Polar, Performance, and Water Ballast

Have you ever wondered what differentiates the 'high-performance' gliders from the normal two-seaters, apart from the price tag and the rather fragile appearance? What are the ridiculously long wings of a Duo-Discus good for? Or more importantly, after you have paid the launch fee, what can you do to stay in the air for longer? The answers to these questions require an understanding of the performance metrics of the glider.

You might have heard of more experienced pilots talking about 'polars', or you might have seen the convex curve which is confusing to get started with. You might also have seen people adding or dumping water into and out of their gliders. Building on the knowledge of glider performance, we can have a closer look at how these tools help cross-country pilots to fly faster and further.

There is a wealth of text, published or online, discussing the topics mentioned above. However, some of these are rather scattered pieces of discussions on the forums, or they can be written in another system of conventions than what is adopted in Cambridge. Some are sloppy about their assumptions and approximations, and some dive straight into the calculus making it impossible to follow. This work aims to present the derivations of the governing equations and the polar functions in a clear and detailed manner, and summarises the implications for those who would rather not follow the mathematics.

The road map of this article is as follows:

1. Consider the forces acting on the glider in unaccelerated flight: lift, drag, and glide ratio.
2. Aerodynamic coefficients: definitions and meanings.
3. Relationship between lift and drag.
4. General method of solution, and assumptions necessary to simplify it.
5. Analytical form of the glide polar.
6. Implications of the polar: minimum sink speed, and best glide.
7. Adjustments to the polar: headwind and sinking air.
8. Adjustments to the polar: change of glider weight, first purpose of water ballast.
9. 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

We start by considering a glider in unaccelerated flight in still air. We shall assume the following:

1. The aeroplane in question is a glider, i.e. it creates no thrust.
2. The flight is unaccelerated, i.e. the glider is flying straight and level without changing its airspeed.
3. The air is still, i.e. there is no macroscopic movement of air in forms such as wind, thermals, ridge lift, etc.
4. The air is homogeneous in its thermodynamic properties, especially, it has a uniform density \( \rho \).

Governing equations from a force perspective

Hopefully you already understand how a glider can remain airborne, but just in case you are in confusion, consider an unpowered glider in unaccelerated flight in still air: three forces act on the glider, namely:

1. Gravity (weight), pointing vertically downwards.
2. Lift, pointing upwards and perpendicular to the flight path.
3. Drag, pointing backwards and along the flight path.

By Newton's first law, in order for the glider to stay unaccelerated, these three forces must balance. Imagine the glider is flying horizontally. If this is the case, then the lift force must point vertically upwards. We then have a drag force pointing horizontally backwards with no force balancing it, because the other two are both in the vertical direction.

Therefore, the only way for the forces to balance is that, the glider cannot be flying in the horizontal direction. The flight path must be at an angle to horizontal. We shall denote this angle as \( \theta \). By experience, a glider in unaccelerated flight in still air keeps descending, rather than climbing. Therefore, we know the flight path is inclined downwards. We shall define this direction as positive \( \theta \).

With this made clear, the gravity (\( W \)) can be decomposed into two components, one to balance the lift (\( L \)), and one to balance the drag(\( D \)). The following relationship holds:

\[ W \sin(\theta) = D \] \[ W \cos(\theta) = L \]

Dividing these two expressions, \(W\) can be eliminated, giving:

\[ \frac{L}{D} = \frac{1}{\tan(\theta)}\]

The quantity \( \frac{L}{D}\) is referred to as the Lift-to-Drag Ratio.

Governing equations from an energy perspective

An alternative way to think about this is from an energy perspective. Because the drag force wants to slow down the glider and take its kinetic energy away, the glider must keep descending, so that it releases its gravitational potential to make up for the loss, otherwise it cannot remain at the same speed. Consider riding a bicycle: if you stop pedalling on level ground, you will gradually slow down and eventually stop, this is because drag force steals your kinetic energy away and you have no means of replenishing it. However, if you cycle downhill, you will not stop even if you do not pedal.

Therefore, we conclude that a glider flies downhill. This is in agreement with the conclusion of the previous section. We can borrow the notation and call the slope angle of this imaginary hill \( \theta \). Geometrically, if we travel for a unit distance on the face of the hill, then in the horizontal direction the distance travelled will be \( \cos(\theta) \) and in the vertical direction the height drop will be \( \sin(\theta) \).

From an energy conservation point of view, the following expression holds (it means the energy that the drag force uses up equals to the energy the gravity must provide):

\[ D \times 1 = W \times \sin(\theta) \]

This is the same result as in the last section.

It is worth noting that the energy approach tells us nothing about the lift force directly.

Glide ratio

The glide ratio is a measurement of the efficiency of the glider. It means 'how many feet can the glider travel forward for every foot of altitude drop?' If a glider has a glide ratio of 50:1 (fifty-to-one), it means the glider is capable of travelling 50 feet forward for every foot of altitude drop, the same thing applies in meters, etc. We want the glider to travel as far as possible while losing minimum altitude, therefore, a larger glide ratio is favourable over a smaller one.

This measurement we can vaguely call 'performance', although strictly speaking this is only one aspect of it. We can say glider A (50:1) has a higher performance than glider B (30:1), for example.

In the last section, we concluded that, for each unit distance travelled on the hill, the horizontal distance covered is \( \cos(\theta) \) and the vertical distance covered is \( \sin(\theta) \). Therefore, the glide ratio is:

\[ \textrm{Glide ratio} = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)} = \frac{L}{D}\]

This is a very important result. It is also worth noting that, up until now, we have made no approximations.

Glide ratio: influential factors

If you are familiar with calculus you will already have noticed that everything derived above is valid in an instantaneous sense. If you are not, take a minute to appreciate that the scale of time does not play a role in the process explained above: a glider with a glide ratio of 50:1 can travel fifty feet while dropping by one foot, it can also travel for fifty miles while dropping by one mile (slightly more than 5000 ft). If we extend this to the other extreme of the length scale such that the time associated is very small, we can see that the glide ratio is defined for any instant of the flight process.

There is no restriction on the glide ratio changing from one instant to another either, otherwise we will have no need to consider the polar if the glider flies the same at whatever speeds. Before proceeding into the more detailed discussions, some of the most important factors are presented here.

The glide ratio is mainly affected by:

1. The aerodynamic design of the glider. The more streamlined, sleek, and aerodynamic-looking a glider is, with long slender wings and smooth gel coating, the more likely it is to have a larger glide ratio.
2. The configuration of the glider. The glide ratio is almost always the highest in the clean configuration, i.e. with nothing sticking out or deployed. Lowered undercarriage, extended brakes and spoilers, deployed and windmilling propellers, opened or lost canopies, attached ropes are things that will reduce the glide ratio. Generally speaking, having the flaps set to other angles than neutral is not good for the glide ratio, but this very much depends on other factors.
3. The way the glider is flown. For a fixed glider mass (which is generally the case with the exception of jettisoning water), the glide ratio is a function of indicated airspeed, giving rise to the polar which is the immediate next topic. Moreover, if the glider is not flown straight (with sideslip), or flown otherwise than normal (e.g. stalled, inverted), the glide ratio can decrease drastically.

Lift and Drag Coefficients

Definitions

In aerodynamics, the lift coefficient (\( C_L \)) and drag coefficient (\( C_D \)) are defined as follows:

\[ C_L = \frac{L}{\frac{1}{2} \rho V^2 S} \] \[ C_D = \frac{D}{\frac{1}{2} \rho V^2 S} \]

Where:

• \(\rho\) is the true density of air.
• \(V\) is the true airspeed of the aeroplane.
• \(S\) is the area of the wing (projected onto the ground), a fixed value for a given glider.
• \( \frac{1}{2} \rho V^2 \) is collectively known as the dynamic pressure, or dynamic head.

A comment on the dimension

You should notice that, for both coefficients, the unit of both the numerator and the denominator is the unit of force (Newton in SI units). The denominator comprises \( \frac{1}{2} \rho V^2 \) which has the unit of pressure, and \( S \) which has the unit of area, so the product yields a force.

Consequently, both \( C_L \) and \( C_D \) are non-dimensional. These quantities have no units. Non-dimensional quantities are the language of aerodynamics: it allows us to study the underlying physics without being distracted by how things are measured. A K-21 is heavier than a Junior, therefore, in unaccelerated glide with the same angle-of-attack, the wings of the K-21 produces more lift than the Junior wings. What causes this? Is it because the design of the K-21 is aerodynamically superior? The answer is not necessarily, as the K-21 can be flying faster, for instance, or has larger wings. The comparison only becomes meaningful when the lift is non-dimensionalised into the lift coefficient.

A comment on dynamic pressure

More information is available under Pressure, Atmosphere and Instrumentation.

The true air density and the true airspeed always appear together as a compound quantity \( \frac{1}{2} \rho V^2 \) which is referred to as the dynamic pressure or dynamic head.

The density of air is not a constant: it depends on pressure (which most notably depends on altitude) and temperature. This causes a major inconvenience as we have to assert a value to it in order to arrive at any numerical results directly useful for flying: e.g. you do not check any non-dimensional quantities in the cockpit, you read the instruments instead which tells you the airspeed in knots or the altitude in feet.

To overcome this problem, we notice that density only appears within \( \frac{1}{2} \rho V^2 \). Therefore, we can define an equivalent density \( \rho_e \) and an equivalent airspeed \( V_e \) such that:

\[ \frac{1}{2} \rho V^2 = \frac{1}{2} \rho_e V_e^2 \]

We can assert a value to \( \rho_e \) and arrive at a value of \( V_e \) such that, when used together, they produce the same amount of dynamic head, therefore, the aerodynamic effect is exactly the same.

The most reasonable value to assign to \( \rho_e \) would be the density of air at some standard conditions. This can then be implemented into some instrument that tells you \( V_e \) (all this instrument has to do is to measure the dynamic head). So long as all the manuals and polar charts express airspeed in \( V_e \) assuming the same value of \( \rho_e \), the change in true air density will not cause these performance guidelines to vary.

In practise, the instrument that tells you \( V_e \) is the air speed indicator (ASI), and \( V_e \) is known as indicated airspeed. Based on the discussions above, you should realise that:

1. Indicated airspeed is directly related to the dynamic head.
2. The dynamic head is the only way true airspeed affects glider aerodynamics (before it disintegrates by overspeeding).
3. Therefore, the glider's aerodynamics is affected only by indicated airspeed, not true airspeed (apart from the never-exceed speed).
4. We should tabulate performance figures and draw polar graphs using indicated airspeed.
5. We do not need to adjust the performance tables or polar graphs to compensate for non-standard atmospheric conditions.

Components of lift and drag coefficients

More information is available under Aerofoils and Wings.

To proceed with the discussions, it is necessary to quote these without proof. Indeed, these formulae cannot be proven. There are complicated aerodynamic theories that derives these, however, while the success in doing so is remarkable, the theories themselves rely on rigorous assumptions and extensive modelling, so the derivations cannot really be called proofs. You are advised to understand the following as experimental correlations.

\[ C_L = C_{L0} + C_1 \alpha \] \[ C_D = C_{D0} + \frac{k}{\pi A} C_L^2 \]

It is, however, necessary to explain the physical rationale in detail.

The lift coefficient \( C_L \) can be decomposed as follows:

1. \( \alpha \) is the angle-of-attack.
2. \( C_{L0} \) is the lift coefficient at zero angle-of-attack. This term equals to zero if the aerofoil is symmetric, greater than zero if the aerofoil is cambered, and smaller than zero if the aerofoil is cambered the wrong way.
3. \( C_1 \) can be thought as an empirical factor. It is rather close to \( 2\pi \).
4. The lift coefficient increases proportionally with the angle-of-attack up to the point where the wing stalls.

The drag is more complex: the drag on an aeroplane has three components:

1. Friction drag, this is the drag caused by the air sticking onto the glider and trying to slow it down. Imagine flying a glider in honey which is rather sticky. The friction drag coefficient \( C_{DF} \) is approximately a constant for a given glider.
2. Pressure drag, this is the drag associated with the glider trailing a wake. This is also known as the form drag because it is related to the form of the glider being not fully aerodynamic. You would intuitively think that a Land Rover Discovery has more drag than a Jaguar fastback: because the Discovery is not streamlined while the fastback is, and this is what pressure drag is about. The pressure drag coefficient \( C_{DP} \) is approximately a constant for a given glider, because its form does not change in flight. Were this approximation not to be made, the following derivation can remain unaltered by pretending this variation is a part of the induced drag.
3. Induced drag, this is the drag caused by having lift. There is no free lunch in aerodynamics and wherever you have lift you must have drag, no matter how good your design is. The induced drag coefficient \( C_{DI} \) takes the following form:

\[ C_{DI} = \frac{k}{\pi A} C_L^2 \] 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 following relationship between \( C_D \) and \( C_L \) is fundamental to the discussions that follow:

\[ C_D = C_{D0} + \frac{k}{\pi A} C_L^2 \]

This is a parabolic function. It is this function that is referred to when talking about a 'parabolic polar': the actual (and more useful) polar curve that we shall derive is not a parabola.

A statement of the task that follows

From the relationship presented above, and making use of the following facts or assumptions:

1. Mass of the glider remains constant.
2. Energy is conserved.
3. The air is still.
4. The density of air is uniform and known, or rather and better, we work in the corrected (indicated) airspeed system.

We will be deriving a one-to-one relationship between indicated airspeed and sink rate.

A General Method of Solution

Before making further assumptions and simplifications, a general method of solution is worth presenting. The algebraic difficulties, as we shall see, is formidable, but it lends itself nicely to numerical methods.

Re-arranging the definitions of \( C_L \) and \( C_D \):

\[ W \cos(\theta) = C_L \times \frac{1}{2} \rho V^2 S \] \[ W \sin(\theta) = C_D \times \frac{1}{2} \rho V^2 S \]

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.

Substitute the parabolic relationship between \( C_D \) and \( C_L \) into the expression above, we have:

\[ f(C_L) = g(V) \]

Where \( f(x) \) and \( g(x) \) are functions that are too cumbersome to typeset. Keep in mind that \( C_{D0} \) is embedded in \( g(x) \).

Solving the above which the author does not believe is analytically possible:

\[ C_L = \frac{W \cos(\theta)}{\frac{1}{2} \rho V^2 S} = h(V) \]

In words, a relationship between the lift coefficient and the indicated airspeed can be arrived at.

The above can be further re-arranged, such that:

\[ \cos(\theta) = \frac{h(V) \rho V^2 S}{2W} \]

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:

1. 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.
2. 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):

1. \( W \), 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.
2. \( \rho \), 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.
3. \( S \), wing area of the glider, known and stays constant (we shall not consider the effects of deploying flaps, etc. on the performance).
4. \( C_{D0} \), this depends on the aerodynamic design of the glider.
5. \( k \), 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.
6. \( \pi \), 3.1415926...
7. \( A \), aspect ratio of the wing, depends on the glider design and a known constant.

Therefore, there are three changing quantities in this equation:

1. \( V_S \), this is the quantity we are interested in, the y.
2. \( V \), this is the quantity we can control, the x.
3. \( L \), 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

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

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 \( V_{MS} \) is proportional to the square root of wing loading. 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, the sink rate scales at the 3/2 power of the wing loading. 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.