# How to obtain the $dC_d/dC_l^2$ value from the drag polar of an airfoil? [closed]

I'm currently trying to do an initial design for a propeller. In order to do this I'm trying to use Xrotor. Xrotor allows the user to enter certain information about both the propeller geometry, flight conditions and the airfoil lift and drag data after which it performs its calculations. Xrotor requires the following values for the lift and drag data (example values are given):

========================================================================
1) Zero-lift alpha (deg):   0.00       7) Minimum Cd           : 0.0070
2) d(Cl)/d(alpha)       :  6.280       8) Cl at minimum Cd     : 0.150
3) d(Cl)/d(alpha)@stall :  0.100       9) d(Cd)/d(Cl**2)       : 0.0040
4) Maximum Cl           :  2.00       10) Reference Re number  : 2000000.
5) Minimum Cl           : -1.50       11) Re scaling exponent  : -0.2000
6) Cl increment to stall:  0.200      12) Cm                   : -0.100
13) Mcrit                :  0.620
========================================================================


This information can be obtained from the drag polars of the airfoil.

$\alpha,C_l$ plot">

$C_d,C_l$ plot">

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The 360 degree polar was made using JBlade. The only value that Xrotor requests that I am not sure on how to calculate is the d(Cd)/d(Cl^2) figure. I'm fairly confident that this can be obtained using the $$C_d,C_l$$ drag polar, but I'm not completely confident on how I should go about it. What is the correct way of determining this value?

• I am voting to close this question because it is asking about aviation engineering rather than the physics of flight. – sammy gerbil Jan 10 at 8:33

This value can be found by performing analyses at multiple $$\alpha$$, after which the $$C_d$$ and $$C_l$$ values at those $$\alpha$$ can be found. Then you can plot the $$C_d$$ on the $$y$$-axis and the $$C_l^2$$ on the $$x$$-axis and then the slope of the graph represents the $$dC_l/dC_l^2$$ value. This should be constant across the linear part of the lift slope and is in the order of magnitude of $$0.004$$.
First, get the best fit for $$C_D=C_{D0}+k_1\cdot C_L+k^2 C_L^2 \tag{1}$$ using the limits of the linear zone of the lift curve for $$c_l$$. Then, $$d(C_d)/d(C_l^2)$$ is just the coefficient $$b$$ of the equation: $$C_D=C_{Dmin}+b(C_{Lo}-C_L)^2, \tag{2}$$ which is just another form of (1). So, to find $$b$$, just expand (2), equal the equations and solve for $$b$$. The results are:
$$C_{Dmin}=C_{D0}-\frac{k_1^2}{4k_2}$$;
$$C_{Lo}=-\frac{k_1}{2k_2}$$ and $$b=k_2$$