Parameter bounds on maximum lift-to-drag ratio

Parameter bounds on maximum lift-to-drag ratio describe how an allowed set of drag polar parameters limits the optimum predicted by that model. For the positive parabolic drag polar, write $c=C_{D0}>0$ andThis is the unconstrained model optimum; an allowable range of lift coefficients can reduce the attainable value.

The derivatives $\partial F/\partial c=-F/(2c)$ and $\partial F/\partial k=-F/(2k)$ show that $F$ decreases with either parameter. Consequently, positive bounds $c\in [c_{-},c_{+}]$ and $k\in [k_{-},k_{+}]$ implyThis interval arithmetic argument is global over the rectangle, without a small-error approximation.

An original example uses $c\in [0.018,0.022]$ and $k\in [0.045,0.055]$. The predicted optimum ratio lies between about 14.374 and 17.568. Its optimizing lift coefficient lies in the different intervalor approximately $[0.5721,0.6992]$. A range of performance values and a range of operating points answer different questions.

A rectangular enclosure can be conservative. For a separate joint-parameter example, suppose fitting constrains $k=0.001/c$ exactly over a positive interval of $c$. Then the product $ck$ is constant and every allowed pair gives $F=15.8114$, even though each parameter varies. Combining independent marginal extremes would include pairs excluded by that relation.

For small random parameter errors, first-order propagation instead givesThe covariance term is needed when parameters are correlated, consistent with the primary metrological propagation law.^[ref] A standard uncertainty is not automatically a hard interval bound. Neither calculation includes failure of the assumed polar outside its fitted regime, and neither by itself identifies one operating point that is optimal for every parameter pair.