 Hello, let us look at the procedure followed for estimating the drag of a military aircraft. So when I say drag I basically mean drag coefficient because once you estimate the drag coefficient then drag is simply a product of half a dynamic pressure into the coefficient. So military aircraft can be of various types they may fly at subsonic speeds especially aircraft which are used for transport or for reconnaissance or they may be operating at supersonic speeds. So they cover the entire range of speeds we are not talking about any aircraft which is flying at hypersonic speeds right now. So therefore there are some differences in estimation of the drag coefficient for such aircraft as compared to that for the transport aircraft which mostly fly in subsonic and some in transonic regime. What are these differences? Let us understand. First of all there is going to be wave drag present when the aircraft flies supersonic. So we need to include methods to calculate the wave drag coefficient. The bluntness of the wing and the nose has a great effect on the drag coefficient of the aircraft and the intake could be either closed or open nose because of the design. So that also creates some additional complications in estimation. Now the procedure that I am going to describe has been taken from the latest textbooks in aircraft design from the AIAA Stable those by Leland, Nikolai and Grant Carrickner. Both these gentlemen have a huge amount of experience working on various types of military and transport aircraft and also on unconventional aircraft and airships. So based on their large experience and database they have come up with a method which has been described in their textbook. So a lot of material that you will see in this presentation is also taken from the textbooks from Nikolai and Carrickner volume 1. The total drag coefficient for a transport can be assumed to be a summation of four components. The one CD naught due to the wing, second due to the body, third is the delta CDO because of so many other parameters and the combination of these three is called as the zero lift drag coefficient and then we have the delta CDL which is the lift induced drag. So when you calculate the exact value or accurate value of the CDO for the wing, body and delta CDO for the zero lift drag and also the way lift induced drag then you can get the value of CD. Let us first look at how we can get the value of CDO wing. So here there will be three cases depending on whether you fly subsonic, transonic or supersonic. First let us look at the subsonic condition. In the subsonic condition the formula is same as that you use for the transport aircraft which you are very familiar with as shown on the screen. The only two small changes here are the value of R the lifting surface correlation factor and CF the turbulence flat plate skin friction coefficient. So these parameters these two parameters they differ from what we have seen for a transport aircraft. So let us look at how these two are determined for a military aircraft. So for a subsonic case the value of R can be easily obtained based on the cause of the sweep of the maximum thickness line of the wing and there are these curves for various Mach numbers. So from the x axis you proceed up to any value of the Mach number that aircraft operates and proceed further on the left hand side to get the value of R. It is as simple as reading a graph. For CF you have to follow the same procedure that we follow in transport aircraft that means you have to evaluate the cutoff Reynolds number using the L by K ratio and the wings Reynolds number using R e is equal to rho v L my mu. So you calculate both these you calculate the wing Reynolds number first and then you calculate the cutoff Reynolds number and then you have to choose the one which is smaller. So for finding the cutoff Reynolds number in case of the wing the characteristic length would be the chord or the mean aerodynamic chord. So therefore L by K would be C by K where C is the mean aerodynamic chord. K is factor that comes from the roughness. So the table here shows the value of K in inches applicable for various types of surface finishes that you normally encounter on a military aircraft and the graph on the y axis is basically just a correlation between the cutoff Reynolds number and the admissible roughness L by K. So for various mark numbers you can get the values. So for determining CF there are two steps. First step is that you calculate the cutoff Reynolds number R e L and you calculate the wing Reynolds number R e and then you choose whichever is smaller. So whatever is the smaller value that one you use in this particular graph and there are two bunches of lines there is one single line for turbulent flow and there are three lines for laminar flow. So depending on the value of the M infinity. So you can use the value of M infinity and calculate the value of CF. So that was for the calculation of CDO wing for the subsonic case. The procedure was quite similar to what you are used to for transport aircraft but when it comes to transonic wing then in transonic flow there are a few changes. Now the transonic flow begins at the critical mark number and the drag rise starts actually at the MDD track divergence mark number and generally it is considered when the rate of change of CDO exceeds with mark number exceeds 0.1 that is the point where you can consider to be the drag divergence. So the CDO wing is going to be CDF plus CDW and CDF can be simplified as the first term that is CF into 1 plus L T by C S weight by SF. Now the this value of CDF is assumed to be constant in the entire transonic range. So what you do is you calculate the value for mark number 0.6 and you assume that that value is applicable in transonic flow. And CDW is obtained through von Karman similarity law for transonic wings. So that I will show you. So this is how you calculate CDW curve for transonic wings. This depends on the usage of the experimental data. So what you do is you have some experimental data that data has to be corrected for the three important parameters the sweep the aspect ratio and T by C. So this particular graph is actually applicable only for unswept wings. So what you do is you apply the corrections for the three parameters which I have mentioned there. So the values of MD, CDW peak and DW peak are corrected by using the cos of the quarter chord line, cos of the sweep of the quarter chord line. So these corrections will help you to get the values. Now when it comes to supersonic aircraft, if you want to calculate CDO wing of supersonic aircraft then you need to use this particular procedure. So the CDO wing will be again the same thing CDF plus CDW. So CDF is the wing supersonic skin friction coefficient and CDW is the wing supersonic wave drag coefficient. So CDF is CFS fed by SF where CF is CFI into CFC by CFI. This is the compressibility effect on the turbulence skin friction. So CFI is calculated based on the minimum values of Reynolds number either the cutoff value or the standard value. So you can see that the value of CFC by CFI this ratio can be obtained for various Mach numbers using this particular line. Now for the wing zero lift drag if you look at supersonic flow depending on the shape of the aerofoil whether it is sharp nose or blunt there are different procedures available. So if you have a sharp nose aerofoil then you use supersonic linear theory. So for sharp nose aerofoils basically there is a supersonic leading edge. So beta into cot of the sweep is going to be more than equal to 1. So from there you can get the value of CDW and CDW uses this particular value of capital B the B factor for sharp nose aerofoils. So this B factor depends on whether the aerofoil is biconvex or double wedge or hexagonal depending on the aerofoil shape the value of B changes. So use this particular table to calculate the value of B and after that beta T by C SE and SRF are already known to you. But if you have a subsonic leading edge then the value of beta cot lambda is less than 1. So here the formula changes and you have the value of cot of sweep of leading edge also coming into picture. If you have blunt nose aerofoils then for supersonic leading edge again the condition remains the same. So you can get this formula can be used to calculate and if it is a subsonic leading edge then the formula changes and the CDLE can be obtained here. CDLE can be obtained here because there is a graph that correlates the mark number with B CDLE by ARLE. Now B is the wingspan AR is the aspect ratio R is the radius. So you know these parameters for an aircraft. So for various values of the delta leading edge you can use and get the corresponding value and from that you can get the value of CDW. So that much is for the CDO wing. Now let us move on to get the CDO body, the body coefficient. Again there will be 3 cases subsonic, transonic and supersonic. In the subsonic case again the formula is very similar to what you are already used to for transport aircraft. The only difference is that the value of LB by D you know depending on what type of body is used we have to use various formulae to get the correct expressions in this equation. For transonic flow the CDO body would be a sum of 4 components CDF that is the skin friction dry coefficient which would be CF into SS by SB or CDP which can be obtained by this expression CDB which is obtained by this expression and CDW which we will see in the next slide. So here, sorry so CDW is here. So with the fineness ratio you can get for various mark numbers the value of the wave dry coefficient CDW. So with this once you get all these 4 parameters you can get the value of CDO. So what you need is you need the wave dry coefficient as a function of the fineness radio. So the same graph that you saw here for improving the clarity it has been redrawn here in a larger frame so that you can easily use it to get the values of CDW for various mark numbers given the fineness ratio. Moving on to supersonic flow for a body. So here again the CDO body would be a sum of these 5 components and each of these components is explained here and now we will see the formulae for each of them. So first is CDAN2, CDAN2 is this term, CDAN2 is this term, body after wave dry. For that you can get the interference drag based on whether the shape has got a blunt body at the end or it has got a closed body at the end depending on that you can use the corresponding graph. Similarly, if you have a ojive nose for ojive nose you can use these parameters to calculate the value and if you have conical nose then the graph is this one and if you have an ojive nose the graph is this one. Finally, you have CDA which again depends upon the shape so either you have a conical after body or you have a ojive after body depending on that depending on the boat tail shapes you can get you can choose the correct graph and read the value on the y axis after giving the input from the x axis. So now we come to delta CDO. So delta CDO is due to various miscellaneous components like canopy, protuberances, nozzle boat tail, wing mounted stores, pile on tank etc. So there are some recommendations given as the what will be the drag area for various types. So incremental drag for external stores and this is the again the incremental drag for canopy, protuberances and nozzle boat tails. So when you have stores you can use this graph when you have canopy, protuberances and nozzle boat tail you can use the one on the top and you have to correct for S ref remember. Finally we come to CDL. CDL now this is a very simple formula for calculation of the lift dependent drag but you can get the more accurate formula for span efficiency factor where this expression can be more detailed. So what is the procedure? First is you calculate the wing CL alpha supersonic. So choose CL alpha according to the taper ratio and there are tables shown in the next slide and for small angles you can assume that the normal force coefficient is equal to the lift force coefficient. So you can see lambda equal to 0.5, 0.25 and 1 for various values of lambda you have these graphs available. So you have to interpolate between them depending on what kind of trailing as you have and what kind of configuration do you have. Similarly if you want to calculate the CL alpha supersonic for a wing body so you can use this particular graph. Now here what you need is you need a factor f okay. So CL alpha wing body is equal to f times CL alpha wing. So CL alpha wing is known to us but CL alpha wing body for that you need f and f will come from this graph depending on the various d by b values and for various aspect ratios and Mach numbers there is some experimental data available through which you can get the value by interpolating in the database. Now let us look at determination of lift induced drag in supersonic condition depends on whether your leading edge is supersonic or subsonic okay. So the induced drag coefficient actually remains the same formula remains the same KCL square. However the value of K will change in the case of supersonic leading edge you have K is equal to 1 upon CL alpha wing body which we already have but this is respect to SRF but in case of subsonic leading edge there is this additional term minus delta n here this can be obtained as shown in the calculations. So for this calculation you need value of K dash and K double dash. So these two parameters are obtained here okay. So they can be obtained here K double dash is here and K dash is 1 by pi E AR. I would like to acknowledge the contribution by Leyland, Nikolai and Grant Karikner in giving us the detailed formulae from their experience and their knowledge okay and a very large database. So this was just a very brief overview of the procedure for doing an example and for understanding I recommend students to go and read this textbook in detail. In the appendices they have given full procedure for calculation. I also want to acknowledge the contribution of Miss Tanvi Prakash my PSD student for help in creating this tutorial. She has meticulously gone through the book by Nikolai and Karikner extracted the various formulae and the procedures and put it together for your convenience. Thank you very much.