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‫Welcome back.

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‫So in order to rewrite this differential thrust in terms of differential lift and differential drank,

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‫we first have to look at this fine angle.

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‫You have this fine angle over here as well.

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‫It's this one here in the same triangle is also here like this.

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‫So for the differential thrust force, you only need to extract the vertical components of both of these

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‫vectors.

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‫And you can see that this deal has a positive vertical component like this and this deal has a negative

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‫vertical component like this, so therefore the T equals the L times.

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‫And in order to get this vertical component of DL, you simply have to multiply this DL by cosine fi.

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‫And to get the negative vertical component of this director, you simply have to put here minus the

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‫the Times and sine fi and that's it.

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‫That's how you write this equation.

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‫Now of course you might ask now why we did that.

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‫The reason why we did that is because from aerodynamics we can take formulas for DL and so for DL.

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‫The aerodynamics gives us this equation.

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‫DL equals C sub L times one half times the air density row times you squared, which is this velocity

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‫vector here, the resultant air velocity vector, then times C times D hour and then the differential

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‫drag equals C D times one 1/2 times the air density times you squared times C times D.R.

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‫So let's very quickly look at these terms and see what they are.

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‫SEECP L is called lift coefficient.

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‫This lift coefficient is a dimensionless quantity that essentially tells you how good the shape of your

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‫airfoil is for lift generation.

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‫A bigger lift coefficient means that your shape is better for lift generation.

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‫And it makes sense, right?

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‫The bigger the s.L, the bigger your deal.

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‫However, S.L is not constant.

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‫It depends on three things.

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‫It depends on the angle of attack, which is this one here.

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‫It also depends on the Mach number and on the Reynolds number, first of all, let's look at the angle

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‫of attack.

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‫I'm going to draw here a typical graph for s.L versus Alpha.

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‫So this is S.L here and this is Alpha and a typical graph looks like this meaning as you increase your

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‫angle of attack, which is this angle here, you also increase your lift coefficient.

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‫However, when the angle of attack becomes too large, then your lift coefficient starts dropping,

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‫and that's when, for example, an airplane enters into a stall.

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‫So essentially, what you want for live generation, you want a smooth flow around the airfoil.

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‫That's what we call laminar flow.

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‫But when you're angle of attack is too large, then your flow will become turbulent.

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‫Something like this, and as a result, you lose lift and gain a lot of drag.

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‫And you will start falling down then.

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‫And in order to avoid falling down, then you have to put your nose down again, meaning you have to

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‫decrease your angle of attack.

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‫And then you have something called the magic number and the magic number is the ratio of your velocity

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‫over the speed of sound.

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‫So your magic number equals your you velocity divided by the sound.

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‫And so if you're below one, if your magic number is below one, then you travel under the speed of

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‫sound.

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‫And if it's above one, then you go supersonic.

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‫And your lift coefficient also depends on the magic number.

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‫And finally, you have something called a renals number, which also comes from aerodynamics.

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‫The Reynold's number is a ratio that is used in aerodynamics to determine if the airflow over the wing

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‫will be laminar or turbulent.

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‫The difference between the laminar and turbulent flow is the following.

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‫This would be a laminar flow.

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‫It goes over the wing in a very smooth way and this is a turbulent flow.

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‫It detaches itself from the wing and it becomes very chaotic and it's very hard to model this kind of

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‫flow.

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‫And so this Reynolds number, it helps us to predict what kind of flow we will have when the air goes

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‫over some kind of wing.

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‫So I mentioned that Reynolds number is a ratio.

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‫And one way to write it is like this rule times, Soviet times all over new roadways, the air density.

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‫Then why is the air velocity that would be Revy and then El is essentially the cross-sectional length

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‫of the object.

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‫It can also be called cord.

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‫And then this mbewe here.

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‫It measures the viscosity of your fluid.

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‫So what is viscosity?

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‫If you think about water versus honey, then you know that honey is more gluey, right?

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‫Honey doesn't change its shape as easily as water.

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‫And so honey is more viscous than water.

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‫And so if you look at this ratio, then essentially the denominator, it contributes more to the fluid

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‫being laminar.

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‫So the bigger this new value, the more it contributes to the flow being laminar.

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‫Why?

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‫Because the viscosity is greater and viscosity is like gluing this and this glue in it will help the

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‫fluid to be more laminar and not become chaotic, like in the case of a turbulent flow.

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‫On the contrary, the numerator, the bigger it is, the more it contributes to the fluid becoming turbulent.

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‫So, for example, the bigger the velocity, the easier it is for the flow to detach itself from the

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‫wing and become turbulent.

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‫And that's why when you have lower Reynold's numbers, then the flow tends to be laminar.

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‫And that is very good for generating lift.

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‫By the way, you want your flow to be laminar.

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‫You don't want it to be turbulent.

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‫You want it to be laminar because then you will have high lift forces and it is a lot easier to model

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‫the airflow when it's laminar.

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‫And so if your value for viscosity is greater, then this ratio becomes smaller.

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‫And so a smaller ratio means a smaller Reynolds number.

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‫And that's why laminar flows happen for smaller, rainless, no values.

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‫But when your numerator becomes bigger relative to your denominator, then your renals number increases

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‫and that contributes to the fluid being turbulent.

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‫And so your lift coefficient also depends on your renals number.

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‫In conclusion, your S.L depends on your angle of attack, your Mach number and your Reynold's number.

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‫So that was the lift coefficient and now that's the air density that you have is just air density in

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‫kilograms per cubic meter.

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‫It's how dense your air is.

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‫Then this you vector, that's your resultant air velocity vector, but not in this formula.

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‫You have to square it.

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‫Then see is the core length, which is this one in green.

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‫And then finally, your D-R, that's your differential with the differential width of your airfoil,

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‫so you can imagine that this Sea Times D.R, it's really your differential area, but not the differential

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‫area of your upper surface.

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‫Rather, if you think about it, then this is your court length, then if you multiply them together.

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‫Then you will have this differential rectangle inside this airfoil.

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‫That looks something like this, and that would be your differential area here.

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‫All right, and finally, let's talk a little bit about the drag coefficient, see some somebody.

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‫And so drag coefficient, just like lift coefficient, it's a dimensionless quantity that tells you

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‫if you're airfoil shape contributes to drag force more or it contributes to it less.

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‫So if you're see Savides bigger than your differential drag force would be bigger and vice versa.

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‫So I've looked up to Shape's on Wikipedia.

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‫One is a sphere, and so if you let air go through it like this, then the drag coefficient is zero

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‫point forty seven.

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‫And if you have a streamlined body like this.

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‫Then your drag coefficient is zero point zero for.

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‫And in aerospace, you want a very low drag force and therefore airfoils, they look like a streamlined

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‫body like this one.

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‫And also your drag coefficient depends on the angle of attack, the magic number and the Reynold's number.

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‫So it's not constant.

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‫Now, what I want you to understand is that s.L and said they are not about size, they are about shape

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‫to objects having the same shape but different size, assuming that they're Alpha and M and the rainless

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‫number, assuming that they're all the same.

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‫Then the two objects with the same shape, but different sizes have still the same s.L and the same

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‫CD.

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‫So that's the point of the lift and drag coefficient.

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‫They help us analyze the shape of an object without considering their sizes.

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‫You want to know how the shape of an object contributes to the lift and drag force without caring about

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‫the size of that object?

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‫Now, you know that this new vector can be rewritten like this, you equals square root.

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‫Or mega times are squared, plus the sub, the squared, and we got that thanks to Pythagoras Theorem.

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‫That means that we can rewrite our differential lift and drag forces like that.

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‫Diesel equals C sub L times one 1/2 times the air density, and now instead of writing you squared,

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‫you can take this thing and you can square it.

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‫And if you do that, then the square root will disappear.

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‫So you will simply have.

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‫Omega are squared, plus Visa V squared, and you put it in the brackets here and then times C times,

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‫these are and you can rewrite your differential drag force like that as well.