Estimation of Lift Coefficient
Let us have a look at how the lift coefficient is estimated. Before we go ahead we must learn to distinguish between the 2 dimensional and the 3 dimensional value of lift coefficient.
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The 2 dimensional lift coefficient is for the aerofoil and it is normally depicted as small c subscript small l alpha or ??? in small case this is usually a larger value as you can see here it is a larger value as compared to the lift coefficient of a wing which is the 3D lift coefficient including the 3d
effects and that is normally depicted as capital C capital L alpha or the ???.
And this particular reduction between the 2D and the 3D value is because of the 3D effects on the wing. So our task is to estimate the 3D lift coefficient of an aircraft whose geometry is made available to us and the simple relationship between the capital ??? and small ??? is expressed in
terms of the wing aspect ratio and the Oswald’s efficiency factor e as shown in this equation.
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Estimation of the span efficiency factor is a very difficult task and the formula available have a lot of variation. One way to estimate the Oswald’s efficiency factor or one formula to estimate it is as listed here in this particular formula the terms that play a role are the wing aspect ratio and the sweep of the maximum thickness line. This is a geometrical value and if you do not know this value then you can assume it to be the sweep at 30% of the chord for low speed aircraft and at nearly half the chord for a high speed aircraft.
Another important requirement is that you normally are given the data for the sweep at the leading edge or sweep at the trailing edge. And if you want to calculate the sweep at any location n or any fractional location n for example 30% or 50% and you know the sweep at the leading edge that is
lambda 0 and the wing taper ratio then this particular formula can be used to estimate the value of tan ?? which is used here as a function of tan ?0 AR and the taper.
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A more accurate a more detailed formula for estimation of the span efficiency e for long range transport aircraft is given by Professor Dennis how in his book here you can see it is a very long formula and it relates the span efficiency or the span efficiency factor e with the Mach number the
aspect ratio the quarter chord sweep the t/c number of engines taper ratio and a factor based on the taper ratio.
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Let us understand the concept of absolute angle of attack before we go ahead. Now there is one angle at which ? there is one angle ? at which lift is equal to 0 that is called as alpha lift equal to 0 ??=0 lift is difficult to keep track of this particular parameter because it is affected by the twist
distribution and by the airfoil camber. So what we do is we define an absolute angle of attack we call it ??. Such that ?? is defined as the geometry angle of attack ?? = ? − ??=0 So when lift equal to 0 then ?? = 0.
So for a typical aircraft the maximum angle of attack alpha max during takeoff is limited to 15 degrees or so because of the fact that if you take off at an angle more than that Or if you angle more than that then that tail is going to hit the ground. So keeping in mind the takeoff and landing
considerations the angle of attack during these operating scenarios is limited to around 15 degrees.
Therefore ????? that is a maximum value of the absolute angle of attack will become ??,??? = ???? − ??=0
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Now let us see how to estimate the CLmax value and before we do that we need to understand what are the drivers of the maximum lift coefficient? The first design driver is the wing geometry increase in the sweep reduces CLMax and increase in the aspect ratio increases CLmax. Similarly aerofoil shape if you have increase in the thickness to chord ratio and if you have larger leading radius you will have a higher acceleration of the air over the aerofoil and therefore you will have higher CLmax values.
Reynolds number surface texture and the interference from fuselage nacelles and pylon are other factors trailing edge flaps and or leading edge flaps geometry and their span also affect the value of the CLmax. If you have a larger chord and a larger span obviously more part of the flap more part of the wing is flap and taking part in the high lift. So CLmax will be higher but if you sweep the flaps then you have lower values of CLmax.
This is one reason why in many transport aircraft you will see a typical configuration of the wing would be that you have the wing like this it will have a sweep but in the central portion you will have flaps which are going to be straight flaps and these tend to be the large chord flaps and then
you have the smaller chord flaps which could also be in parts. So the reason why we go for this kind of a flap with trailing edge sweep 0 is because swept flaps have a lower CLmax value. So at least this flap and this portion of the flap the inboard flaps are going to have higher values of CLmax.
Thanks for your attention. We will now move to the next section.
Estimation of Maximum Lift Coefficient
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Remember we have to proceed from 2D to3D. So the information that we have seen so far is mainly for 2D. If you look at 3 dimensional effects generally what happens is that if you have low quarter chord sweep if you have reasonably high aspect ratio and if you have a taper of nearly 0.5
and large flaps then the loss because of the 3D effects is only around 10%. So therefore the wing ????? would be approximately 0.9 of the ????? of the aircraft.
And most airline airliners will fall in this category. So the most airliners will you can assume for most airliners that the 3D effects are only above 0.9. Now when you use partial span flaps then it is possible to use this formula assuming that 0.9 is what you will get but it this particular additional
term helps you to identify the effect of flapped.
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And unflapped area and just to show you that the flapped and the unflapped area are the areas which are under the influence of the flap. So if you have a trailing edge flap then the area ahead of it also becomes a part of the flapped area and if you have leading edge flaps then the area behind
it also is a part of the flapped area. So you have to use these ratios and notice if you have almost full span flaps. For example suppose you have a wing where you have full span flaps then you know the ratio between the 2 is going to be almost equal to 1.
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In military aircraft we sometimes see strakes mounted at the root of the fuselage of the wing near the fuselage and these strakes create a vortex so we call them as either as leading extensions or strakes and the effect of the strakes is to create an increase a nonlinear increase in the lift curve
slope. So with no strikes even you have a linear curve with strakes it becomes a little bit nonlinear and it is beneficial. So the ??? with strakes can b simply estimated as the ??? without strakes that you already know multiply by the summation of the strake area and the wing reference area upon the wing reference area this formula can give you an information regarding the strakes (Refer Slide Time: 03:01)
But this is only this is only applicable for low angle of attacks at which the stakes are not very effective because then we assume them to be like additional area of the wing itself. Now the presence of tail whether the horizontal tail or a canard also affects the lift that is produced. So that
delta ?? because of horizontal tail is equal to Δ???(??? ?? ℎ????????? ????) = ???,? (1 −????)??? where ???? is the effectiveness of the flap and or it is the influence.
So this is obtained using a formula which relates to the geometry of the flap this effect is created because there is always a downwash acting behind a wing. And so the angle at which the air comes onto the tail is not the freestream. But there is some kind of a downwash. And this particular
downwash angle changes with the angle of attack ? also.
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So the effect of high lift devices most flaps they increase the ? at ??=0 but they do not change ??? ??. Basically it is like an equivalent increase in the alpha that is why if you noticed the line was parallel.
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For full span flaps there is hardly any effect. So the Δ?3? is same as Δ?2?.
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But if you have partials and flaps then you can use this formula to get the value of the delta alpha increase. And notice that.
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These formulae are going to help you to acquire the in information that you need for your calculations. And notice that the Δ?2? is 10 degree at takeoff a 15 degree at landing as we saw in the curve in the figure behind. Thanks for your attention. We will now move to the next section.
Lecture – 56
Flaps as High Lift Devices
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Flaps as we all know are using high lift devices, and during landing we have a very large deflection as can be seen here there is a very large deflection typically between 30 to 60 degrees, and the ?? at landing is normally the ????? and the aim is to lower the landing distance during takeoff the
flaps are deflected at lower angles as you can see here, the deflection angle of these flaps are lower than that you see in the landing typically, the angle of flap deflection would be 15 to 30 degrees.
So, one can assume that the ?? at takeoff is going to be 80% of ????? , because the deflection of the flaps is at a lower angle and the purpose of using flaps during takeoff is to have a better climb performance.
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But, there are also many different types of flaps and each of them has got a different effect on the
lift coefficient. These are some of the standard flap sections that you would use. So, for example, if you have a basic flap, basic aircraft suppose you have a basic wing whose ????? is equal to A this is our baseline let us see how the usage of flaps and that 2 different type of flaps increases the
value of ????? from A. So, with a plain flap you can have 50% higher value of ?? just by deflecting a plain flap.
If you have a split flap you can get slightly better up to 60% higher, this is because we are not spoiling the flow on the upper surface we are only creating a deflection of the flow on the downward surface as compared to the plain flap. If you have a slot in the flap then you are allowing
the air from here to actually go and flow over it. So, you can get slightly higher you can get maybe 65% higher value of the ?? and if you use a Fowler flap in which not only does the flap deflect down with a gap.
But also moves backward resulting in an effective increase in the surface area then you can almost double the ?? coefficient of to order 1.9. And if you have a basic wing section like this, if this is the left corner of the basic wing section, using a flap is going to lead to a parallel line and at all
angles is going to affect the increase in the lift. Similarly, if you look at the ?? versus ?? curve, so, if you have a basic wing line as shown here, then you go for a tilt when you include flaps in the aircraft.
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Now, if we want to sequentially look at the effect of different type of flaps. So, if the basic airfoil you know this is the same information but now placed in a more detailed fashion. And as I mentioned a Fowler flap can lead to almost 90% increase in the in the lift coefficient. Notice that
these are only approximations because the actual increase depends on the geometry of the airfoil also. If you go further and if you start putting now slots in fowler flaps, you can go for a higher increase.
Let us look at now some leading edge devices. So by using a Kruger flap, which is basically a flat curved plate with some kind of rounded leading edge, you can get 50%, if you put a slot you get 40%, a fixed flat with a gap in between can give you around 50% increase and as you keep on moving and as you keep on increasing the complexity of the system you get more and more benefits. For example, if you look at this configuration, where you have a slat and you have a double slotted Fowler flap, you can get around 120%; increase in the lift coefficient. But this is basically a complicated system. So, you have to be careful about use of flaps.
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Because improvements do not come without any problem improvements always come up with some kind of a compromise. So, this table sums up the typical values of the maximum lift coefficient. So, if you do not have the actual data available, and if you just know the type of the flap for example, if you are told that the aircraft is using double slotted flaps and slats, you can just use this value as a good starting value to calculate the to estimate the max lift coefficient.
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This graph from Raymers textbook shows 2 things. It shows that different flaps types have higher values of ?????. And it also shows how these values reduce with the increase in the quarter chord sweep. So you can use this graph for the wings of moderate aspect ratio only. And with this graph,
you know you can probably get something like if the sweep is 40 degrees, and the flap type is Fowler, then you can get the value of ????? that you can take from the graph. Thanks for your attention. We will now move to the next section.