= Zero-lift angle of attack
{title2=$\alpha_{L=0}$}
The \b[<zero-lift angle of attack>] is an incidence at which the relevant <lift coefficient> is zero. For a two-dimensional <airfoil> in its approximately linear attached-flow regime, write
$$
c_l\simeq a(\alpha-\alpha_{L=0}).
$$
The slope $a$ and intercept $\alpha_{L=0}$ are different properties. A cambered section can produce positive <lift> at zero geometric <angle of attack>, so its zero-lift incidence can be negative. Flow conditions and the distinction between a section and a finite wing must accompany an experimental value.
In <thin airfoil theory>, $a=2\pi$ per <radian>. With $\xi=x/c=(1-\cos\theta)/2$ and mean-line slope $z'(\xi)=dy_c/dx$, the camber contribution is
$$
\alpha_{L=0}=\frac1\pi\int_0^\pi z'\!\left(\frac{1-\cos\theta}{2}\right)(1-\cos\theta)\,d\theta.
$$
The weighting means that the full <airfoil mean camber line>, rather than only maximum <airfoil camber>, determines the ideal incidence. The originating lecture notes derive this result from the first two <Fourier coefficients> of the tangency condition.<F3 Lecture Notes: Thin-Airfoil Analysis Problem, continued by MIT OpenCourseWare>[pp. 2–3, camber-slope coefficients, lift calculation and zero-lift angle]{ref}
For an original parabolic mean line $y_c/c=4m\xi(1-\xi)$, the slope becomes $4m\cos\theta$. Since $\int_0^\pi\cos\theta\,d\theta=0$ and $\int_0^\pi\cos^2\theta\,d\theta=\pi/2$, direct <integration> gives
$$
\alpha_{L=0}=-2m.
$$
At $m=0.02$, this is $-0.04$ radians, or about $-2.292\,{}^\circ$, and the modeled zero-incidence lift coefficient is $4\pi m\simeq0.2513$. This parabolic example is not the piecewise mean line of <NACA 2412>. Substituting the actual piecewise mean line of <NACA 4412> instead gives about $-4.1545\,{}^\circ$ within the same ideal theory.
An experimental estimate varies incidence while recording a force balance or integrating measured surface <pressure>, then locates the zero crossing in a suitable flow regime. For two stipulated linear data points $c_l=0.10$ at $-1\,{}^\circ$ and $c_l=0.30$ at $1\,{}^\circ$, linear interpolation or extrapolation gives $a=0.10$ per degree and $\alpha_{L=0}=-2\,{}^\circ$. These invented observations illustrate the reduction, not a particular profile measurement. <Reynolds number>, <Mach number>, surface condition and incidence calibration belong in a real comparison.
Zero <lift> does not require zero <drag> or zero <pitching moment>. Nor does extending a linear fit beyond <aerodynamic stall> establish another valid zero-lift prediction. The definition concerns the chosen lift component under specified conditions.
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