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water ballast
# Analytical form of the glide polar.
# Implications of the polar: minimum sink speed, and best glide.
# Implications of the polar: water ballast
# Adjustments to the polar: headwind and sinking air.
# Adjustments to the The non-dimensional polar: change of glider weight, first purpose of water ballast.# More effects of water ballast and recommended readings. This article is a major project which will take me at least a month to complete. I cannot save a draft on WiKi, so if you accidentally come here and see this page in its very much incomplete form, please bear with me and come back after some time.
== Glider in Unaccelerated Flight in Still Air ==
== Lift and Drag Coefficients ==
In the discussions that follow, only the incompressible flow regime is considered. This is justified by the low speed that gliders fly at.
=== Definitions ===
\[ V_{BG}=(\frac{4k \omega^2}{\pi A \rho^2 C_{D0}})^{\frac{1}{4}}=(3)^{\frac{1}{4}} V_{MS}=1.3161 V_{MS} \]
Therefore, the best glide speed is always 31.6% higher than the minimum sink airspeed according to our analysis. Slight discrepancies may arise in reality due to the approximations we have made, mainly the aerodynamic ones.
From the expression for \( V_{BG} \) given above, it is evident that:
# \( V_{BG} \propto \sqrt{\omega} \), such that the best glide speed will increase as the wing loading increases, by means such as using water ballast.
# Increasing the aspect ratio can reduce the best glide speed.
# Decreasing \( C_{D0} \) can increase the best glide speed.
It is also of interest to calculate the best '''possible''' performance of the glider, which, by definition, happens at the best glide speed. The algebra proceeds as follows:
\[ V_{BG}^2 = \sqrt{\frac{4k}{\pi A C_{D0}}} \frac{\omega}{\rho} \]
This is to be substituted into:
\[ (\frac{V_S}{V})_{\text{best}} = \frac{\rho}{2 \omega} C_{D0} V_{BG}^2 + \frac{2 k \omega}{\pi A \rho V_{BG}^2} \]
To yield:
\[ (\frac{V_S}{V})_{\text{best}} = 2 \sqrt{\frac{k C_{D0}}{\pi A}} \]
Or, alternatively (to give the large number like 40 or 50 that we are familiar with):
\[ \text{Best Glide Ratio} = 0.5 \times \sqrt{\frac{\pi A}{k C_{D0}}} \]
This is a '''very important''' result, as it gives all the factors underpinning the best performance of a glider (in a particular configuration):
# '''The wing loading does not change the best performance'''. Therefore, a 50:1 glider will be 50:1 with a light pilot or a heavy pilot, or with or without water ballast. This is, however, based on our model, and in reality more factors may come into play. For example, if the wing loading is high, then the best glide speed increases accordingly and the change in Reynolds number may have some effect. Alternatively, the different structural deflections of the wings may produce subtle differences in the aerodynamic geometry. Nevertheless, this is the rationale underpinning the use of water ballast: it does not degrade aerodynamic performance.
# Increasing the aspect ratio of the wing is an effective (and, in fact, easiest) way to improve the best performance, as the best glide ratio scales with \( \sqrt{A} \). This is the reason why high performance gliders have slender wings.
# Improving aerodynamic design, such that \( C_{D0} \) or \( k \) is reduced, can improve the best glide ratio as we would intuitively expect. However, modern advancement in aerodynamics has been agonisingly slow and you realise that there is not much potential to be released by comparing a fibre glass glider built in the 1980s with a modern one. What differences do you spot?
From a geometric point of view, the above solution process is equivalent to finding a ray from the origin that is tangent to the polar curve. You should ask an instructor to demonstrate this to you to reinforce the understanding. This geometric method is useful when more factors are taken into account, such that an analytical solution cannot be obtained easily.
=== Water Ballast ===
Water ballast has no effect on the glider best performance, but it makes the best glide speed faster, so the pilot can cover a certain amount of cross-country distance faster. This is the first reason for using water ballast.
In fact, the use of water ballast '''does not change the shape of the polar at all''', not only for the best performance point. To see this, please read the next section on non-dimensional polar. The shape of the polar is dictated only by the '''best glide speed''' and the '''sink rate at best glide''', but, as shown previously, both quantities are proportional to \( \sqrt(\omega) \). Therefore, as the wing loading changes, the polar curve '''scales''' around the origin with \( \sqrt(\omega) \) but keeps its shape. Because the best glide is a tangent to the polar, and that the polar is scaled around the origin of the ray, the slope of the ray (best performance) is invariant.
The second reason for using water ballast is to improve the performance in headwind and sinking air. This is difficult to prove mathematically as the workings in the next section will show, but geometrically this can easily be demonstrated. Because the polar curve is scaled to be larger, any shift in origin due to headwind and sinking air is '''comparatively smaller'''. This makes the new tangent to the polar closer to the best glide line in stationary air, such that the degradation of performance is less.
Conversely, it can be demonstrated graphically that water ballast is detrimental to performance (in terms of covering ground distance) when there is tailwind or rising air. However, gliders are not usually flown downwind for meaningful distances, and when rising air is present, a pilot will attempt to stay in it and soar, rather than moving to another place, so these effects are unimportant.
Experienced pilots sometimes argue that carrying water ballast improves thermalling performance. A mathematical establishment cannot be made unless a model exists to characterise the behaviour of a thermal (which indeed exists, but the validity is questionable in the author's opinion). It is worth pointing out that, very hand-wavingly we can say, if there is any benefit in carrying water ballast when thermalling, it will come from thermalling at a larger radius, rather than at a higher speed.
== Adjustments to the Analytical Polar: Headwind and Sinking Air ==
You may have been instructed that you need to fly faster in headwind or sinking air to cover ground efficiently. This section briefly demonstrates the underlying mathematics, but you are encouraged to use the geometric method to prove to yourself that it is indeed the case.
=== Effects of Headwind ===
We firstly approximate true airspeed with indicated airspeed. By doing this, the following construction is possible:
\[ V = V_g + V_w \]
Where \( V_g \) is ground speed and \( V_w \) is the headwind speed. This expression is to be substituted into the analytical polar. We should also bear in mind that it is the most efficient covering of '''ground''' distance that is of interest, therefore:
\[ \frac{V_S}{V_g} = \frac{\rho C_{D0}}{2 \omega} \frac{V^3}{V-V_w} + \frac{2k \omega}{\pi A \rho}\frac{1}{V(V-V_w)} \]
This quantity must be differentiated with respect to \( V \) to find the optimum. Performing the differentiation and, after considerable simplifications:
\[ 2V^5 - 3V^4 V_w - 2CV + CV_w = 0 \]
Where:
\[ C = V_{BG}^4 \]
This is a fifth-order polynomial. It is a fact established by Abel in 1820 that a fifth-order polynomial cannot, in general, be solved by radicals. However, we can 'prove' that the solution lies in \( V > V_{BG} \) by considering the following:
# Substitute \( V=V_{BG} \) into the equation above, and show that the value is negative.
# Differentiate the expression and substitute \( V=V_{BG} \) into the differentiated expression, and show that the value is positive.
# Hence, we have a function that is presently negative, but it is increasing, so we would expect a root (the required solution) at a larger \( V \) than the present \( V \) which is \( V_{BG} \)
The above arguments are far from watertight: The differentiated expression only gives a positive value if \( V_w \leq \frac{2}{3} V \). While this is usually the case, the above cannot constitute a proof. A more rigorous analysis on the locations of the maxima and minima is required.
The problem is much simpler if the geometric method is used: to use the geometric method, imagine setting up a ground speed zero which is different from the airspeed zero. The polar is plotted with respect to the airspeed zero but the tangent ray needs to start from the ground speed zero. Because the ground zero is located in \( V>0\), the tangent is steeper and intersects the polar at a larger \( V \).
=== Effects of Sinking Air ===
If the glider is flying in some air that is sinking with a uniform downward speed \( V_{SA} \), then the polar equation should be adapted into the following form:
\[ V_S = \frac{\rho C_{D0}}{2 \omega} V^3 + \frac{2k \omega}{\pi A \rho} \frac{1}{V} + V_{SA} \]
Using the differentiation method to find the optimum airspeed for covering ground (notice that, because there is no headwind or tailwind, the indicated airspeed is equivalent to ground speed. This is not to say we approximate TAS with IAS, but there is a monotonic relationship between the two which is dictated by the altitude, which is a free variable in our problem.)
\[ \frac{d}{dV}(\frac{V_S}{V}) = 0 \]
\[ \frac{\rho C_{D0}}{\omega} V^4 - V_{SA} V - \frac{4k \omega}{\pi A \rho} = 0 \]
This equation has the solution of \( V=V_{BG} \) if \( V_{SA} = 0 \) as expected, but if \( V_{SA} > 0\), then the solution is \( V>V_{BG} \). The proof of this is left as an exercise for the reader.
=== Effects on the Minimum Sink ===
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.
[[Category:Theory]]
