 So, the problem is that the moment you encounter drag divergence mark number, the drag is increasing because of a shock wave and its effects. So, we call this as a wave drag, it is not because of viscosity effects, it is not because of friction, it is because of the presence of this shock wave. So, what is it let us have a look, it is a component of drag at transonic or supersonic speeds. Why do I say transonic speeds? Because the critical mark number normally is normally why, always it is the mark number below 1 because if the freesty mark number itself is 1, then sonic conditions are going to be there in the leading edge only and that is not the critical mark number because much below that already we have had acceleration of flow. So, it is independent of viscous effects that is the important point. So, it is seen as a sudden rise in the drag with mark number and the shock wave radiates considerable energy and that is what is manifesting itself in terms of drag. So, let us see why we have this phenomenon. Now, if you go to supersonic flow and let us say we took a wedge body. So, on the upper surface we will have P more than P infinity, on the bottom we will have P less than P infinity. So, the wave drag will be the net drag due to the higher pressure behind the shock wave. The shock wave will manifest itself right from the leading edge in supersonic flow and the angle of the shock wave will be a function of the mark number, sand inverse 1 by m. So, you know, you may not know, but you can assume right now. This is a topic that is covered normally in basic gas dynamics, pressure, temperature and density are going to increase, but velocity and mark number are going to decrease, the total pressure and temperature are going to remain constant. So, you should take a flat plate and give it as little bit of angle and submerge that in supersonic flow and infinity more than 1. So, what will happen is that flow comes along these lines and because it is going to encounter bending downwards there will be an expansion wave through which pressure is going to reduce. On the bottom there will be a shock wave, there will be a shock wave on the bottom and then at the end again the flow has to straighten, so there is going to be an expansion wave. So, an expansion wave can be considered to be as something like an opposite or a corollary of the shock wave. So, if you see the pressure distribution, the pressure distribution will be such that you will get a net upward pressure and that is what is the generator of lift and this particular net pressure also has got a component on the back side which is the drag, okay. So, if you actually minimize the shock wave or the drag because of the shock wave you should use a very thin leading edge or a very thin profile or you should use a sharp leading edge because that results in weaker shock waves and hence lower drag, okay. Now, let us look at thin aerofoils. For example, in this particular aircraft called as the F Lockheed 104 star fighter, okay. So, the value of Cl in supersonic flow can be approximated as 4 alpha by root of n square minus 1 where alpha is the angle of attack and the wave drag will be 4 alpha square. Now, these two expressions can be derived by you looking at simple gas-animic equations applicable for supersonic flow, I do not want to do that, I wanted to talk more about some other aspects so I am not spending time in the class on that but you can look up yourself, okay. So, let us see how we tackle the wave drag, this is of more importance to us. So, if there are thin objects like a thin aerofoil or a thin wing and if you fly at supersonic conditions because they are thin they lead to a weaker disturbance and thicker objects like me are going to have higher M infinity, I mean higher effects, thicker shock waves, stronger shock waves which you fly at supersonic conditions. So, what is the solution? The solution is that you change the angle at which the air is facing the free stream, okay. So, you have an example of three wings in front of you. So, when you provide sweep, we will discuss in detail about sweep. The other way is that you use thin wings, so the wings which are used on very high speed aircraft actually have a very small t by c, t by c ratio would be 4%, 5%, never more than that, perhaps 8%, then you can also go for shaping of the fuselage which also helps in reducing wave drag as we will see very soon. And finally, you could do proactive reduction in wave drag by putting some bodies or some, they are called as Cookman carrots or Cookman bumps, these are given by an aerodynamicist who was the chief designer of the Concord aircraft, the chief aerodynamicist of Concord aircraft. He has given these bodies and he has found out that if you present these bodies and if you shape them properly, then actually you can lead to lower drag in transonic and supersonic flight. So, we can also use anti-shake bodies and finally, we can use supercritical aerofoils which are designed to reduce the wave drag by giving a geometry, reflex trailing edge. So, these are the solutions, let us see one by one how these solutions are implemented. Just a video to show you how one can have a transonic flight even at low altitude. So, why are you able to see this cloud of air over the wing and below the wing? What is the reason? Yes, why are we able to see? Why do not we normally see this kind of a thing when you have an aircraft flying and why could you see in this particular flight? Yes. I am Pratik. Pratik, yes. Because of the humidity. So, if you fly in Mumbai city, you will always see clouds over the aircraft. Compared to the sea level. Yeah, so Mumbai is very humid, right? We have 70, 80 percent humidity which means you go to Santa Cruz airport and stand every aircraft in Mumbai is going to have a cloud over and above. Is it true? No. It is not just humidity. Okay, that is what I wanted to clarify. Not humidity alone. Okay, anybody else would like to add? Yes, please pass on the mic. Yes. So, my name is Ashwari. Yeah. And it is due to the fact that the local pressure is quite low. So, the condensation of the water vapor which is present inside the air just condenses and we see it in the form of that cloud. Okay. So, does this condensation take place only at low altitudes? Sir, it may take place at various places like where humidity is present basically and more often it likely to take place at low altitude while taking takeoff and landings. Okay. Where humidity is present basically. Yeah. So, then we come to the same point. Mumbai always has high humidity. So, sir, we can also encounter it here also. In fact, in our previous lecture we saw a video in which aircraft was landing and there was a cloud on that. Correct. But my question is if it is only because of humidity and only because of low pressure then we should always see it when we have an aircraft taking off and landing in Mumbai. No, sir. It should have sufficient velocity so that the local pressure should be low. Correct. Do you think this aircraft is flying at a very special speed that normally planes do not fly? Do you think that aircraft at Mumbai do not take off and land at such speeds? So, mere presence of humidity is not the reason. There is something else. Think about it. Think about it. Okay. Maybe we can get the answer on Moodle. Do you understand my question? I am not saying your answer is wrong. I am saying by your argument we should always see it in Mumbai which has got an extremely high humidity. But when I go to the airport I take so many flights. Very rarely I see it. So, there are certain conditions under which I see. Okay. The reasoning is not correct. The information is correct. Yes, it is because of loss of lower pressure. Yes, it is there because of the condensation of water vapor in the atmosphere. These two are correct. But the reasoning that you are giving that it is because of humidity then I can counter and say that we should always see it. So, it is fine. It is better that we get the answer on the Moodle. Okay. Oh, there lies the answer. So, it depends on also the temperature that is there in the surrounding area. If you have humidity but high temperature such as we have in Mumbai. In Mumbai, what is the condition in Mumbai? ISA plus 15. Right. We saw it in the presentation. If you go to a place where there is humidity but temperature is ISA minus 10 or ISA minus 15. There is a greater chance of seeing it. So, the surrounding temperature also is important for you to get the condensation. Condensation takes place always in low temperature. Right. So, maybe if you go to lay airport in winter, okay, you might be able to see on a humid day or if it is raining or there is a lot of water vapor in the atmosphere on a very cold place. But then lay has a problem of high altitude where density is less. So, then it becomes a very complicated thing. Okay. So, the issue is that if you have very high drag because of the wave drag then you have high fuel consumption and the solution is swept wings which we will see. The other solution is area rule or giving a very interesting shape to the fuselage waste. Now, how this was obtained and what is the science behind it is explained in a nice video. So, I thought I will just show you the video. The sound barrier had finally been broken. But there it was what I call a brute force approach in the sense that your rocket just rammed that airplane through the speed of sound. But the drag was so high that they used up all the fuel in just about five minutes. So, it was not practical supersonic flight but it did accomplish the breaking of the barrier. There needed to be a more efficient way to break the speed of sound. Dick Whitcomb set out to find a way. Whitcomb found that when a plane reached near supersonic speeds the drag around the wings would increase by as much as a factor of five. He saw that much like a bullet the fuselage was extremely aerodynamic without the wings but when the wings were added an aerodynamic bump was causing incredible amounts of drag that was slowing the plane down. It became obvious to him that he had to find a way to take the bump out of the equation. Whitcomb's test showed that when he added the entire area of wings and fuselage together the drag or aerodynamic bump was exactly the same as the drag of a fuselage with wings. He worked tirelessly to find a solution when one day as he was thinking about the problem the solution hit him like a bolt of lightning. He must indent or pinch in the waist of the fuselage. This new shape of the fuselage would closely resemble the shape of a coke bottle. Whitcomb was astonished to find that by changing the shape of the fuselage he took the bump out of the equation and allowed the plane to become as aerodynamically smooth as a fuselage without wings. This very simple fix came to be known as the area rule. I had the idea then we built some models to try and demonstrate it. We built airplanes within coke bottle shaped fuselages and lo and behold the drag of the wing just disappeared. So this is Dr. Richard Whitcomb from NASA who gave us Whitcomb aerofoils as well as the area rule. So what he said just now is that when they were trying to design aircraft to fly faster and faster they found that as you up reach sonic speeds the fuel consumption increases so much so that you would consume the entire fuel in only 5 minutes of flight. That means there is impractical to design an aircraft. So he found that if you take just the fuselage it is very good aerodynamically in supersonic flight. But the moment you add wings it creates huge amount of wave drag. So the first experiment he did was suppose I distribute the area of the wings over the fuselage and make it like a bulged fuselage. So he found that the drag of a fuselage with the bulged center equal to the area of the wing is equal to the drag created by the aircraft plus the wing. So that way he figured out that it is something to do with the sudden increase in the area of the cross section as you come towards as you go along the length. So with that he got this idea one day not through any scientific reasoning or through any calculations but just looking at the results and he says what do I do so that I do not get a sudden bulge in the area of cross section as I go along the length. So that the answer is that you reduce the area of the fuselage or the cross section area of the fuselage. So he just figured out that the place where you have a wing or a tail if at that place locally you reduce the area of cross section of the fuselage so that the net area of cross section is not changing rapidly then you will not get high wave drag and then they verified it by wind tunnel testing and finally we had the area rule and it was applied then to the design of many aircraft and once that was done it was possible to create realistic aircraft like this Convair F-102 which could fly comfortably supersonic for a reasonable amount of time. So this one discovery opened the way or opened the path for high speed flight in aircraft. The other important thing is sweeping the wings. So let us see how sweeping the wings is going to be beneficial. For the sweepback we have to thank Dr. Adolf Buzumann and Dr. Adolf Buzumann is the person who first gave the idea that we can encounter or we can... So he was the first one to observe the creation of shock waves he was an experimentalist and then he realized that the air over the wing or the drag effects of air over the wing are dominated not by the free stream velocity but by the velocity perpendicular to the wing. This is an observation. So he just figured out that what is important is not the free stream Mach number or the free stream speed but the speed normal to the wing and if the wing is unswept then those two speeds are equal. So with this he got the idea that can we take the wing behind so that the free stream component or the free stream velocity can have two components a normal component and a span wise component. So this is called as sweep. The dotted line is unswept or normal aircraft wing and the one on the back is the swept wing and then when needed you can bring it forward. So along the lateral axis you provide an angle called as a sweep angle and also remember it need not be only back it may also go forward. So what we have to provide is an angle between the lateral axis and the wing center line or wing reference line it could be behind sweep back or ahead sweep forward both of them are aerodynamically identical. That means both of them are going to reduce the wave drag because both of them are going to create a component along the normal to the wing span lesser than the free stream component but the difference is in a sweep back the component along the wing is going to be outwards and in sweep forward the component along the wing will be inwards and that makes a big difference plus there are other interesting differences also which we will talk about.