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I am using a reflexed airfoil designed for positive pitching moments to achieve a pitching angle matched to the changing of free-stream velocity (Aerodynamically tailored). I ended up with a pitching moment equation $$ dM = c*Cm_{a.c} - (C_{l_0}+C_{l_α}*α)*Χ_{a.c} $$ where $c$ the chord length, $Cm_{a.c}$ the pitching moment coefficient about the aerodynamic center of the airfoil $Cl_0$ the zero lift pitching coefficient, $Cl_a$ the lift coefficient at a certain angle of attack and and $Xa.c$ an offset distance about aerodynamic center.

I want to design a propeller and the data i have is free stream velocity, RPM, and geometry of the blade. I assume a thin airfoil so the lift coefficient ($(C_{l_0}+C_{l_α}*α)$) is roughly $2π*α$ and $α$ varies with free stream velocity and rotational speed: $α = Δβ - tan^{-1}(V_{inf}/V_r)$ and $Δβ$ the pitch angle.

The idea is to find the ideal angle of attack $α$ that satisfies the pitching moment equilibrium condition $dM = 0$. However, pitching moment coefficient $Cm_{a.c}$ is also unknown.

*Let's say $X_{ac}$ value is given.

So, my question is how this equation can be solved? Can i get any other data from the airfoil's profile that help to find the solution? Are reflexed airfoils have any characteristics that i should consider in such situation?

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  • $\begingroup$ I don't understand the chord with the $C_M$, that would make the dimensions not right. $\endgroup$
    – Koyovis
    Jul 27, 2017 at 11:11
  • $\begingroup$ @Koyovis This comes from the equation $$dM = dM_{ac} - dL*X_{ac} = 1/2ρV^{2}c^{2}Cm_{a.c}dr - 1/2ρV^{2}cC_LX_{ac}dr = 0$$ $C_L = (C_{l_0}+C_{l_α}∗α)$ $\endgroup$
    – george
    Jul 27, 2017 at 11:27
  • $\begingroup$ Yes indeed. I would expect a length divided by a length, a dimensionless entity, to accompany the $C_M$ $\endgroup$
    – Koyovis
    Jul 27, 2017 at 11:39
  • $\begingroup$ @Koyovis Imagine schematic shown in this link: aviation.stackexchange.com/questions/40910/… $\endgroup$
    – george
    Jul 27, 2017 at 11:44
  • $\begingroup$ Do you want to solve the equation for $Cm_{ac}$ or for $\alpha$? If you have one equation you cannot solve for both. From the title of your question it appears that $\alpha$ is an unknown, however if free stream velocity, rotational speed and twist are known, then $\alpha$ is known and you can solve for $Cm_{ac}$ $\endgroup$
    – Koyovis
    Jul 28, 2017 at 7:03

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If you have access to a model of your airfoil, a moment balance, and a wind tunnel, you may determine $C_{M_{ac}}$ experimentally: the moment the wing experience may be normalized to calculate $C_{M_{ac}}$

Then solving for alpha is trivial. Otherwise a tool like XFLR5 may be useful to find $C_{M_{ac}}$ for your airfoil.

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  • $\begingroup$ I believe he's using a symmetrical aerofoil and a NACA profile of which data is published. $\endgroup$
    – Koyovis
    Jul 28, 2017 at 3:25
  • $\begingroup$ @Koyovis I am using a an EPPLER airfoil with reflexed profile. For symmetrical airfoils i found that pitching moment coefficient is zero, so it doesn't work $\endgroup$
    – george
    Jul 28, 2017 at 9:16
  • $\begingroup$ If it is published the pitching moment at ac is usually published $\endgroup$
    – user74671
    Jul 28, 2017 at 11:15
  • $\begingroup$ @user74671 Yes. It is published, but the only thing i can find is a relation between $Cm$ and $α$. airfoiltools.com/airfoil/details?airfoil=e328-il (last plot) *Pitching moment has to be zero for static stability $\endgroup$
    – george
    Jul 28, 2017 at 11:23
  • $\begingroup$ Ok then still find alpha where L=0, look at graph, Cm there is Cm at ac $\endgroup$
    – user74671
    Jul 28, 2017 at 11:28

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