Gliding flight is pretty fascinating. Gliders are aircraft that fly without the use of thrust. This article is mostly relevant to aircraft designed to glide - like paragliders, hang gliders and sailplanes. However, any powered aircraft turns into a glider when you turn its engine off. Gliding is a magical experience that lets aircraft of all kinds soar the same way birds do.

One thing that’s important to most gliders is how far they can fly (without the aid of rising air from thermals, etc). Gliders (and most aircraft) when flown without power at a constant speed in still air will lose a certain amount of altitude given some amount of forward movement. The ratio of this forward distance covered to altitude lost is referred to as the glide ratio (\(E\)).

$$\begin{equation} E = \frac{d_{forward}}{d_{down}} \equiv \frac{d_{x}}{d_{y}} \label{eq:1} \tag{1} \end{equation}$$

Apologies in advance for the hand-wavey math. The aerodynamics of gliding can’t really be distilled into simple equations in two-dimensional space. However, I’m hoping that it’ll be enough to gain a layperson’s intuition about the factors that afffect glide ratio during flight.

Straight and level flight

Modern recreational paragliders have glide ratios of about 8. This means that in still air (flying at trim speed), a glider will cover 8 feet for every foot of altitude lost. This is illustrated in the diagram below. Try adjusting the glide ratio to see how it affects the glider’s flight.

For reference:

• A typical skydiving canopy will achieve a 3:1 glide ratio.
• An idling Cessna 172 will have a 9:1 glide ratio.
• Hang gliders have glide ratios of up to 16:1.
• A training sailplane (ASK 21) has a best glide of 34:1

The illustration above should be intuitive to most people, but it forms a useful foundation for some of the things we’ll discuss later in this article. Particularly, take note of \(t\) (how long it takes for the glider to reach the red airmass), and \(d_x\) (the distance travelled over the ground before the glider reaches the red airmass). These should remain fairly similar (ignoring slight numerical errors) even as glide ratio is adjusted (since the glider’s airspeed doesn’t change). However, \(d_y\) (the vertical distance travelled before the glider reaches the red airmass) should vary with glide ratio.

Into the wind

While it’s nice to only consider gliders flying through still air, the reality is that the air that we end up flying through is often in motion relative to the earth. Even standing on the ground we are very rarely in completely still air. We can almost always feel the air moving around us. This movement (that we call wind) doesn’t just move horizontally across the earth, but also has a vertical component that has a big influence on how aircraft move through the air.

Let’s first limit ourselves to the horizontal component of moving air. We will oftentimes find ourselves either flying into a headwind or flying downwind with the wind at our backs. Let’s consider how that affects our glide ratio over the ground.

In these cases, the airmass is moving as we move through it. As a result, the amount of distance we cover over the ground may decrease or increease:

It follows from Equation \(\eqref{eq:1}\):

$$\begin{equation} E = \frac{d_x}{d_x} = \frac{v_x \times t}{v_y \times t} = \frac{v_x}{v_y} \end{equation}$$

The vertical descent rate can therefore be expressed as follows: $$\begin{equation} v_y = \frac{v_x}{E} \tag{2} \label{eq:2} \end{equation}$$

We’re only considering a headwind or tailwind for now, so it’s safe to assume that vertical descent rate will remain constant as the windspeed changes. Let’s express this mathematically, equating the vertical descent/ascent rate in calm and windy air. $$\begin{aligned} {v_{y (calm {\text -} air)}} &= {v_{y (windy {\text -} air)}}\\ \implies \frac{v_{x (calm {\text -} air})}{E_{calm {\text -} air}} &= \frac{v_{x (windy {\text -} air)}}{E_{windy {\text -} air}} \end{aligned}$$

This allows us to derive the glide ratio in windy air, and express that in terms of wind speed (\(v_{wind}\)), airspeed (\(v_{x (calm {\text -} air)}\)), and the glider’s glide ratio in still air (\(E_{calm {\text -} air}\)): $$\begin{aligned} E_{windy {\text -} air} &= {E_{calm {\text -} air}} \times \frac{v_{x (windy {\text -} air)}}{v_{x (calm {\text -} air)}}\\ &= (v_{x (calm {\text -} air)} + v_{wind}) \times \frac{E_{calm {\text -} air}}{v_{x (calm {\text -} air)}} \end{aligned}$$

This is what it looks like in practice:

If you play around with the wind speed you’ll see how it affects the simulated glider’s glide ratio. In the simulation the glider has a trim airspeed of 10. You’ll note that increasing the windspeed to 10 reduces the glider’s glide ratio to 0 and anything beyond that causes the glider to fly backwards over the ground. Giving the glider a tailwind effectively boosts the glider’s glide ratio - allowing it to fly further than it otherwise would have in still air. This is why it’s much easier to make progress flying downwind than it is penetrating into a headwind.

Ups and Downs

How about flying through lift and sink? How does that affect our glide?

Let’s go back to Equation \(\eqref{eq:1}\) and use it figure out what the glide ratio of the glider will be if it’s flying through air that has both vertical and horizontal components to it: $$\begin{equation} E = \frac{v_x}{v_y} \end{equation}$$

Because the airmass that the glider moves through is moving upwards/downwards we need to add its speed (\(v_{sink}\)) to the speed with which the glider naturally falls through the air (\(v_x\)). Similarly, we add the speed of the wind (\(v_{wind}\)) to the glider’s $$\begin{align} E_{sinky {\text -} air} &= \frac{v_{x (calm {\text -} air)} + v_{wind}}{v_{y (calm {\text -} air)} + v_{sink}} \\ &= \frac{v_{x (calm {\text -} air)} + v_{wind}}{\frac{v_{x (calm {\text -} air)}}{E_{calm {\text -} air}} + v_{sink}} \tag{3} \label{eq:3} \end{align}$$

This can be visualized below:

Note how just a little sink (-1) nearly halves the glide ratio. Flying in even a miniscule amount of sink can be just as detrimental as flying into a headwind that’s half as fast as your trim speed.

Speed to fly

So far all of this discussion has assumed that the glider always flies at trim speed. Now let’s consider what happens if the glider speeds up or slows down.

Looking at Equation \(\eqref{eq:3}\) we can see that the glider’s airspeed (or groundspeed in calm air) is a component in both the numerator and denominator. We can see that large values of \(v_x\) will cause \(v_{wind}\) and \(v_{sink}\) to have less of an impact on overall glide ratio. This might imply that we’d want to fly faster through sink and into headwinds; but slower in lift and downwind. (The last part wasn’t always intuitive to me)

Sadly glide ratios do not remain constant as the glider’s airspeed changes. Rather, it tends to decrease as the velocity approaches the stall point - and likewise decreases once velocity surpasses the glider’s speed of “best glide”. The glide ratio at each speed is unique to an individual aircraft and is typically obtained through experimentation. If we plot the glide ratio against airspeed for a given aircraft - we obtain what’s known as its “polar curve”. Below is a polar curve for an intermediate paraglider (EN/LTF-B) - copied over from my friend Rob Barlow.

(The polar’s curved shape is due to the effect airspeed has on the glider’s lift to drag ratio (L/D) - which is numerically equivalent to glide ratio.)

Note how adjusting the glider’s speed affects its glide ratio in moving air. You’ll also note that flying the glider faster through sink or into a headwind can still help improve glide ratio despite the reduced \(E_{calm{\text -}air}\). This is something that gliders exploit to cover greater distances over the ground. A glider that is capable of flying faster (without big compromises to glide ratio at higher speeds) will typically be capable of covering more distance over the ground than one that either lacks the ability to fly faster or is unable to fly faster without big penalties to glide ratio. Skilled glider pilots are able to advantage take of the wider airspeed envelope to make sure that they are always flying at the correct speed-to-fly.

Hope this is helpful to someone else!

If you’re in the SF Bay Area and are interested in learning to fly gliders, here’re some schools that I recommend for:

• Paragliders: Lift Paragliding
• Sailplanes: Williams Soaring Center
• Hang Gliders: Norcal Hang Gliding, Berkeley Hang Gliding Club

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