 Zero-lift angle of attack

The 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, writeThe 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 $z^{\prime }(\xi )=d{y}_c/dx$, the camber contribution isThe 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.^[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 givesAt $m=0.02$, this is $-0.04$ radians, or about $-2.292{}^{\circ }$, and the modeled zero-incidence lift coefficient is $4\pi m\simeq 0.2513$. 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.