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→Implications of the Analytical Polar Curve: Minimum Sink and Best Glide
== Implications of the Analytical Polar Curve: Minimum Sink and Best Glide ==
=== Shape and General Features of the Analytical Polar ===
The polar equation is a combination of a third order term which is monotonically increasing throughout the domain of definition, and a hyperbolic term which, in the domain of \( V>0\), decreases monotonically. Therefore, a global minimum is expected. This will be (confusingly) referred to as the global maximum because conventionally, the Y-axis (\( V_S \)) is turned upside-down, such that going down means higher sink rate.
The analytical form of the polar curve applies to the speed range from several knots above stall to \( V_{NE} \) (\( V_{NE} \) must be converted to its indicated value). It does not apply close to stall, which is because the aerodynamics of the glider changes considerably before the onset of stall such that the drag ceases to be a parabolic function of the lift.
The analytical polar given above can be plotted by any computer code, or a plot can be found in any gliding textbook. You can also ask an instructor to draw you one. Conventionally, the Y-axis (\( V_S \)) is turned upside-down, such that going down means higher sink rate. When plotted this way, this curve is convex and has a single global maximum. The curve is monotonic on both sides of the maximum.
=== Minimum Sink Airspeed ===
\[ V_{MS} = (\frac{4kW^2}{3 \rho^2 S^2 C_{D0} \pi A})^{\frac{1}{4}} \]
It is common practice to define a quantity called wing loading as \( \omega = \frac{W}{S} \) which quantifies how much weight each meter squared wing area is carrying. With this definition in place, notice that : \( [ V_{MS} \) is proportional to the square root of wing loading. propto \sqrt{omega} \] The implication is, the minimum sink airspeed is not fixed: with the glider loaded heavier it will become higher. It is worth noting that, by flying at this airspeed: \[ V_S(V_{MS}) \propto \sqrt{omega} \]
If a glider thermals at the minimum sink airspeed, carrying water ballast will enable the glider to fly faster and likely at a larger radius. This can prove to be beneficial as some experienced pilots will say, but a mathematical proof is not possible in the absence of a model to characterise the behaviour of the thermal. Water ballast is usually carried on good thermal days but not on days with marginal conditions. You will sometimes hear pilots say that the water doesn't work, the author's interpretation to which is that, because the thermals are not strong and big enough, the increased minimum sink by carrying water ballast outweighs the possible benefits if any.
