Aerodynamics and flight mechanics

Why a wing flies and where its profile comes from

A paraglider is an unpowered aircraft. It has no engine: it lives on the energy of the air and on the height of the launch. Everything that happens in flight is the work of a soft wing in the oncoming flow.

Cut the wing across the span, like a loaf of bread. The cut face is the profile — the shape of the wing in section. The line from the leading edge to the trailing edge is called the chord. On class A and B paragliders the profile is asymmetric biconvex: the upper surface is more curved than the lower.

Now put the profile into the flow. The air over the upper surface travels faster than the air under the lower one. Bernoulli’s law applies here: where the flow is faster, the static pressure is lower. Above the wing a suction appears, below it a positive pressure. It is exactly this pressure difference between the large (upper) and the small (lower) surface of the profile that causes the lift. The main work is done by the suction on top. The wing does not so much rest on the air below as get sucked upwards — like a lid pulled outwards by a vacuum.

The link between speed and pressure shows up in the pipe experiment. Send gas through a cylindrical pipe at a constant flow rate and widen its diameter. The flow speed falls and the pressure rises. Narrow the pipe and the opposite happens. The upper surface of a wing works like such a narrowing: the flow accelerates and the pressure falls.

Where does a soft wing get its profile from? There are no rigid parts inside: no spars, no hard skin. The wing is sewn from airtight fabric, the upper and lower surfaces are joined by ribs (internal walls), and along the leading edge run the cell openings. In flight the oncoming air enters through them and inflates the sections from inside. It is that internal overpressure that holds the shape: the wing works like an air mattress, except that the headwind is the pump. The practical conclusion: as long as the wing is moving at sufficient speed, it is “rigid”. Let the angle of attack drop and the pressure no longer holds the shape, and the wing collapses.

Now let us put lift into numbers. The formula is short: Y = Cy · (ρ · V² ÷ 2) · S. Let us take it letter by letter.

• S — the wing area in square metres.
• ρ (the Greek letter “rho”) — air density. At the ground in standard conditions it is 1.225 kilograms per cubic metre.
• V — the speed of the wing relative to the air.
• Cy — the lift coefficient. Everything is gathered in it: the shape of the profile and the angle of attack. It is measured in a wind tunnel and in flight.

The part ρ · V² ÷ 2 is called the dynamic pressure. It is the “force of the wind in your face”. Note the square: double the speed and the dynamic pressure grows fourfold. Two practical conclusions follow. First: at low speed the wing holds only weakly, and any mistake costs more. Second: the brakes, big ears and every other control are several times harsher at high speed than at low.

A worked example, step by step. All-up weight 100 kilograms, wing area 25 m², speed 36 km/h (that is 10 metres per second). Step one: the weight in newtons — 100 × 9.81 ≈ 981 N. Step two: the dynamic pressure — 1.225 × 10² ÷ 2 = 61 Pa. Step three: the coefficient needed — 981 ÷ (61 × 25) ≈ 0.64. That is an ordinary working value for a paraglider profile. Work the same wing out at 48 km/h and you need half as much coefficient — about 0.36. The wing gets that by dropping its nose: the angle of attack decreases.

Angle of attack — the main lever of control

The angle of attack is the angle between the chord of the profile and the oncoming flow. It is not the angle to the horizon and not the angle to the slope. It is measured from the flow, and in a glide the flow comes from ahead and below.

The angle of attack is the only quantity the pilot controls directly. Speed, sink and glide ratio follow from it as consequences. There are three tools:

• The brakes — they pull the trailing edge down. The angle of attack grows, the speed falls, the drag rises.
• The trimmers — they let out or take in the rear rows and set the basic angle of attack.
• The speed system — it pulls the front rows in and the leading edge drops. The angle of attack decreases, the speed rises.

Learn this by heart: as the angle of attack increases, the aircraft’s horizontal speed falls. The wing’s aerodynamic drag rises, the wing seems to rear up and moves forward more slowly. The angle of attack cannot be increased indefinitely. Beyond a critical value the flow separates from the upper surface and the lift falls away sharply — that is a stall.

How wide is the working range? It is narrow. Take full speed system as a notional zero. Above it comes the trim setting, then best glide, then minimum sink, and somewhere past fifteen degrees the flow separates. The exact critical angle is a property of the particular profile, and it is not written in the wing’s data sheet. What matters is something else: your whole working margin fits into about fifteen degrees. Instruments do not show this angle. The pilot judges it indirectly.

• By the pressure in the brakes. Rising pressure means a rising angle of attack.
• By the noise. The lines whistling more loudly means the angle of attack has dropped. Gone quiet means the speed has gone.
• By your body. Pressed into the harness means the wing is going back. Feeling weightless means the wing is running away forwards.
• By where the canopy is. Wing behind your back — a large angle of attack; wing ahead — a small one.

The main thing a beginner has to understand: the angle of attack changes without your hands moving at all. The air changes it. The wing flies into rising air — the flow comes from below and the angle of attack instantly grows. The wing comes out into sink — the angle of attack drops.

A worked example, step by step. You are flying at 10 metres per second. The wing enters rising air of 2 metres per second. Step one: take the ratio of the vertical component to the speed — 2 ÷ 10 = 0.2. Step two: take the arctangent — that is roughly 11 degrees. That is how much the angle of attack gained in a fraction of a second while you did nothing. With a working margin of some fifteen degrees, that is an enormous amount. That is why thermic air loads the wing so much, and why wind shear at the edge of a thermal can stall it with no help from your hands at all.

Forces in flight, balance and the pilot as a pendulum

Only two forces act on a flying paraglider. The first is weight: the pilot, the harness and the wing. The second is the total aerodynamic force: everything the air does to the aircraft. The total aerodynamic force is resolved into two components relative to the oncoming flow:

• lift — perpendicular to the flow;
• aerodynamic drag — directed exactly against the motion.

So the total aerodynamic force is the vector sum of lift and drag. These are not three different forces but one force and two of its projections.

The aircraft flies steadily and in a straight line, without accelerations, when the total aerodynamic force is equal in magnitude to the weight and opposite to it. On an unpowered aircraft that balance is only achieved while descending. Part of the weight goes into overcoming drag: you pay for moving forwards with height.

Can lift be directed towards the ground? In normal flight, no. A paraglider does not fly canopy-down, and the lift always points away from the ground. In a turn it tilts with the wing, but it does not turn downwards. Sink and descending air change nothing here: you lose height, and the lift still points upwards.

The pilot as a pendulum. You hang on 6–8 metres of line below the wing. This is a classic pendulum: the centre of gravity (you) swings below the point where the aerodynamic force is applied (the wing). The pendulum gives stability — the system returns you under the wing by itself. It also produces pitch oscillation: the wing goes forward, the pilot lags, then the wing slows and goes back. Damping that oscillation is the pilot’s direct job.

Resolve the balance along the axes and you get numbers. Call the glide angle θ — the angle between the flight path and the horizon. Then lift equals weight × cos θ, and drag equals weight × sin θ. From which comes a neat consequence: tan θ = 1 ÷ glide ratio.

A worked example, step by step. Glide ratio 9. Step one: tan θ = 1 ÷ 9 = 0.111. Step two: θ ≈ 6.3 degrees — that is how flat your path is. Step three: cos 6.3° = 0.994, so lift is 99.4 % of the weight. Step four: sin 6.3° = 0.110, so drag is about 11 % of the weight. At an all-up weight of 100 kilograms that is roughly 108 newtons, that is, “eleven kilograms” of constant braking. That is what your height pays for.

Pendular stability and damping

A paraglider is stable not because it is clever but because of the way its geometry works. Let us take it in layers.

Layer one: the pendulum. The aerodynamic force acts at the canopy. The weight acts at the pilot, six to eight metres lower. Displace the pilot sideways and gravity immediately pulls him back under the wing. That is a restoring moment, and it is stronger the longer the lines. That is exactly why a paraglider forgives rough mistakes: the system itself tends to return to “pilot directly under the wing”.

Layer two: two pendular effects. Specialists distinguish them like this. The upper pendular effect — the wing swings relative to the pilot: a gust hits the canopy and the canopy goes forward or sideways while the pilot stays put through inertia. The lower pendular effect — the pilot returns beneath the wing under his own weight. Because of the large distance between canopy and pilot, the lower effect is the stronger one. That is why, after a shaking, the system eventually settles by itself.

Layer three: roll stability. In flight the canopy is bent into an arc — that is its arc shape. Tilt such a wing sideways and its two halves end up at different angles to the vertical. The lower side begins to lift itself back with its own lift. The arc works instantly, before the pendulum even comes into play.

Layer four: pitch stability. The wing runs ahead — the angle of attack drops, the lift drops, and the wing falls back by itself. The wing goes back — the angle of attack grows, the drag grows, the wing slows and returns. The system catches itself. The price is oscillation: every return overshoots the balance point.

Damping is what kills the oscillation. It is killed by air resistance and by the change of angle of attack during the rotation. In roll the damping is good: the arc and the large area work immediately. In pitch it is weak, the oscillation dies away slowly, and all that time the wing wanders in angle of attack from almost zero to close to critical. Both edges are dangerous: a collapse at the bottom, a stall at the top.

The surge: its phases and how to damp them

A surge is the wing moving forward relative to the pilot. It is the commonest and the most dangerous phase of an oscillation: it is on the forward run that the angle of attack falls to zero and below, and the wing collapses.

Let us take a full cycle phase by phase — as it is described in the controlled-oscillation exercise.

1. Pulling the brakes. The angle of attack and the drag grow, the wing slows and goes back. The pilot swings forward through inertia.
2. Keeping them pulled. The speed and the lift fall almost to zero and the wing hangs behind your back. In the exercise this phase is held for two or three seconds.
3. A sharp release. The wing accelerates by itself — like a dog let off the leash.
4. The forward throw. The wing goes far ahead. Here the angle of attack is at its minimum, and here too is the risk of a frontal collapse.
5. Damping. At the furthest forward point the pilot pulls the brakes briefly, and the movement turns back.

Now the same thing through the pilot’s eyes in real flight. There are three phases: recognise — stop — release.

• Recognise. The brakes have gone empty, your body has gone weightless, the wind noise is rising, the canopy is moving forward out of your field of view.
• Stop. A quick, measured movement down on both brakes. The dose depends on the strength of the surge: the sharper the wing goes forward, the deeper the movement.
• Release. Hands up. Not as fast as you pulled, but definitely up. Let the wing fly.

The key thing in a surge is not force but timing. The very same movement of the hand, made in the right second, damps the oscillation; half a second later it amplifies it. Hold the brakes too long and you get a secondary stall. Do not pull enough and you get a collapse. Both outcomes come from the same movement made at the wrong time.