= Section lift-curve slope
{title2=$a_0$}
The \b[section lift-curve slope] $a_0$ is the local derivative of a two-dimensional section <lift coefficient> with respect to its <effective angle of attack>, at stated flow conditions:
$$
a_0=\frac{\partial c_l}{\partial\alpha_{\rm eff}}.
$$
The local linear approximation is $c_l\simeq a_0(\alpha_{\rm eff}-\alpha_0)$, where $\alpha_0$ is the <zero-lift angle of attack>. <Lifting-line theory> uses this section relation while separately solving for the induced flow from the finite wing.<Prandtl's Lifting Line Introduction (16.100) by MIT OpenCourseWare>[Page 2, local lift law and fundamental lifting-line equation]{ref}
The angle unit matters. In incompressible <thin airfoil theory>, the ideal slope is $2\pi$ per radian, or $2\pi(\pi/180)\simeq0.1097$ per degree.<F3 Lecture Notes: Thin-Airfoil Analysis Problem, continued by MIT OpenCourseWare>[Page 3, section lift and zero-lift incidence]{ref} A tabulated slope of 0.10 per degree is about 5.73 per radian. Treating 0.10 as a per-radian value introduces a factor of about 57.3.
For an original section model with $a_0=6.0$ per radian and $\alpha_0=-2\,{}^\circ$, an effective incidence of $3\,{}^\circ$ gives
$$
c_l=6\left(5\frac{\pi}{180}\right)\simeq0.524.
$$
The relevant angular increment is five degrees, not three. With sectional <dynamic pressure> $q=500$ Pa and <airfoil chord> $c=0.30$ m, the resulting lift per unit span is $L'=qc\,c_l\simeq78.54$ N/m.
The <finite-wing lift-curve slope> is different because increased lift changes the <downwash>. For an ideal untwisted elliptic wing with uniform section slope, $\alpha_i=C_L/(\pi AR)$ gives
$$
C_L=a_0\left(\alpha-\alpha_0-\frac{C_L}{\pi AR}\right),\qquad
a=\frac{a_0}{1+a_0/(\pi AR)}.
$$
Taking $a_0=2\pi$ and <wing aspect ratio> $AR=8$ yields $a\simeq5.027$ per radian. This particular reduction is a consequence of the stated ideal loading, not a universal correction for arbitrary swept or separated wings.
A proposed <wind tunnel> measurement fits nearby section-force measurements against incidence while recording <Reynolds number> and <Mach number>. The fitted interval should remain within the approximately linear regime; <aerodynamic stall> can make one constant slope inadequate.
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