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Lecture - 52
Drag Estimation of Military Aircraft
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 dynamic pressure into the coefficient.
(Refer Slide Time: 00:39)
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. Secondly, 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 Leyland Nicolai and Grant Carichner. 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 Nicolai and Carichner volume 1.
(Refer Slide Time: 02:42)
The total drag coefficient for a transport can be assumed to be a summation of 4 components, ?? = ??0???? + ??0???? + Δ??0 + ???
Let us first look at how we can get the value of ??0????.
(Refer Slide Time: 03:30)
So, here there will be 3 cases depending on whether you fly subsonic, transonic, or supersonic. (Refer Slide Time: 03:41)
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 2 small changes here are the value of R the lifting surface correlation factor and ?? the turbulence flat plate skin friction coefficient. So, these 2 parameters, they differ from what we have seen for a transport aircraft. So let us look at how these 2 are determined for a military aircraft.
(Refer Slide Time: 04:18)
So, for a subsonic case, the value of R can be easily obtained based on the cos 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 the aircraft operates and proceed further on the left hand side to get the value of R it is as simple as reading a graph.
(Refer Slide Time: 04:48)
For ??, 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/k ratio and the wings Reynolds number using ?? =????. 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.
(Refer Slide Time: 05:16)
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/k would be c/k where c is the mean aerodynamic chord and 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/k. So, for various Mach numbers you can get the values.
(Refer Slide Time: 06:06)
So, for determining ?? there are 2 steps, first step is that you calculate the cutoff Reynolds number ??? and you calculate the wing Reynolds number ??? and then you choose whichever is smaller. So, whatever is the smaller value that one you use in this particular graph and there are 2 bunches of lines there is 1 single line for turbulent flow and there are 3 lines for laminar
flow. So, depending on the value of the ?∞, so, you can use the value of ?∞ and calculate the value of ??.
(Refer Slide Time: 06:46)
So, that was for the calculation of ??0???? for the subsonic case that the procedure was quite similar to what you are used to for transport aircraft, but when it comes to transonic wing.
(Refer Slide Time: 06:59)
Then in transonic flow there are a few changes not the transonic flow begins at the critical Mach number and the drag rise starts actually at the MDD drag divergence Mach number and generally it is considered when the rate of change of ??0 exceeds with Mach number exceeds 0.1 that is the point where you can consider to be the drag divergence. So, the ??0????
= ??? + ??? and ??? can be simplified as
??? = ?? [1 + ? (??)] (????????) Now, this value of ??? is assumed to be constant in the entire transonic range. So, what you do is you calculate the value for Mach number 0.6 and you assume that that value is applicable in
transonic flow. And the ??? is obtained through Von Karman’s similarity law for transonic wings.
(Refer Slide Time: 08:06)
So, that I will show you so, this is how you calculate ??? 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 3 important parameters the sweep, the aspect ratio and
t/c. So, this particular graph is actually applicable only for unswept wings. So, what you do is, you apply the corrections for the 3 parameters which I have mentioned there.
So, the values of MDD, ??????? and ???????? 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.
(Refer Slide Time: 09:01)
Now, when it comes to supersonic aircraft, if you want to calculate ??0???? of supersonic aircraft. (Refer Slide Time: 09:08)
Then you need to use this particular procedure. So, the ??0????
will be again the same thing ??0???? = ??? + ??? So, ??? is the wing supersonic skin friction coefficient and ??? is the wing supersonic wave
drag coefficient. So, ??? = ?? (????????) where ?? is ?? = ??? (??? ???)
This is the compressibility effect on the turbulent skin friction. So, ??? 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 (???
???) this ratio can be obtained for various Mach numbers using this
particular line.
(Refer Slide Time: 09:56)
Now, for the wings zero lift drag if you look at supersonic flow depending on the shape of the airfoil whether it is sharp nose or blunt, there are different procedures available. So, if you have a sharp nosed aerofoil then you use supersonic linear theory. So, for sharp nosed airfoils 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 ??? and ??? uses this particular value of capital B the B factor for sharp nose airfoils.
So, this B factor depends on whether the airfoil is biconvex or double wedge or hexagonal depending on the airfoil shape the value of B changes. So, use this particular table to calculate the value of B and after that beta t/c Se and Sref are already known to you. But, if you have a subsonic leading edge, then the value of ? cot Λ?? < 1. So, here the formula changes and you have you have the value of cot of the sweep of leading edge also coming into picture (Refer Slide Time: 11:11)
If you have blunt nosed airfoils, then for supersonic leading edge again the condition remains the same. So, you can get this used this formula can be used to calculate and if it is a subsonic leading edge then the formula changes and the ???? can be obtained here because it there is a graph that correlates the Mach number with b ????/???? . Now B is the wingspan AR is 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 there you can get the value of ???. So, that much is for the ??0????. Now, let us move on to get the ??0???? a body coefficient.
(Refer Slide Time: 12:09)
Again there will be 3 cases subsonic transonic and supersonic.
(Refer Slide Time: 12:15)
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/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.
(Refer Slide Time: 12:39)
(Refer Slide Time: 12:41)
For transonic flow the ??0???? would be a sum of 4 components ??? that is the skin friction drag coefficient. So, with the fineness ratio you can get for various Mach numbers the value of the wave drag coefficient ???. So, with this once you get all these 4 parameters, you can get the value of ??0 .
(Refer Slide Time: 13:18)
So, what you need is you need the wave drag coefficient as a function of the fineness ratio. 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 ??? for various Mach numbers given the fineness ratio.
(Refer Slide Time: 13:40)
(Refer Slide Time: 13:44)
Moving on to supersonic flow for a body so, here again the ??0????
would be a sum of these 5 components and each of these components is explained here. And now, we will see the formula for each of them.
(Refer Slide Time: 13:56)
So, first is ????2, ????2 is this term body after body wave drag. For that you know 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.
(Refer Slide Time: 14:24)
Similarly, if you have a Ogive nose for Ogive nose, you can use these parameters to calculate the value.
(Refer Slide Time: 14:34)
And if you have conical nose, then the graph is this 1 and if you have an ogive knows the graph is this 1. (Refer Slide Time: 14:42)
Finally, you have ??? , which again depends upon the shape so either you have a conical after body or you have an ogive 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 given the input from the X axis. So now we come to delta ??0 .
(Refer Slide Time: 15:05)
So, delta ??0 is due to various miscellaneous components like canopy, protuberances, nozzle, boat tail, wing mounted doors, pylon 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 or novel boat tail you can use the one on the top and you have to correct for Sref remember. Finally, we come to ??? .
(Refer Slide Time: 15:44)
??? now there 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.
(Refer Slide Time: 16:03)
So, what is the procedure first is you calculate the wing ??? supersonic. So, choose ??? 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.
(Refer Slide Time: 16:17)
So, you can see a lambda = 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 edge you have and what kind of configuration do you have.
(Refer Slide Time: 16:37)
Similarly, if you want to calculate the ??? for supersonic for a wing body so, you can use this particular graph now, here what you need is you need a factor F. So, ??? wing body is equal to F times ??? wing. So, ??? wing is known to us but ??? wing body for that you need F and F will come from this graph depending on the various d/b values and for various aspect ratios and Mach numbers there is some experimental data available through which we can get the value by interpreting the database.
(Refer Slide Time: 17:14)
Now, let us look at the determination of lift induced drag in supersonic condition depends on whether you are leading edge is supersonic or subsonic. So, the induced drag coefficient actually remains the same formula remains the same ???2 however, the value of k will change in the case of a supersonic leading edge you have k is going to be ? =1 ???????−???? which we already have, but this is referred to Sref.
But in case of subsonic leading edge, there is this additional term minus ΔN here this can be obtained as shown in the calculations.
(Refer Slide Time: 17:53)
So, for this calculation you need value of ?’ ??? ?’’. So, these 2 parameters are obtained here.
So, they can be obtained here, ?’’ is here and ?’ =1????
(Refer Slide Time: 18:11)
I would like to acknowledge the contribution by Leyland Nicolai and Grant Carichner in giving us the detailed formulae from their experience and their knowledge 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 a full procedure for calculation.
I also want to acknowledge the contribution of Ms. Tanvi Prakash, my PhD student for help in creating this tutorial. She has meticulously gone through the book by Nicolai and Carichner extracted the various formulae and the procedures and put it together for your convenience.
Thank you very much.

Lecture-53
Tutorial on Drag Polar Estimation of Military Aircraft
Hello everybody let us look at how we estimate the drag polar of a military aircraft. And to illustrate this procedure we have chosen F16-C Fighting Falcon.
(Refer Slide Time: 00:30)
As the base aircraft on which we will do these calculations.
(Refer Slide Time: 00:37)
So the colour scheme in this presentation is that the general instructions like these will be given in brown colour. If there are any values which are specified in any reference source they will be shown in black colour any values that we assume will be in blue colour. The places where you should do calculations will be highlighted in red colour with question marks and there will be this symbol the pause button.
So wherever you see this button you have to remember that that is a place where you should stop the video and do some calculations and then match with the values obtained by us. The calculated values will be shown in this dark blue colour. And wherever we compare our numbers with existing
aircraft we are going to generally use the green colour.
(Refer Slide Time: 01:34)
Let us look at the source of the data and comparison for this particular tutorial. As I said we are going to use F16-C and the textbook by Brandt Stiles Bertin and Whitford contains a detailed description of the procedure for estimation of the drag polar and that procedure has been borrowed
by us and used in this tutorial.
(Refer Slide Time: 02:01)
Many aircraft have a non parabolic drag polar here is an example. So the C D for such aircraft can be expressed in terms of ?? = ??0 + ?1??2 + ?2??
There may also be actually some higher order terms but those are normally neglected. So this becomes a non parabolic drag polar because there is a linear term ?2?? they are the first order term ?2??. So we know that here ?? stands for the drag coefficient.
??0 stands for the parasite drag coefficient also called ??0 and ?? is the lift coefficient ?1 is a coefficient which is obtained in terms of the aspect ratio AR and the Oswald’s efficiency factor ?0 which can be determined using this formula in terms of the wing aspect ratio and the leading edge
sweep.
(Refer Slide Time: 03:11)
Let us look at the second parameter ?2. Now ?2 is used to model the effect of camber on drag polar among other things there are many aircraft which may generate a minimum drag not at an angle at which ?? = 0 but at a slightly different angle. There are many reasons for this because an aircraft is actually not just a wing aircraft is basically wing plus fuselage plus tail. So there are certain situations in which the angle at which the aircraft flies and has the minimum drag does not correspond to the condition where the ?? = 0.
So here is an example of an airfoil which has a non parabolic drag polar so that is the top blue line there is a non parabolic drag polar which has got 2 components there is a parabolic component which is modeled by the coefficient ?1 and then there is a non parabolic component that is modeled
by this equation. So in such a graph the profile drag is minimum at some small positive value of ?? generally.
So this additional ?2?? term in this equation models this particular effect. Otherwise if this term was absent then you would actually get a parabolic drag polar. So we will see today how we can get the equivalent parabolic drag polar also for an aircraft. Now if you look at this expression ??
in terms of ?1??2 and ?2?? and let us say if you want to find the value at which the Drag is minimum.
So if you differentiate this expression with respect to ?? take those ?? by those ?? put it equal to 0 then take the second derivative and confirm that that number is negative then you can easily derive the condition that ?2 will be equal to ?2 = −2?1??,???? So that is how the coefficient ?2 can be calculated but to do that we need to get the value of ??,???? that means we need to know ?1 we already know from the previous slide but to know ?2 we need to know the ?? at which the drag is minimum.
(Refer Slide Time: 05:40)
So for that we have to make some assumption. So we assume that this minimum drag occurs the airfoil has a minimum drag at ? = 0. And we look at the equivalent skin friction coefficient that will be ??? = ??0? ???? and the parasite drag coefficient will be then obtained as ??0 = ??? ???? ?
this just by reverting the formula where S is the wing reference area and ???? is the aircraft wetted area.
(Refer Slide Time: 06:16)
So now let us see how we get the equivalent parabolic drag polar. So this is what we would like to have we would like to have ??0 = ????? + ?1??,????2 this is the equivalent Drag polar. So how do we obtain the value of ?????? that is the question? So what we do is since the airfoil is assumed to generate minimum drag at alpha equal to 0 therefore what we assume is that the induced drag actually does nothing but moves this ?? which gives minimum drag to a mean value.
And this value is assumed to be halfway between 0 and the value of ?? when ? = 0. So ??? = 0 and half of it if you take that would be the assumption of moving of ?????? . In other words we know that the ?? at ? = 0 will be absolute angle of attack into the lift curve slope and this absolute angle of attack is actually going to be minus of the lift equal to 0. So that means the angle at which lift is equal to 0.
So knowing the airfoil you can get this value and you know also the value of ??? . In fact we have a separate tutorial on calculating ??? of an aircraft. So therefore now we can get ????? which is the requirement here as ???
???? ?.
(Refer Slide Time: 07:56)
Let us look at the typical values of the equivalent skin friction coefficient that is ??? for jet bomber and civil transport it is point 0.0030 for military jet transporter is 0.0035 for F jet fighter 0.00035 again for a carrier based Navy jet fighter it is slightly higher. For a supersonic cruise aircraft it is
lower and for a single seat propeller aircraft it is very much higher. For light twin propeller aircraft it is a little bit lower and for propeller seaplane it is the highest value. So in our case and of course there is also a jet seaplane. So we do have in our case the problem that we are looking at is the Air
Force jet fighter. So the typical value of equivalent skin friction coefficient would be 0.0035.
(Refer Slide Time: 08:46)
So let us start the calculation for an aircraft in level flight where all the four forces are in balance but while operating at a cruise mach number so we are going to go through calculations at cruise mach number under the ISA.
(Refer Slide Time: 09:01)
Sea level conditions here is the aircraft geometry and this geometry is then we just note on the value of various parameters. So for this particular aircraft we know that the wing span is 30 feet as specified and the conversion is 9.144 meters. The wing reference area is specified as 300 square feet which converts to 27.87 square meters. Similarly the tail span is specified as 18 feet which converts to 5.49 meters. The tail reference area 108 square feet which converts to 10.033 square meters. Strake surface area 20 square feet strikes 1.858 square meters.
Root chord at the wing at the center of this fuselage 16.5 feet which is 5.03 meters tip chord is 3.5 feet or 1.07 meters
(Refer Slide Time: 09:58)
So what we do is we look at some more parameters for example the sweep of the hinge line of the flaps is 10 degrees the leading edge sweep is 40 degrees the sweep of the quarter chord line is 30 degrees and the sweep angle of the maximum thickness line is 24 degrees. So the aerofoil maximum thickness line is at 24 degrees sweep.
(Refer Slide Time: 10:27)
We also need some data from the side view for example we need the distance from the quarter chord of the wing to the quarter chord of the tail the so called tail arm and we also need the value of the lateral or vertical displacement of the horizontal tail from the plane of the wing which is 1
feet 0.3048 meters in this case.
(Refer Slide Time: 10:47)
So with armed with this information we can now do the estimation of the wetted area of this aircraft you can estimate this by either making a CAD model and then the CAD software gives you the wetted area and that is what can be done by using a software such as open VSP we have already
covered detailed description of the software open VSP and we hope that by now you have already tried our hands on open VSP. The other option that we have is that you convert the whole geometry into some standard and simple shapes like cylinders cones rectangles half cylinders etc and then
calculate the wetted area of each component.
(Refer Slide Time: 11:33)
So we have done the same thing. So first we looked at CAD model by using open VSP and we took a model available from the VSP hangar this is that model but if you notice this model has these 2 bombs already loaded and also these 2 additional missiles loaded. So we do not need this.
So this is how you render the shape of the model.
(Refer Slide Time: 11:59)
And then you can see there are so many components there are so many components which are there including the missile stabilizer and pods etc.
(Refer Slide Time: 12:12)
But in our case we have to modify the missile by removing the pods or the external tanks and the missiles. So we get a simple clean model and for this simple clean model we just have these various components which are going to be considered.
(Refer Slide Time: 12:30)
So using this particular CAD model it is possible to actually get the wetted area and also the parasite drag coefficients directly in the software.
(Refer Slide Time: 12:40)
So this is the wetted area of each component as calculated by open VSP and the total wetted area if you add these numbers comes out to be 180.2 square meters.
(Refer Slide Time: 12:55)
Now let us look at the second method where we look at the aircraft geometry and convert that into standard simple shapes. So here is a 3 view diagram of the aircraft.
(Refer Slide Time: 13:06)
And what we do is we look at the geometrical data as specified in the book by Brandt, Stiles, Bertin, and Whiftord
(Refer Slide Time: 13:15)
And then what we do is we approximate the geometry in simple shapes. So for instance you have three view of the aircraft. So you can see that in the top view you just create some simple surfaces and also in the side view you also try to create some simple surfaces try our best possible to match
the geometry with that given and then you can do rendering of those surfaces to remove the internal details and hence you can get some idea about the various components.
(Refer Slide Time: 13:52)
Now what we do is we look at this geometry and calculate the value of S wet for each component using some equations. Now these equations and procedures are already explained in the textbook by Brandt et al the total area comes out to be 1418 square feet but we are going to now explain to
you one by one.
(Refer Slide Time: 14:17)
How this is done. So what we did is that we first tried to compare the values between the numbers that we got using open VSP with the Standard Model available and the values which are quoted in square feet and then in converted in square meters. So we notice that there is a large amount of
error around 25% 26% for wings and 19.25% for vertical tail around the same value almost for that but huge variation in being strakes horizontal tail and canopy.
As a result there is a 37% increase in the value of a S wet as against quoted values. So now the problem is that if you continue with this S wet in your calculations then you are going to introduce this much error automatically in.
(Refer Slide Time: 15:14)
All the calculations. So what we did is we calculated this wetted area using the geometry. So for any surface like wing or any stabilizing surface. So the formula is given here for Sexposed and then using that to get the value of Swet. So for the wing which is surface 1 and surface 2 we know the
wingspan from the geometry we know the root chord we know the tip chord and we know the t/c max. So we can calculate the value of Swet for each of the trapezium and then multiply by 2.
Similarly for the horizontal tail we have 2 small trapezia which are identical in geometry because the same aspect ratio same sweeps and then we look at the strakes. So these are also simple triangular devices. So maybe you can do half of bass into height once the geometry is known to you vertical tail is also a simple trapezium because the root chord is known to you tip chord is known to you and the span is known to you.
The dorsal fin happens to be the fin which is mounted on the below the vertical tail that is also a standard geometrical construct. And then there are 2 hidden surfaces 9 and 10 which also contribute slightly to the geometry.
(Refer Slide Time: 16:37)
So similarly if we now look at the fuselage and the canopy or the mid part we can replace it with the these we can calculate the wetted area using the standard formula depending on whether the cross section is elliptical or rectangular. So for a fuselage for example it is a rectangular cross section so you can get the cylinder height cylinder width. So it is a mix between it is actually more towards elliptical and then we have fuselage sides these are 2 semi circular cylinders.
Then we have a mid part of the canopy which is a pure cylinder and then we have a fuselage bottom. So all these are the formula which can be used by you to calculate the actual wetted area of the aircraft. So these numbers have been given here only for comparison purposes for you actually you should be doing these calculations yourself.
(Refer Slide Time: 17:39)
Moving ahead if you look at the nose the canopy front and the rear part because the mid part is a cylinder and the nozzle we get cones or frustrum of the cones and for that we can use standard formula available. So nose can be considered to be conical and with a nose cone of length height
and width the end and the rear portion can be a frustum of a cone. So there are 2 cones here with l1,h1,w1,h2,w2 then for the canopy front it is half cone or a small cone.
And for the rear can be also it is considered to be conical. So using these numbers you can calculate the areas of various components.
(Refer Slide Time: 18:29)
And after that here is the full table and as per this full table if you add these numbers the number comes out to be 139.31 square meters.
(Refer Slide Time: 18:40)
The value of Swet given is 1418 square feet and spread calculated is 131.73 by converting this number. So what has happened is throughout in these calculations we have done rounding off because each quantity given in feet has been converted into meters rounded off and used in the
calculation that is why there was an error of approximately you know eight and a half in 130.So we will go ahead assuming this to be the value because we do not want to introduce error.