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→Adjustments to the Analytical Polar: Headwind and Sinking Air
It should be obvious by now that the above adjustments to the polar have no effect on the minimum sink speed: the difference only arises when \( V \) is divided over to the left side, i.e. a glide ratio is sought after. Physically this makes sense: the minimum sink speed is purely an interaction between the glider and the surrounding air, and if we disregard all relativity to the ground, then the air in which the glider flies can move in whichever possible way (so long as it is not accelerating) and the glider can perform the same macroscopic motion with it without altering the detailed aerodynamics.
== The Non-Dimensional Polar and the Determination of the Polar in Practice ==
The polar equation can be abstracted into the following form:
\[ V_S = AV^3 + \frac{B}{V} \]
With \( A=\frac{\rho C_{D0}}{2 \omega} \) and \( B=\frac{2k \omega}{\pi A \rho}.
At best glide, by our calculations from the previous sections, the best glide speed is given by (notice the change in notation for reading convenience):
\[V_i = (\frac{B}{A})^{\frac{1}{4}} \]
\[V_{Si} = 2(AB^3)^{\frac{1}{4}} \]
These can be substituted into the abstract polar equation, such that:
\[ 2\frac{V_S}{V_{Si}} = \frac{AV^3}{(AB^3)^{\frac{1}{4}}} + \frac{B}{(AB^3)^{\frac{1}{4} V} \]
This can be simplified into:
\[ 2\frac{V_S}{V_{Si}} = (\frac{V}{V_i})^3 + (\frac{V_i}{V}) \]
This is the '''non-dimensional polar'''. It tells us that the polar curve is deterministic from only two quantities: the best glide speed, and the sink rate at the best glide speed. These two quantities both depend on the wing loading, so the additional requirement is that they be measured with the same level of wing loading.
Aerodynamic coefficients such as \( C_{D0} \) are difficult to determine and measuring such quantities require sophisticated equipment and techniques include wind tunnel testing and flight tests. However, the polar can be determined simply by test flying the glider and plugging the measured airspeed and sink rate into the non-dimensional polar as coefficients. This is a useful method to determine a polar of a glider which you may not have a manual for.
The above, nevertheless, assumes that the parabolic relationship between \( C_L \) and \( C_D \) holds true, which is something we have been doing throughout this article. This relationship has its limitations and such limitations lead to most of the deviations from the analytical polar as observed in flight.
=== Further Reading ===
Readers are recommended to ''The Paths of Soaring Flight'' in which a more thorough mathematical treatment is available. This book can be downloaded as PDF by simply Googling the title.
