 So, the next theory, the skipping stone theory or the theory in which momentum is transferred I also call it as a bullet theory, so you fire a bullet, there is a reaction, so let us see. So, what is this bullet theory? The bullet theory says that you have an airfoil type body at a particular angle or no angle but you have curvature, in this case the bottom surface is almost flat, the top point is curved and we have also put it at some angle, now we are firing these bullets what are the bullets? The bullets are the flow particles, the fluid particle, so the particles above or little bit away from their airfoil are going straight undisturbed because they do not hit the body. The particle which is hitting the body is being deflected downwards and the deflection depends upon the curvature of the local part, so in the bullet theory we have bullets hitting the bottom of the wing transferring momentum to it which is upward and in airfoil it is true that the air molecules that hit the bottom wing they transfer the upward momentum to it, nothing wrong in these two, these two the analogy is correct and a large part of lift a non-trivial part of lift is generated by this principle also, so we are not saying it is wrong but we are saying that this is not the only way of doing it and now let us see how we can debunk this theory for many, many scenarios. The first scenario is that in this particular example the bullet do not hit above the wing, they go above the wing therefore according to this theory the particles which flow just above the airfoil should not be disturbed, they should go straight but we have seen that the particles that are flowing above the wing do not go straight, they also bend down, the particle that go below the wing also bend down, the particles which go above the wing also bend down, the particles which are very far away from the wing they do not bend down but a particle which is just above the wing also bends down, so this is one place where the theory bullet theory fails. Secondly the shape of the wing does not matter to the bullets except for the angle of reflection whereas the air pressure on the top of the wing is only a small percentage lower than the pressure on the bottom, so there will be a difference but it does not matter as far as the bullet is concerned. The bullets are all going parallel they do not hit each other but air particles are not always flying parallel, air particles are actually colliding with each other and you know there is something called as a kinetic theory of gases which tells you about the general phenomena of Brownian or normal motion. In the Brownian motion the air particles are colliding themselves huge number of times, so they are not a bunch of bullets moving independently, they are actually a huge number of bullets which are colliding with each other and also moving around, so this is something that is not taken care. Now bullets are going to weigh reasonably large some grams, 5 grams, 10 grams etc., so they have a huge momentum because of mass okay, air molecules are very very light okay, they may be large in number but they are very light, so if momentum is the only way, the momentum of the air transferred to the wing is actually quite small, so just by momentum you cannot get a lift, so you cannot fly an Airbus A380 with 1000 passengers just by the momentum transfer of some air particle simply below the wing, too much to expect that just by momentum of air you will get such a large amount of lift okay, so the bullets that miss the wing above below they are undeflected but you know that far below and far above the wing okay, the in fact it is generally said that if the wingspan is 10 meters then roughly 10 meters above and 10 meters below the aircraft the fluid is disturbed, the level of disturbance is highest near the surface and as you go up the level of disturbance reduces but as for the bullet theory if you miss the wing you are unaffected okay, so what does skipping stone theory says, skipping stone theory says that take sand, take a plank or take a kind of a dish put sand on it, represent sand as molecules of air and you take this plank okay, this is a plank you have sand here and you just move it like this, so as you move it like this and if you give a small angle, so you give a small angle and move it like this, so the particles here will be moved aside, the particle below will be moved down and the plank will move forward, this is called as a skipping stone theory or the theory, so in this particular case the sand particles are simply pushed aside okay, so here also the same argument is given that as you move this particular plank the momentum moves the reaction of the motion of this plank into this bed of sand actually deflects the sands downwards okay, but the same does not happen when you move a plank through fluid okay, because air and water okay, they are not sand, air and water are not sand they are different, sand is not fluid, sand consists of distinct particles, fluid consists of much more than just sand, so the behavior, now in certain situations flow of sand can also be similar to flow of a fluid, I am not denying that, but I am just saying that in general you cannot equate water and air or fluids with sand, so in fact when you move a plank through water or a wing through the air, what happens is not what we saw for the motion of the sand in a plank, so Tanyi tell me what is the difference, if you move a plank into sand and if you move it through water, is there any difference in the flow, in case of sand the sand is simply pushed aside, in case of water or air something more happens, let us see what happens, so when you swirl, when you move a plank through water, you can try it out, you can try it in the bath tub or any experiment, you will see that there is some swirling which takes place, especially in the front portion, the water will actually move from below the plank to above the plank in the front okay, as you move it and this is because there is a diffusion, force fluid undergo diffusion and some kind of a circulation rounded flow gets created, so you can try the experiment next time, you just move your hand in water, you will not see that the water gets directly pushed behind, ahead it starts curling, so there is something different, the phenomena is a bit different when you move it through sand or when you move it through fluid like water, so let us see what this particular thing is, this particular behavior or this particular phenomena is called as circulation and circulation is basically a mathematical quantity, circulation is used by the aerodynamicists to try and describe the amount of rotatory motion created in the fluid when you move a body or presence of body in a fluid stream or motion of a body in a stationary fluid or whenever there is relative velocity, the phenomena that causes curling of fluid is called as circulation and mathematically it is defined as a line integral, so what you do is around the body you just take the integral of the local flow velocity into ds which is a direction vector, so you go around the body, so basically you put a string around the body and now you traverse on that string and just do the line integral, you do this not line integral, the contour integral of the velocity and ds value and if you find that the net value of the integral is 0 that means the amount of water that curls this way is same as the one that curls this way exactly the same when I say amount I do not mean numerical value I mean the integral v dot ds then you can say circulation is 0 because whatever rotational circulation has been imparted same has been cancelled by the other direction but when you actually do this integral you find the value is non-zero there is a net circulation still remaining and therefore we come up with the definition of a condition called as a kata condition and this kata condition is basically a condition that when you actually put a body in flow in a fluid flow and you have these particular flow we observe that the flow leaves trailing edge smoothly now this is something that is experimentally verified it leaves the strailing edge smoothly now it cannot actually unless there is some value of circulation because you if you do not define some circulation then this is what will happen and that means the flow is going to have some infinite velocity at the trailing edge so at the sharp trailing edge the flow will undergo infinite velocity because from a direction this way it has to now flow that way so this cannot happen and for this not to happen or for the flow to leave the trailing edge smoothly there is a necessity that there is some physical quantity called circulation which acts in such a way that it actually pushes the point of flow reversal up to the trailing edge so that the flow leaves trailing edge smoothly so this is basically and this is enforced by friction this condition is imposed by friction if you have frictionless flow which is theoretical or if you have on paper some kind of a theoretical flow where you can assume friction to be 0 then you may encounter such flow patterns but in reality there is friction present there is viscosity present and therefore you will see that it leaves smoothly so one way of encountering this or one way of imposing this condition is that the velocity just above and just below the trailing edge v1 up and or vtip up and vtip low is exactly the same this is one way in which we can numerically impose and there are other ways also in which we can be imposed so the question for you is what is the Kutta condition imposed mathematically and this question I want you to answer via moodle because I want you to read up about Kutta condition I want you to understand why this condition has come about who is Kutta who is the person who gave this condition how did he do it how are we going to apply it and what happens if we do not if we if we try to investigate a flow without putting this condition what happens did you get no lift do you get less lift so do some research on this condition look at the historical development and educate us using the moodle okay let us proceed further now another important observation which I would like you to make is that it is not just the air below the wing that is pushed down in the skipping stone theory what you expect is that air above is not going to be disturbed too much but the air below is going to be pushed down actually you can see that there is circulation because of which all across the flow field there is some kind of a circulation setup the other thing which is very important is that this particular video so you see this particular plane is made to fly through a stationary almost stationary stream and what you observe is that as the plane goes ahead the air above the wing quite from quite far away is also being pushed down so most of the air is pulled down from above the wing so there is some kind of a suction created above the wing also the the air below the wing anyway is being pushed down but the air above the wing is also being pushed down and we observe another interesting thing in this small small chip file is that at the tips something interesting is happening at the tips the air is being pulled down but it also goes into some kind of a vortex so it is not that just the air is being pushed down the air is pushed down above the wing there is pushed on below the wing at the tips the air is curled and goes down and these curls actually move down this particular phenomena that you are seeing at the wingtip is called as the wingtip vortex and this wingtip vortex is responsible for one type of drag called as the induced drag we will read about it I think the next lecture is going to be on drag and various components of drag so for any lifting vehicle or any lifting aircraft induced drag is a very large component and the numerical value of induced drag will depend upon the strength of this vortex so if you want to reduce the induced drag you must do something so that the strength of this vortex or the speed at which the air is curling and the size of this particular vortex it has to be curled both the speed and the dimension of this vortex and to do that we use some devices at the wings called as the wingtip devices there are various types of wingtip devices many of them are called as winglets even in winglets there are many types there is a just a simple winglet there is a H shape winglet there is a L shape winglet all kinds of winglets are available whatever they may be the main purpose of the winglets is to somehow reduce the strength of these vortices and some smart people have also suggested that this curling motion of air at the wingtip why can't we use it to generate power by putting some small turbine type devices there and you are getting this circulatory wind at very high speeds so use it to generate some power use it to drive a small fan so people have proposed many things they are possible but then what should look at the cost complexity and weight because of providing those devices so most people have decided to use passive devices like winglets to minimize the tip vortex strength so skipping stone theory would not be able to explain this particular phenomena similarly there is a Venturi theory also as I mentioned to you says that because of the top curvature there is a nozzle type effect that is not true because actually there is no nozzle compared to the free stream up and free stream below the thickness of the wing is very small and if that is the only reason then the effect should be very very small so kilometers above the wing and kilometers below the wing there is free stream and just by that if you say nozzle effect so now let us look at actually what is happening and now we try to answer the question of how lift is really generated so we look at a new idea called as a streamline curvature okay and that particular idea should explain what we are trying to understand so as per this particular explanation of streamline curvature the the primary point is that because of certain reason because of the presence of the body if the flow streamlines undergo some turning and some curvature whatever be the reason presence of the body in free stream causes change in the curvature of the streamline so we are going to look at a concept where presence of the body in the flow stream has led to change in the streamline curvature and whenever it happens in flow then two important equations come into play okay so if you look at inviscid flow equations okay I am calling it inviscid because I want to right now neglect the viscous effects viscous effects are actually more predominant when you look at creation of drag or the numerical value of drag so along the stream wise direction that means along the direction s the for so the flow particle which moves along this curving streamline so do you agree that if I remove the body the streamlines are straight okay in assuming a steady flow and when I bring the body then something happens because of which the streamlines undergo some change in curvature so if they undergo change in curvature the reason for that is a presence of the body so the presence of the body in the flow stream has caused change in curvature if the change in curvature happens if and only if it happens then we have to look at the inviscid flow equations both along the stream wise direction between the s direction so the flow particle has a velocity v as you know the flow particle the streamlines are always drawn along perpendicular to the local flow velocity so we have a velocity v along direction s which is a stream wise direction and we also have a direction r which is perpendicular to the stream wise direction called as a radial direction so you can apply equation and you can get this particular value rho v dv by dou v by dou s is equal to minus dou p by dou s this is along stream wise direction and along the normal direction rho v square by r which is the something like when you have a when you have a body moving along a curvature you have on that some kind of a force acting called as a centrifugal force which is acting on it correct so that is rho v square by r and that is a function of the change in the pressure in the radial direction so in the normal direction so these two equations we have to keep in mind right now and we have to now apply to them apply to the flow field whenever we have any curvature so I am replacing once again reproducing the equation rho v dou v by dou s is equal to minus dou p by dou s along the stream wise direction notice that if I assume rho is constant if I assume and you know condition rho is constant we get the Bernoulli's equation so this is nothing but Bernoulli's equation in the incompressible domain now the normal equation which I showed you this is similar to circular motion whenever a body goes along a curvature it undergoes those forces for example if there is a car that is moving along a curvature then the car experiences some force and v square by r along the center of the curvature similarly a fluid particle is going straight on the streamline there is no curvature now you bring in the body whether you bring in a flat plate or symmetric aerofoil or unsymmetric aerofoil or thin aerofoil it doesn't matter if because of the presence of the body there is going to be a change in curvature there is going to be a force acting radially and the magnitude of this force is going to be directly related to the rate of change of pressure so basically what is happening is suppose you have a fluid with the pressure p infinity on top and bottom and you introduce this small shape this curved flat plate with some curvature r so there is some pressure p l lower below the body there is some pressure p u above the body so on the upper surface the partial variation of the pressure with r is equal to rho v square by r and because r is non-zero because there is some positive curvature r therefore rho v square by r is non-zero so if rho v square by r is non-zero that means that the value of p infinity minus p upper is more than equal more than or equal to 0 more than 0 actually not equal to 0 therefore p infinity is more than p u okay that means the pressure on the upper surface is lower than that in the free stream okay look at the lower surface in the lower surface also you have some pressure p l and in the area below the curved body dou p l by dou r is equal to rho v square by r again it is non-zero because the value of r is non-zero okay so therefore there also you have p l minus p infinity is more than 0 so if p if p infinity is more than p u and p l is more than p infinity then you can always say that p l is more than p u okay so if p l is more than p u there will be a pressure difference across the body and that is going to create a force on the body so interestingly so the velocity change does not cause pressure change in fact the pressure change causes velocity change so the argument actually is reversed so in short what is happening is presence of any body that leads to streamland curvature creates a pressure difference and that pressure difference is both above and below the body okay so now the bottom line is lift will be created if and only if there is going to be some change in the curvature of the streamlines when the body is introduced and the direction of this force will be up or down depending on in which direction the curvature is created if you have equal creation of streamland curvature above and below such as a symmetric aerofoil then the net lift will be 0 because the top half will give one force and the bottom half will give exactly equal and opposite force so the net force will be 0 so from now on if you apply this particular principle of streamland curvature the first thing you will look for is when you brought in the body was there any change in the streamland curvature if there was calculate it along each stream line and then you get the value of the net force that is created so this particular second thing is that in general what you can assume is that the transfer of momentum of the free stream flow that occurs on the body is also explainable by curvature effect because of the transfer of momentum if the streamland does not get curve then you may not get a lift so the bottom line is if you look at the flow pattern so let us go back now look at this picture whether lift is generated or not does not depend upon the shape of the body only it only depends on whether this particular stream line does it undergo a change in the curvature or not similarly here do streamlines undergo change in curvature or not larger the change in the curvature larger will be the doh P by doh R and that will give you the value of the force created in any direction depending on the situation so this particular system or this particular approach can easily explain the phenomena of producing lift I will again repeat if there is no change in the streamland curvature there will be no lift if there is change in the curvature there will be lift