometrical data okay you can get the value the value of Cp will change depending on thin medium and thick and it will increase as per the frontal cloud rule and it will reach the critical value at some particular point. So you can apply this uniformly okay. So let us see this we will just do little bit of maths to get the value of Cp critical. So we know that the pressure coefficient is defined as the difference of pressure between the local pressure and the free stream pressure non dimensionalized by the free stream dynamic pressure q naught q infinity okay this is just revisiting last time one. So now what I do is I do some mathematical jugglery so that I can get p infinity by q infinity outside so it is a ratio of p by p infinity minus 1. This is our first equation Cp is equal to p infinity by q infinity times ratio of the pressure local by free stream minus 1 okay we also know that dynamic pressure is defined as half rho v square therefore q infinity will be half rho infinity and v infinity square. Now since Mach number m is defined as v infinity by a infinity local Mach number therefore you can reproduce or replace and say that q infinity is equal to half rho infinity m infinity square a infinity square. Now you just plug it in that equation so if you define gamma is the ratio of specific heats so the sonic speed is defined as root of gamma rho by q. So from these three equations at one place you have a infinity square replace it with a gamma rho naught rho infinity by q infinity so you can get q infinity as half m infinity square gamma p infinity okay. So we substitute that in the previous expression now you will explain you will explain this better to yourself when you derive it yourself. I am going to upload the slide so you will be able to derive it and then you will be able to get a feel much better it is just simpler replacement but now p by p infinity or p0 by p infinity is equal to 1 plus gamma minus 1 by 2 m square power gamma by gamma minus 1 for isentropic flow relation. So now we have to assume if the flow is isentropic then replace inside now so p by p infinity will be what it will be p by p infinity will be just replace m by m infinity okay. So therefore you put it inside the expression take the ratio of p and p infinity and put it inside finally if you define cp equal to cp critical by m infinity equal to 1 okay so you can get cp critical as 2 by gamma m infinity square ratio correct where the two conditions m is at the point where you reach critical mark number and that point m is equal to 1 by definition so therefore you can get an expression. So this is a neat expression it only contains gamma which is the gas constant unless the gas dissociates and gamma changes from 1.3 to some other number it is valid and that is true in isentropic flow so all you need is m infinity so now where is the aerofoil in this where is the thickness where is the camber nothing it is just a function of m infinity so you can call it as a universal expression for cp critical okay just put the value of m infinity and you will get cp critical remember it is only cp critical not m critical just the value of cp so the value of pressure coefficient at critical mark number correct or at the place where sonic conditions are first reach is always the same it is just a function of m infinity so if your critical mark number is higher m infinity is higher so cp critical is going to be got by this expression. So as m infinity becomes n critical as m equal to 1 so it is a straight forward expression now problem is how do you get m critical how do you know at what free stream mark number the flow will accelerate so that sonic conditions are first reach because at a velocity higher than n critical there will be larger areas of low exposed to sonic flow so what is the point at which it happens for the first time for that we need a method so there are two methods which are available we will discuss both the methods one is a simple graphical method the other is using the equation which we just now derived so graphical method is actually very elegant and simple on the y axis you plot cp critical on the x axis you plot mark number okay so obtain a plot of cp critical versus mark number how do you get this plot yes exactly where are you guys sleeping from the equation that we just derived what is the equation I will show you again from this equation I just now told you that cp critical is a function purely of m infinity for any aerofoil so we have this expression just put the value so using this expression you can get the plot a simple plot okay it is a quadratic plot because there is m infinity square okay no questions now the next is obtain the value of cp not that means cp at lower mark numbers are usually this is given usually it is available from the aerofoil data but suppose you do not know okay suppose you do not know then you will have to do a guess then you plot cp versus m infinity from the Pantel Glouder rule so what you do you get the cp not which means you get the value at which you cut the y axis because the cp not is going to be a function it will not be the same for all aerofoils so cp not has to be got from either experimental data or from some online calculator or some way you have to get cp not but how does cp vary with m infinity from cp not is by the Pantel Glouder rule so you will get one more line like this now at the intersection of this you can get the value of cp critical because the red line is a locus of cp critical for all wings or aerofoils which fly at a particular mark number and the blue line is the locus or the value of variation of cp with mark number for a particular aerofoil starting from a value which is the intersection on the y axis wherever they intersect that is the critical mark number this is one way of doing it there is also a method to do the whole thing mathematically which means just solve these two equations simultaneously okay so what you can do you can say okay this is a Pantel Glouder rule and you can notice here that cp not is going to increase as m infinity increases or as cp not increases the m infinity will increase or vice versa so at sonic conditions so you just equate them at m equal to 1 and by equating them you can actually solve and get the expression it is the same thing either you do it graphically or you do it by simultaneous equation. Now some people have this misconception that it is very easy to get a critical mark number location just look at the place where the thickness of the aerofoil is maximum that will be the place where you will have the maximum acceleration so that is the place where the mark number will be equal to 1 so as the free stream mark number increases sonic conditions will first be reached at the maximum thickness point correct so many people say it is very simple that is not true I will show you an example for example look at this aerofoil okay this is a simple NACA 0012 symmetric aerofoil 12% thick so you notice that the cp is maximum at x by c equal to 0.11 that means at 11% location of the card from the leading edge the cp is maximum but thickness is not maximum at 11% the thickness is maximum at 30% so even though the maximum thickness occurs at 30% of the card the highest cp has already been achieved at an upstream location so do not have this misconception that the location of the maximum thickness is the location where the sonic conditions are first reached or where the location where the cp is maximum so it depends on local acceleration you have already drawn these plots last time in the tutorial okay so you have to be careful so now if you want to know more about why this happens I would encourage you to go to the main source Anderson which is the basic textbook we are following and it is nicely explained why this happens so that is a self study for you I do not want to talk about it more here so they are not corresponding and this is a very interesting observation so to get the value to get the location where the velocity is maximum you actually have to look at the complete geometry of the aerofoil and not just the maximum thickness location okay so critical Mach number is a Mach number at which sonic conditions are first reached why do we care that is not what we really care the designers or the pilots are mostly interested in this particular Mach number which is called as the MDD the drag divergence Mach number and the name itself tells the whole story it is a Mach number at which there is a shooting up of drag and that is the reason why people thought there is a sonic barrier now my question to you is do you think that the drag of an aerofoil will start shooting up at a Mach number where at a critical Mach number or will it be before it or will it be after it what do you think at what free stream Mach number will you have excessively higher drag buildup before M critical at M critical after M critical or none of the above we have a fourth choice also what do you think after why after why not at M critical why not at M critical because at M critical you have M equal to 1 with results in a weak wave a weak shock wave and weak shock wave will not give you the maximum drag as the Mach number goes slightly beyond this M critical you start getting higher drag ok. So, the Mach number at which drag shoots up and shoots up means what rapidly it may become 10 times the drag at M critical that Mach number is the critical Mach number but that is not the definition ok the definition is the point at which the graph suddenly undergoes a change in the slope and this particular sharp increase occurs not only because of shock waves but also because of the flow separation that is induced by the shock wave. So, typically this drag divergence Mach number will be M 0.02 beyond M critical typically. So, if the critical Mach number is 0.83 it will be approximately 0.85 approximately it can change slightly also depending on the geometry of the aerofoil but generally it is 0.02 Mach number so slightly beyond. So, sonic conditions appear and then very soon you start getting a fantastic increase in the drag so that is the problem as I said shock wave is one reason but that is not the only reason shock wave is only a generator of problems it is not the problem in itself because of shock wave very soon you will experience flow separation and that comes because of the adverse pressure gradient ok but there could be a weak shock wave which may not lead to flow separation. So, therefore it will create additional drag but not make it so bad that the flow separates. So, across the shock wave we know that the pressure is going to increase and the velocity is going to decrease.