 Hello, today we will look at how to estimate the lift coefficient of a military aircraft. And as an example, we have taken F16C fighting Falcon to illustrate the procedure. Before you actually go ahead and watch this tutorial, I would advise you to have a look at the procedure for lift coefficient estimation that we have already discussed. Because tutorials are meant for students who have already seen the lecture, they know the procedure and here we only show you how that procedure is implemented for a practical aircraft. So, if you have not watched the lecture on lift coefficient estimation for military aircraft, I would recommend that you first watch those lecture clips and then only do this tutorial. We are going to follow a color scheme in this presentation. The general instructions are going to be given in brown color, which is the basic theme of this presentation. If there are some values which are specified either online or in the specifications of the aircraft, they will be shown in black color. There are certain values which we will assume based on literature or any other source book, those will be shown in this blue color. The places where you have to do some calculations will you will be alerted with red color question marks and when you see this pause button, I would recommend that you actually pause the video at that stage and do the calculations and then only proceed further. I would like to reiterate that aircraft design is learnt best by doing the calculations. Just by listening to a video and nodding your head, you will not be able to really appreciate the calculations and hence you will not get a feel for aircraft design. The values which are calculated will be shown in this dark blue color and towards the end, we will compare the values that we have obtained with some values mentioned in documents or open sources. So, those will be shown in the green color. Let us look at the source of the data and comparison for F-16C analysis. Our principal source for this particular tutorial is this excellent textbook by Brandt, Stiles, Burton and Wittford. It is a strongly recommended to go through this particular textbook. It is a very wonderful textbook which gives a general introduction to aeronautics but with a design perspective. So, we have looked at the third edition of this book and there is an example of a whole aircraft drag estimation and lift estimation. So, for this particular tutorial, we are following the lift estimation procedure explained by the authors in this particular textbook. This is the top view of the F-16C aircraft and this information is presented in the textbook in the form of geometrical data. So, we see that the leading edge sweep of the wing and the tail is 40 degrees but the quarter cot sweep is 30 degrees. This aircraft has a wingspan of 30 feet and the other numbers are already all mentioned there. So, in the textbook what they have done is the geometry has been presented in this way in the textbook. The data is given in the FPS system and we are going to use SI system in our tutorial. So, let us have a look at some of the useful parameters which we will utilize in this particular tutorial. First is the wingspan which is given as 30 feet or 9.144 meters. The wing reference area which is the plan form area in the top view including the area inside the fuselage as you can see in the figure is 300 square feet which is 27.87 square meters. The tail span or the distance between the tips of the two tail, two horizontal tails is 18 feet, 5.49 meters. The tail reference area or the area of the two trapezius that represent the tail is 108 square feet or 10.033 meter square. This aircraft is fitted with leading edge strakes near the wing root wing fuselage junction and these strakes are a total of 10 square feet each so that is 20 square feet area, 1.858 square meters. The root chord of the wing on the fuselage centerline is 16.5 feet which is 5.03 meters. The tip chord is 3.5 feet or 1.07 meters and the hinge line of the flaps are swept at an angle of 10 degrees. And also the maximum thickness line, the line along which we have the maximum thickness of the aerofoil approximately 40 percent of the chord the angle of that line is 24 degrees. So these are some of the parameters that we will need in our calculation. So it will be a good idea for you to either take a picture of this slide and keep it with you when you do the calculations or you can even note down these values on in a small document. Similarly, there are some information that we need regarding the side view of the aircraft. For example, the distance from the quarter chord of the wing and to the tail, the so called tail arm this is 14.7 feet or 4.48 meters and the location of the horizontal tail below the plane of the wing or above the plane of the wing as in this case is 1 feet or 0.3048 meters. So this is the another geometrical information that we will need. The first thing we will do is we will estimate Oswald efficiency factor E0. For that we will use some data for example, the sweep of the maximum thickness line. First let us calculate the wing aspect ratio. As you know aspect ratio is defined as span square by area. So we know the span 9.144, we know the area 27.87. So therefore aspect ratio is span square by area. This is the first place where you have to stop the video and do these calculations. It is very straightforward 9.144 square divided by 27.87. This number comes out to be 3. Similarly for the horizontal tail, we need to calculate the aspect ratio. For that we know that the tail span is 5.49 meters and tail area is 10.303 square meters. Once again aspect ratio is 5.49 square divided by 10.03. The second place to pause do the calculations. Turns out that the tail aspect ratio also is 3. So in this aircraft the tail is nothing but a scaled down version of the wing with similar geometry and similar aspect ratio, same aspect ratio. Now here is a formula from the book by Brandt et al for calculating the Oswald efficiency factor. There is a much more elaborate formula which we have used for transport aircraft. But for a military aircraft since we are following a standard reference, we would like to calculate the value of E0 also using that same formula. So as per this formula you need to know the aspect ratio of the wing or the tail and the sweep of the maximum thickness line. These numbers are known to us. So therefore when we insert these numbers in the formula, you now have to pause the video and calculate the value of E0. So do it first for the wing. The value turns out to be 0.703 and that is also applicable for the tail because the tail in this case has the same aspect ratio and the same sweep of the maximum thickness line. So both wing and tail have the same Oswald efficiency factor of 0.703. Let us estimate the lift curve slope now. We are told that this aircraft is fitted with NACA 64A204 airfoil in the wing as well as in the tail. The two-dimensional lift curve slope of the airfoil which is small cl alpha or also dcl by d alpha that is the definition. Here is a figure from one of the NASA NASA reports, NASA archive that talks about the aerodynamic data of the flood plate and the 64A204 airfoil. So from this figure we see that at angle of attack of 0 we have cl1 equal to 0 and at the angle of attack of 9 degrees we have cl1 equal to 1. So with these two points if we fit, if we want to get the linear lift curve slope we just fit a straight line between these two points and for that we can use a simple formula cl alpha will be dcl by d alpha that will be equal to cl2 minus cl1 and divide by alpha2 minus alpha1. So it turns out that cl2 is 1.0, cl1 is 0, alpha2 is 9.0, alpha1 is 0. So it is a simple calculation 1 divide by 9 but it is a good idea to pause and do the calculation yourself. The number is 0.11. So the lift curve slope of the airfoil is 0.11 per degree. Let us now use this information to calculate the lift curve slope of the entire wing, the so called 3D effects. So the 2D lift curve slope we have just calculated was 0.11 per degree. The 3D lift curve slope will be called as capital C, capital L alpha and here is a graph that shows how the cl alpha of the airfoil and wing are different. We notice that the wing has in general a lower lift curve slope compared to the airfoil and another thing we observe is that the angle at which the airfoil attains the maximum cl which was 9 degrees in our case is lower than the angle at which the wing will obtain its highest value. So the highest angle at which the wing will obtain the maximum cl is not 9 degrees but probably around 15, 16 degrees but there is a usable value and that number for this aircraft can be assumed to be 14 degrees. So remember we will use this number when we go ahead. So therefore the formula to be used as given in the book by Brent et al for the wing lift curve slope is equal to the airfoil lift curve slope divided by a quantity that takes care of the Oswald efficiency factor and the wing aspect ratio. So putting in the numbers for our case cl alpha for the airfoil is 0.11 the aspect ratio is 3 and the Oswald efficiency factor is 0.703. So therefore we can get the value of cl alpha for the wing. Please pause the video and do these calculation. The value comes out to be 0.056 for cl alpha for the wing and in this case it will be the same for the tail also because the tail aspect ratio is the same as the wing aspect ratio. So therefore they will have it will have the same geometry and hence it has a similar same value of the lift curve slope. Let us continue ahead. We will like to look at what is the effect of strikes on the lift curve slope because as I mentioned this aircraft is fitted with two strikes which are mounted at the wing fuselage junction ahead of the wing. These two strikes total area is 1.858 square meters it was 20 square feet and the wing reference area was 300. So it is to 27.87 square meters. So the book by Brandt et al gives a simple formula for calculating cl alpha of an aircraft with strike. If you know the value of cl alpha without the strike is nothing but scaling up the area of the wing in terms of the additional area of the strike. So it will become 300 plus 20 upon 300 in FPS system multiplied by the lift curve slope. In our case we know that the lift curve slope of the wing was 0.056 and we just have to multiply it by the ratio of ring reference area plus strike area divided by the ring reference area or we just scale up the value. So we find that the cl alpha with the strike becomes 0.06 per degree higher than 0.056 because of the presence of 20 square meter, 20 square feet of those two strikes. Let us look at the effect of the tail on the lift curve slope. So for that we use the data that the lift curve slope with strikes we have just calculated is 0.06 and these two distances L H of 4.48 the distance between the two quarter cords wing and the tail and the distance of the horizontal pole below or above the wing. So this information will be useful for us to calculate. Of course we also need wing tip cord, wing root cord, wing span. So first we calculate the wing taper ratio which is nothing but the tip cord by the root cord or 1.07 divided by 5.03. It is a good time to pause the video and calculate this number. The answer is 0.21. For the average cord we just take the average of these two numbers. Again you can pause here. The value is 3.05. So the parameter dou epsilon by dou alpha which is the rate of change of the downwash angle with the change in the angle of attack. It can be estimated using this formula given in the book by Brandt et al. So what we do is we just insert the values of the parameters that we know. For example CL alpha with strikes is 0.06. See average is just calculated L H is also known to us 10 minus 3 lambda by 7 and 1 minus Z H by B. So once you do all these calculations you can get dou epsilon by dou alpha. Please pause the video at this stage, calculate this value turns out to be 0.5. Moving ahead, so we know the lift curve slope with strikes as 0.06. We have just calculated the lift curve slope of the tail also as 0.056. The tail reference area was 10.03 meter cubes and we also know the wing reference area and we have just calculated dou epsilon by dou alpha as 0.5. So therefore CL alpha for the aircraft will be CL alpha with the strike which we have already calculated plus the tail CL alpha which is in this case same as the wing CL alpha into 1 minus dou epsilon by dou alpha into ST by S because this is a in a way a measure of the tail efficiency. How much of the tail is actually efficient? So putting in the numbers we get this expression 0.06 remember was the value of lift curve slope with strikes and 0.0563 was the slope of the tail lift curve of the tail. So pause at this stage and calculate this value we see that the CL alpha of the aircraft is 0.070. So in other words contributions to CL alpha have been taken from the tail from the strikes and from the wing. The net value is 0.070 per degrees. Now we are ready to do the estimation of the CL max of this aircraft during takeoff and landing. Takeoff is shown in the figure on the left and landing is shown in the figure in the right. If you notice carefully the angle of the aircraft both during takeoff and during landing seems to be very similar. So this is the actually the maximum usable angle of the aircraft. Now this angle is a function of at what what is the maximum angle you can deflect the aircraft along the main wheels during takeoff or landing without hitting the tail on the ground. This number is approximately 15 degrees but we keep a 1 degree margin so it will be 14 degrees and that will be called as the absolute maximum value of the absolute angle of attack of the aircraft 14 degrees and it will be the same in case of both takeoff and landing in this case. Estimation of maximum lift coefficient can now be attempted. CL max will be basically the CL alpha of the aircraft into the change in the absolute angle of attack which is attained during the flight. Now the change in the absolute angle of attack delta alpha is a function of what change is available if you use only the airfoil or the two-dimensional value then you multiply it by the flap area to the wing area or the flabbed area ratio and then cos of the sweep of the hinge line. So we notice that if you do not use full span flaps only a part of the wing is affected with the flaps and hence that part is the one that gets an improvement in the effective absolute angle of attack increase and if you have a hinge line which is swept then that causes reduction in its efficiency so that effect is brought in using the term cos of lambda HL. SF by S is the flap area ratio which is the ratio of the area covered by the flaps upon the total wing reference area and lambda HL as you know is the flap hinge line sweep angle. So first let us have a look at the flap areas and the unflapped areas. So all the area of the aircraft which is under the influence of the flaps whether leading edge flaps or trailing edge flaps that area the total area that is under the influence of both of them is called as the flapped area and as far as F16 is concerned you know this would be the total flapped area because the trailing edge flaps start from almost at the junction of the wing and the fuselage you can see almost and they go up to this place with the sweep of 10 degrees for the hinge line and there are leading edge flaps which start almost near the wingtips and go right up to here. So actually if you take the area which is influenced under both of these flaps you will get these two trapezia and that is going to be the flapped area. So let us calculate the flapped area ratio for F16C this is the area that we have to this is the area which is under the influence of the trailing edge devices and this is the area which is under the influence of the leading edge devices. So the net area is the one that is hashed and looking at the geometry of the aircraft we have determined that the dimensions of the flapped area are that of a trapezium of the larger side 4.1 the parallel side 1.07 and the distance between the two sides as 3.5 meters. So the area of the flaps will be the twice the area of each of these trapezium each of these trapezia. So the area of one trapezium would be half of the sum of the two parallel sides into the distance between them. So please pause the video and calculate the value of each trapezium area that is going to be 9.05 square meters. So each of these areas is 9.05 square meters the total flapped area is actually going to be double of this so 2 into 9.05 please calculate the value the answer is 18.1 square meters. So the flapped area is 18.1 square meters and remember that the total area is 27.87 square meters. So with this we can get the ratio of S flap by S or the flapped area ratio again I would suggest you pause the video and calculate this number the number turns out to be 0.65. Hence for this aircraft the flapped area ratio is 0.65 only 65 percent of the whole wing is actually under the influence of the flaps 35 percent of the flaps is not under the influence of the flaps and that area is mostly the area inside the fuselage actually over the fuselage. So the flapped area ratio is known as 0.65 hinge line of the flaps is swept at an angle of 10 degrees from literature given in the textbook or you can say back calculated from some number in the textbook we have determined that the angle of attack 2D at takeoff is 7.5 degrees and as I mentioned to you the maximum usable angle of attack at takeoff is 14 degrees. So an aircraft lift crop slope was determined as 0.07 per degree. So the delta alpha takeoff is actually delta alpha 2D with SF by F and cos lambda H multiplication. So we have put the numbers here for your convenience and now you should pause the video and do the calculations for the value of delta alpha during takeoff that number is 4.8 degrees. So what it basically means is that the presence of the flaps of this particular area ratio of a particular area and a particular sweep of the hinge line effectively results in increasing the angle of attack of the aircraft by 4.8 degrees that is the meaning of this calculation. Effectively when you deflect the flaps you get the increment in CL corresponding to what you would get if the aircraft had gone for an angle of attack increase of 4.8 degrees. So therefore the CL max flap will be the slope of the lift curve slope of the aircraft DCL by D alpha times whatever is the maximum angle of attack that you can operate plus what is the additional effective angle of attack created because of the flap deflection. So that number is 0.07 into 14 plus 4.8. 14 degrees as I mentioned is the maximum angle which is usable during takeoff and landing both in takeoff and landing and 4.8 is the effective increase in the angle. So the CL max is going to be 1.32 in the case of the takeoff condition. Let us repeat the same calculations for the landing condition. During landing the geometrical parameters actually remain the same the only change that happens is that the angle of attack 2D at landing because of the flaps in landing configuration is slightly higher because flaps are more effective in landing. So therefore we have the same formula delta alpha landing will be delta alpha 2D times SF by S and cos lambda H. The values of SF by S and cos lambda H remain the same as previously. This number was 7.5 earlier now it is 11.5. So you can simply calculate this value by dividing the previous delta alpha of 4.8 by 7.5 and multiplying by 11.5 or you can just do this calculation of multiplying these three terms the value comes out to be 7.4 degrees. I hope you are pausing the video and doing the calculations yourself. Once again the CL max of the flap aircraft is going to be the lift curve slope of the whole aircraft into the maximum usable angle of attack and the additional angle created by the flap deflection. So 0.07 is the lift curve slope of the entire aircraft with the effect of strike, effect of tail, effect of wing included. 14 is the maximum angle of attack absolute angle of attack that is available and 7.4 is the increment in the angle of attack effectively due to landing. So with this the value of CL max during landing comes out to be 1.50. Now let us just see what is the difference between our estimates and the data which is quoted in the source. So for comparison purposes we have utilized the data given in the book by Brands, Stiles, Burton and Whitford. So the first parameter is the lift curve slope. They have quoted the value of that as 0.1 whereas we have estimated the value to be 10 percent higher of 0.11. This is for the aerofoil itself. Now if there is a difference in the aerofoil lift curve slope itself obviously the numbers are going to change for the parameters that follow. So it is no surprise that the aircraft lift curve slope also comes out to be around 8 percent higher than the quoted value of 0.065. This is the value that is the actual value for the aircraft as quoted in the textbook. Then the maximum lift coefficient we have estimated 1.32 whereas the the quoted value is 1.27. So there is a 4 percent difference in the estimated value and the quoted value and at landing we have estimated the lift coefficient to be 1.5 whereas the value given is 1.43 so there is a 5 percent error. In other words if there is a fundamental error in the basic aerofoil lift curve slope itself then we expect these errors to come but still we are actually very much comparable to the value quoted in the literature. I would like to acknowledge the items I have borrowed from the book by Brandstiles, Burton and Whitford. I also would like to thank Daniel Rehmer for his seminal textbook on aircraft design from where also we have drawn some information and last but not the least I would like to sincerely thank my teaching assistant Naman Abdin for help in creating this tutorial. Thank you.