Physics · Mechanics · Grade 10

Aeroplane wing

Where does lift come from, and why does an aeroplane need a longer run-up at high altitude?

The situation

An aeroplane wing is not a plain board: its upper side bulges. Air flows faster over the wing and slower beneath it, and the pressure difference holds a hundreds-of-tonnes airliner in the air. Will the same aeroplane take off at the same speed from an airport by the sea and from El Alto airport (4 061 m) in the high Andes? Where does lift come from, and what does it depend on?

Topic

Fluid flow, Bernoulli's equation. Aeroplane wing: where does lift come from

Key ideas

Bernoulli's equation
Lifting force
Air density

Formulas

S_{1} v_{1} = S_{2} v_{2}, p + \frac{ρ v^{2}}{2} = const

Continuity and Bernoulli equations: when a stream narrows, its speed increases; if the speed along a streamline increases, the pressure falls.

C_{p} = \frac{p - p_{\infty}}{ρ V^{2} / 2} = 1 - {(\frac{v}{V})}^{2}

Pressure coefficient: blue — low pressure, red — high pressure.

L^{'} = ρ V Γ, Γ = 4 π a V \sin (α + β)

Kutta–Joukowski theorem: lift per 1 m of wing; $Γ$ is the circulation (Kutta condition).

L = C_{L} \frac{ρ V^{2}}{2} S, C_{L} \approx 2 π \sin (α + β)

Lift is proportional to the square of the speed and to the wing area ( $α$ is the angle of attack).

v_{u z} = \sqrt{\frac{2 m g}{ρ S C_{L}}}

Lift-off speed: the speed at which $L = m g$ . It is greater if the air is thin.

Theory

Bernoulli: fast flow — low pressure

In an ideal fluid $p + ρ v^{2} / 2$ stays constant along a streamline. Above the wing the streamlines crowd together — the air speeds up and the pressure falls (blue). Below, the air slows down and the pressure rises (red). The difference between the upper and lower pressures multiplied by the wing area is the lift. On the $C_{p} (x)$ graph, the higher the upper line is above the lower one, the greater the force.

Continuity and the Euler equation

If the flow is steady (the speed at each point does not change in time), the streamlines show the paths of the particles. No fluid passes through the wall of a “tube” between two neighbouring streamlines; in an incompressible fluid the same volume passes through every cross-section each second: $S_{1} v_{1} = S_{2} v_{2}$ — the continuity equation. That is why the speed is high where the streamlines crowd together. The tunnel itself works this way: the wide contraction cross-section is ≈ 3.5 times larger than the test section — the air there is 3.5 times slower. The Bernoulli equation follows from Newton's law for an ideal fluid — the Euler equation: $ρ d \vec{v} / d t = - \nabla p + ρ \vec{g}$ ; integrating it along a streamline gives $p + ρ v^{2} / 2 + ρ g h = const$ , and for $v = 0$ it gives hydrostatics, $p = p_{0} + ρ g h$ , where $h$ is the depth (“Pascal's law and hydraulics”). Air is compressible, but if $M = v / c < 0.3$ ( $c \approx 340$ m/s is the speed of sound) the density changes by less than 5 % — in the tunnel $M \leq 0.18$ . In fast flow the gas flows one-dimensionally along the duct: $ρ v S = const$ and $d S / S = (M^{2} - 1) d v / v$ — a subsonic flow speeds up when the duct narrows, a supersonic flow when it widens; a rocket nozzle (de Laval nozzle) therefore first narrows and then widens. Sound waves are covered in the “Sound: speed, pitch, ultrasound” experiment.

Circulation, potential flow and the Kutta condition

The flow in the tunnel has been calculated exactly: the Joukowski transformation $z = ζ + 1 / ζ$ turns a circle into a wing profile, and the flow around a circle is known. This flow is potential — it has no vortices ( $rot \vec{v} = 0$ ), so $\vec{v} = \nabla φ$ , where $φ$ is the velocity potential, and in an incompressible fluid $Δ φ = 0$ (Laplace's equation). The circulation of velocity $Γ = \oint \vec{v} \cdot d \vec{l}$ is non-zero only on a contour enclosing the wing. In an ideal flow there are many such solutions; real air flows smoothly off the sharp trailing edge (the Kutta condition) — this fixes $Γ$ and gives $L^{'} = ρ V Γ$ . A spinning ball or cylinder also creates circulation and receives the same sideways force — the Magnus effect (the water tunnel in the “Viscosity and flow” experiment). In the simulation $C_{L}$ has also been checked by integrating the pressure over the surface — the result is the same.

The Pitot tube and Torricelli's formula

Bernoulli's equation is widely used in engineering. The air speed in a tunnel is measured by a Pitot tube: at the hole facing the flow the air stops and the pressure rises by $ρ V^{2} / 2$ , while the side holes sense the static pressure. The difference is shown by a water U-tube (manometer): $ρ V^{2} / 2 = ρ_{w} g Δ h$ , that is $V = \sqrt{2 g Δ h ρ_{w} / ρ}$ — look at the manometer in front of the tunnel on the stage. An aeroplane's airspeed indicator works in the same way. Torricelli's formula has the same form: water spurts from a hole at depth $h$ in a vessel with the speed $v = \sqrt{2 g h}$ (Bernoulli: $p_{0} + ρ g h = p_{0} + ρ v^{2} / 2$ ) — the same as the speed of a drop falling freely from height $h$ . The Venturi tube (flow meter), the atomiser, the carburettor and the injector of a gas stove all use the low pressure in a fast flow.

Stall — flow separation

In ideal theory $C_{L}$ grows without limit with the angle of attack. Real air is viscous: beyond about 15° the boundary layer above the wing separates from the surface, eddies appear and the lift falls sharply; when the flow separates, the drag also rises sharply. This process is absent from the potential-flow model — in the simulation the drop beyond 15° is an empirical model, and the smoke moves chaotically. Viscosity, laminar flow and Poiseuille's formula — the “Viscosity and flow” experiment; surface tension and capillarity — the “Surface tension and capillarity” experiment.

Take-off and air density

An aeroplane leaves the ground when $L = m g$ : $v \propto 1 / \sqrt{ρ}$ . At an altitude of 4 km the air density is ≈ 0.81 kg/m³ — the lift-off speed is ≈ 23 % greater; the engine thrust also falls, so the take-off run becomes much longer. Simplifications: 2D ideal flow in the tunnel, no wall effects, the smoke slowed 60 times. The aeroplane is a point mass; $C_{L, 3 D} = C_{L} λ / (λ + 2)$ ( $λ = 7.5$ is the aspect ratio of the wing), the thrust falls with speed and is proportional to $ρ$ , and the atmosphere is the ISA standard atmosphere.

Did you know?

The Kutta–Joukowski theorem is named after the German mathematician Martin Kutta (1902) and the Russian scientist Nikolay Joukowski (1906) — they were the first to explain lift through circulation.
El Alto airport in Bolivia lies at an altitude of about 4 060 m above sea level. The air density there is about a third lower than at sea level, so aeroplanes need a long runway.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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