Shifted parabolic drag polar ($C_D=C_{D,\min }+k(C_L-C_{L,\ast }{)}^2$)

A shifted parabolic drag polar models drag coefficient with its minimum at a possibly nonzero lift coefficient:Here $m=C_{D,\min }$ and $h=C_{L,\ast }$ in the header notation. The shift separates the coefficient at minimum drag coefficient from the zero-lift drag coefficient, which is $C_{D0}=m+k{h}^2$.

For an original model, take $m=0.018$, $k=0.050$ and $h=0.20$. The coefficients at $C_L=0$, $0.20$ and $0.60$ are 0.020, 0.018 and 0.026. The smallest drag coefficient occurs at $C_L=0.20$, but that point does not maximize the lift-to-drag ratio.

Writing $f(C_L)=C_L/C_D$ and differentiating givesFor positive lift coefficients the unrestricted optimum is thereforeThe last equality follows by multiplying by $s+h$ and using $s^2-h^2=m/k$. In the example, $s\simeq 0.6325$ and the maximum ratio is about 23.12. An allowed lift interval must still contain $s$ before that model optimum is physically attainable.

The model can also be fitted as the ordinary quadratic $C_D={\beta }_0+{\beta }_1{C}_L+{\beta }_2{C}_L^2$. When ${\beta }_2>0$, completing the square gives $k={\beta }_2$, $h=-{\beta }_1/(2{\beta }_2)$ and $m={\beta }_0-{\beta }_1^2/(4{\beta }_2)$. This makes the relation between a fitted quadratic and the shifted parameters explicit.

A wind tunnel can sample both sides of the proposed minimum while keeping flow conditions and normalization fixed. A small residual at sampled points does not establish accuracy through aerodynamic stall. Nor should $k(C_L-h{)}^2$ automatically be identified as pure induced drag: the usual ideal finite-wing relation depends on $C_L^2$, and a fitted shifted polar may combine several effects.^[ref]