= Symmetric airfoil
A <symmetric airfoil> has upper and lower surfaces that are reflections across its <airfoil chord>. Its <airfoil mean camber line> coincides with that straight reference, so its <airfoil camber> is zero. Symmetry constrains shape; it does not require zero <lift> at every <angle of attack>.<Wing Geometry by NASA Glenn Research Center>[Airfoil Geometry, reflection symmetry and mean camber line]{ref}
In chord-aligned coordinates, a graph representation satisfies $y_l(x)=-y_u(x)$. Reflection changes the sign of incidence and transverse <force>. For an otherwise reflection-symmetric flow problem and corresponding reflected solutions, the section <lift coefficient> is odd in incidence, $c_l(-\alpha)=-c_l(\alpha)$, while <drag> is unchanged. At zero incidence, the symmetric solution has zero <lift>. An asymmetric nearby wall, control-surface setting or flow history can break the symmetry of the complete problem.
Under the attached-flow assumptions of <thin airfoil theory>, the <zero-lift angle of attack> is zero and $c_l\simeq2\pi\alpha$, with $\alpha$ in <radians>. For an original ideal calculation, take $\alpha=3\,{}^{\circ}$, <mass density> $\rho=1.2\,\mathrm{kg/m^3}$, speed $U=30\,\mathrm{m/s}$ and <airfoil chord> $c=0.5\,\mathrm m$. Then
$$
c_l\simeq\frac{\pi^2}{30}=0.328987,\qquad
L'=\frac12\rho U^2c\,c_l\simeq88.83\,\mathrm{N/m}.
$$
The reflected incidence gives the opposite <lift> per unit span in this model. These are illustrative section calculations, not measured data for a manufactured wing.
<NACA 0012> supplies a standard geometric example. A <wind tunnel> comparison can measure <lift> at equal positive and negative incidences while controlling <Reynolds number> and surface condition. Comparing the reflected measurements tests the symmetry of the whole setup as well as the section; it does not extend <thin airfoil theory> through <aerodynamic stall>.
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