0

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?

george
  • 77
  • 3
  • 9
  • I don't understand the chord with the $C_M$, that would make the dimensions not right. – Koyovis Jul 27 '17 at 11:11
  • @Koyovis This comes from the equation

    $$dM = dM_{ac} - dL*X_{ac} = 1/2ρV^{2}c^{2}Cm_{a.c}dr - 1/2ρV^{2}cC_LX_{ac}dr = 0$$

    $C_L = (C_{l_0}+C_{l_α}∗α)$

     – george Jul 27 '17 at 11:27
  • Yes indeed. I would expect a length divided by a length, a dimensionless entity, to accompany the $C_M$ – Koyovis Jul 27 '17 at 11:39
  • @Koyovis Imagine schematic shown in this link: https://aviation.stackexchange.com/questions/40910/how-to-define-pitch-angle-for-a-passive-variable-pitch-propeller?noredirect=1#comment106318_40910 – george Jul 27 '17 at 11:44
  • 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 $Cm_{ac}$ – Koyovis Jul 28 '17 at 07:03
  • @Koyovis the two variables depend on each other..and if i find one...other can be found from the polars of the airfoil. But, how α can be found by knowing only freestream velocity and rotational speed? Let's say there is no twist at the moment so angle of twist = 0 – george Jul 28 '17 at 09:16

1 Answers1

0

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.

user74671
  • 193
  • 4
  • I believe he's using a symmetrical aerofoil and a NACA profile of which data is published. – Koyovis Jul 28 '17 at 03:25
  • @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 – george Jul 28 '17 at 09:16
  • If it is published the pitching moment at ac is usually published – user74671 Jul 28 '17 at 11:15
  • @user74671 Yes. It is published, but the only thing i can find is a relation between $Cm$ and $α$. http://airfoiltools.com/airfoil/details?airfoil=e328-il (last plot)

    *Pitching moment has to be zero for static stability

     – george Jul 28 '17 at 11:23
  • Ok then still find alpha where L=0, look at graph, Cm there is Cm at ac – user74671 Jul 28 '17 at 11:28
  • with L you mean lift? If so, why am i looking for L=0? – george Jul 28 '17 at 11:30
  • If you look at your equation in your other question the over all moment coefficient is made up of CMac, and the moment produced by the lift. (Drag would also produce a moment but its small). When L (or CL) is zero, the only moment acting on the foil is that at the aerodynamic center – user74671 Jul 28 '17 at 11:35
  • The problem is that L (or CL) varies with the freestream velocity. So i suppose that it cannot be zero at any speed condition. However, my idea is similar; to give an initial value to CL and then find corresponding coefficients and α. But, is there any value can be used as Cl starting point for small propellers (≈15 cm radius)? – george Jul 28 '17 at 11:49
  • Cl varies with alpha. It can also vary at different reynolds numbers. Choose the reynolds bumber you anticipate it to operate at then look up alpha when cl is 0 – user74671 Jul 28 '17 at 12:14
  • @user74671 Ok, thanks..So alpha will be kept constant at any speed condition and is the pitch angle (Δβ) that will change throughout a range of speeds right? – george Jul 28 '17 at 12:19
  • Sorry i was speaking in a general airfoil sense. I believe i mean pitch angle. I mean the angle of attack of your airfoil on the propeller with the air. It is a little more complicated with a propeller since there is relative velocities between the spinning and the free stream velocity – user74671 Jul 28 '17 at 12:29
  • Yes, as i mentioned before, angle of attack depends on pitch angle minus the inflow angle: $α=Δβ−tan^{−1}(Vinf/Vr)$. So α will be constant..and Δβ will change according to Vinf and Vr. Is that what you meant? – george Jul 28 '17 at 12:37
  • I think considering the pitch angle and relative velocities is unnecessary to find the cmac For the airfoil. At first just consider the airfoil in a 2d sense. You will need to figure out the range of Re #s that it will experience by determining the range of velocities it will see – user74671 Jul 28 '17 at 16:43