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# 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 ===
\[ C_{DI} = \frac{k}{\pi A} C_L^2 \]
Where \( A \) is the aspect ratio of the wings (how slender the wings are), and \( k \) is a factor that depends on the wing design. This drag component increases quadratically with \( C_L \).
By the explanations above, it should be evident that:
\[ C_{D0} = C_{DF} + C_{DP} \]
== Relationship Between Lift and Drag: the Parabolic Polar ==
The two expressions can both be squared and added together. Notice that \( \cos^2(\theta) + \sin^2(\theta) =1 \), the following is arrived at:
\[ W^2 = (C_L^2 + C_D^2) \times (\frac{1}{2} \rho V^2 S) ^2 \]
Or rather, in the more insightful form:
\[ C_L^2 + C_D^2 = \frac{W^2}{(\frac{1}{2} \rho V^2 S)^2} \]
The right hand side of the expression above is a function of indicated airspeed only, because air density is a constant for it to be compatible with the indicated airspeed.
This is a relationship between the glide slope and the airspeed. From here on, determining the sink rate from the glide slope and airspeed is a trivial geometrical task, so the required relationship between airspeed and sink rate is essentially derived.
== An Approximate Method of Solution: The Analytical Polar Curve ==
We shall attempt a derivation of the analytical polar curve again but using a slightly different algebraic approach than what is used in the previous section. We will see that, by adopting this approach, and by making a simple approximation, the algebra becomes simple enough for us to explicitly express the analytical form of the polar equation (an equation relating sink rate to indicated airspeed).
Firstly, the definitions of \( C_L \) and \( C_D\) shall be substituted into the parabolic relationship between \( C_L \) and \( C_D \), giving:
\[ \frac{D}{\frac{1}{2} \rho V^2 S} = C_{D0} + \frac{k}{\pi A} \frac{L^2}{\frac{1}{4}\rho^2 V^4 S^2} \]
Both sides of the equation above need to be multiplied by \( \frac{1}{4} \rho V^4 S^2 \) (notice that this is the square of \( \frac{1}{2} \rho V^2 S\)), then divided by \( \frac{1}{2} \rho V S \), giving:
\[ DV = \frac{1}{2} \rho V^3 S C_{D0} + \frac{2kL^2}{\pi A \rho V S} \]
To proceed, the conservation of energy must be invoked. We realise that the kinetic energy of the glider is not changing because the glider is flying unaccelerated. Therefore, the release of gravitational potential, the rate of which equals to the power of the gravitational force, must balance the rate at which the mechanical energy of the glider is being dissipated by aerodynamic drag, which is the power of the drag force.
By definition, the power of a force is given by:
\[ P = F \times V \]
In words: the power is the product of the magnitude of the force and the speed of the subject in the direction of the force. By using this relationship, we realise that: the power of the gravitational force is given by \( W \times V_S \) (weight times the sink rate), and the power of the drag force is given by \( D \times V \) (drag times the airspeed). This relationship can also be obtained by a geometrical argument using basic trigonometry.
It should be noticed that the above argument is not watertight: this is because \( V \) is the indicated airspeed which generally differs from the true airspeed, and it is the latter that must be used to calculate the drag power. There are two ways to think around this:
# You can think of this as an approximation that is being made: true airspeed is being approximated with indicated airspeed. As a result, some systematic error will be introduced into the results.
# If you can understand the relationship between \( W \times V_S \) and \( D \times V \) from a geometrical perspective, you can think the following: because we are working in the indicated system where \( V \) is the indicated airspeed, the corresponding \( V_S \) obtained geometrically is the indicated sink rate. It needs to be converted to the true sink rate via the compound quantity \( \frac{1}{2} \rho V^2 \).
By using \( D \times V = W \times V_S \), the last equation becomes:
\[ WV_S = \frac{1}{2} \rho V^3 S C_{D0} + \frac{2kL^2}{\pi A \rho V S} \]
This expression should be examined in detail. The following quantities are known (either set, from design, or can be measured):
# \( W \), weight of the glider, depends on the design, cockpit loading, and amount of water carried, but can be known and usually does not change midway in flight.
# \( \rho \), density of air, because we work in the indicated system, this becomes the air density value used in the ASI, which is a fixed number.
# \( S \), wing area of the glider, known and stays constant (we shall not consider the effects of deploying flaps, etc. on the performance).
# \( C_{D0} \), this depends on the aerodynamic design of the glider.
# \( k \), this depends on the wing design of the glider, a highly complex series expansion to obtain a numeric value exists, but for all practical purposes this is a constant.
# \( \pi \), 3.1415926...
# \( A \), aspect ratio of the wing, depends on the glider design and a known constant.
Therefore, there are three changing quantities in this equation:
# \( V_S \), this is the quantity we are interested in, the y.
# \( V \), this is the quantity we can control, the x.
# \( L \), lift on the glider, what is it?
You should realise that, the existence of \( L \) in the equation above prevents us from obtaining a deterministic relationship between \( V_S \) and \( V \) which is the polar equation we desire. \( L \) can be related to \( W \) by using \( V_S \) and \( V \) and geometrical arguments, but this will complicate the equation and prevent us from arriving at an '''explicit''' relationship. In other words, doing so is the equivalence of reverting to the method in the last section.
Instead, we shall introduce the following '''approximation''': '''the weight of the glider equals to the lift force acting on the glider'''. This sounds intuitively true, but there is an error associated with it, whose magnitude is given by \( 1-\cos{\theta} \). Fortunately, this error is gracefully small at typical glide angles. If the glide ratio is 30:1, the error is 0.056%, which becomes even smaller if the glide ratio is higher.
We have shown that this is a good approximation. Therefore, we can replace the \( L \) in the existing equation with \( W \), and the '''analytical polar curve''' is arrived at:
\[ V_S = \frac{1}{2W} \rho S C_{D0} V^3 + \frac{2kW}{\pi A \rho V S} \]
== 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.
=== Minimum Sink Airspeed ===
We seek an indicated airspeed that will give us the minimum sink rate. This is the indicated airspeed to fly at only if you want to stay in the air for as long as possible with a certain amount of altitude drop. It is useful when thermalling. Flying at this speed may not be able to get you anywhere (in extreme cases you can go backwards rather quickly), so caution and thought is needed.
To find this airspeed, the analytical polar is differentiated to reveal the maximum:
\[ \frac{d V_S}{dV} = 0\]
This gives:
\[ V_{MS} = (\frac{4kW^2}{3 \rho^2 S^2 C_{D0} \pi A})^{\frac{1}{4}} \]
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:
\[ V_{MS} \propto \sqrt{\omega} \]
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:
\[ V_S(V_{MS}) \propto \sqrt{\omega} \]
Therefore, by loading the glider heavier, the minimum sink rate '''possible''' is also higher. This implies that gliders with low wing loading can make use of weaker thermals with a limited rising speed.
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.
=== Best Glide ===
The best glide ratio achievable for a given glider in a particular loading and configuration can be determined from the analytical polar. Recall that the glide ratio is given by the horizontal distance covered over the vertical altitude drop. For analytical purposes, it is necessary to make a small angle approximation such that the horizontal distance is approximately given by \( V \times t \) where \( t \) is time. The accuracy of this approximation is shown in previous sections.
Using the small angle approximation, the inverse of the glide ratio is given by:
\[ \frac{V_S}{V} = \frac{\rho}{2 \omega} C_{D0} V^2 + \frac{2 k \omega}{\pi A \rho V^2} \]
This compound quantity is to be differentiated with respect to \( V \) to reveal the minimum:
\[ \frac{d}{dV}(\frac{V_S}{V}) = \frac{\rho}{\omega} C_{D0} V - \frac{4k \omega}{\pi A \rho V^3} =0\]
This yields:
\[ V_{BG}=(\frac{4k \omega^2}{\pi A \rho^2 C_{D0}})^{\frac{1}{4}}=(3)^{\frac{1}{4}} V_{MS}=1.3161 V_{MS} \]
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]]
