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I can't understand where the equation in the blue circle comes from or is derived from, I know it says the sum of the perpendecular components should be zero, but can you explain visually? and what exactly does the B angle, the flow angle, represent? $$-u\sin B+v\sin B=0$$ Airfoil Theory

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  • $\begingroup$ It's written just before the equation: "The velocity component perpendicular to the airfoil profile is zero". What's not clear? The equation itself? Or why it is zero? $\endgroup$
    – sophit
    Aug 25, 2023 at 12:29
  • $\begingroup$ The equation itself is not clear, I mean where did this equation come from? $\endgroup$
    – Dazai
    Aug 25, 2023 at 12:43
  • $\begingroup$ Ah ok, that equation just translates the fact that the speed perpendicular to the surface must be zero since the airfoil is "airtight" i.e. no air can flow though the wing. $\beta$ is simply the slope of the airfoil so that $u sin \beta$ and $v cos \beta$ are the speed components perpendicular to the airfoil and their sum must be zero. Mathematically this is called "boundary condition". $\endgroup$
    – sophit
    Aug 25, 2023 at 12:55
  • $\begingroup$ thank you for your comment, can you explain visually if possible $\endgroup$
    – Dazai
    Aug 25, 2023 at 13:30

2 Answers 2

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You have to look at the velocity decomposition at location $y_u$. Below I made a velocity diagram at location $y_u$:

enter image description here

If you decompose $u$ and $v$ into the red velocity (tangential to the surface) and the blue velocity (perpendicular to the surface), which formulae do you get?

Then the text goes on to say that the velocity perpendicular to the airfoil surface is 0, so what happens when we apply that knowledge to our just-derived equations?

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  • $\begingroup$ Thank you for your attention and answer. It made me feel deprived of math :d. My mistake was to assume the y-axis is "perpendicular" $\endgroup$
    – Dazai
    Aug 25, 2023 at 13:46
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That equation just translates the fact that the speed perpendicular to the surface must be zero since the airfoil is "airtight" i.e. no air can flow though the wing. This condition is normally termed "non-penetration".

$\beta$ is simply the slope of the airfoil and equals, by definition, $dy_u/dx$ and $dy_l/dx$.

Being $u sin \beta$ and $v cos \beta$ the speed components perpendicular to the airfoil, the non-penetration condition simply gives that their sum must be zero. Mathematically this is called "boundary condition".

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