Difference between revisions of "Polar, Performance, and Water Ballast"

Revision as of 13:14, 13 December 2019

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.

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.