The lift curve, and why it is a straight line

Plot the lift of a wing against the angle it is held at, and three things are immediately visible. The line is straight. It does not go through the origin. And its slope, in the right units, is about six and a quarter — which is to say, about $2\pi$.

None of the three is obvious, and all three follow from the circulation the trailing edge selects.

Why straight #

The circulation that satisfies the Kutta condition is

$$\Gamma = -4\pi a U \sin(\alpha + \beta)$$

and the lift is $\rho U \Gamma$. So lift goes as $\sin(\alpha + \beta)$, and for the angles an aircraft actually flies at — up to fifteen degrees or so — the sine of an angle is the angle to within a couple of percent.

That is the whole explanation. The line is straight because a sine is straight near zero, and it stops being straight only at angles where the wing has already stalled for unrelated reasons.

It is worth noticing how little went into that. No empirical fit, no aerofoil data, no wind tunnel. The straightness is a consequence of the geometry of a circle and the condition at a sharp edge.

Why it misses the origin #

The offset is the $\beta$, and $\beta$ is camber.

A symmetric section has no camber, $\beta = 0$, and produces exactly no lift when held exactly level. A cambered section has $\beta > 0$ and is already lifting at zero incidence — which is what camber is for.

To get zero lift out of a cambered section it has to be tilted nose-down through the angle $\beta$. That negative angle is the zero-lift angle, and it is a property of the shape alone: change the camber and it moves; change the thickness and it barely does.

So camber does not make a wing lift harder. It slides the entire line sideways, which is a much more specific claim, and it is why two wings can have the same slope and quite different lift at the same attitude.

Why the slope is near 2π #

This is the part that looks like a coincidence and is not.

In the limit of a thin section, the theory gives

$$\frac{dC_L}{d\alpha} = 2\pi \text{ per radian}$$

The $2\pi$ comes from the circumference of the circle that the Joukowski transform maps into the aerofoil. It is a geometric constant that has survived a conformal mapping into an aerodynamic one, and there is nothing empirical about it at all.

Real sections come in slightly above it, because they have thickness. The computed slope for the ten-percent-thick section drawn here is 6.84 per radian against $2\pi = 6.28$ — about nine percent high, which is the standard thickness correction and is itself predicted.

A number from the geometry of a circle turning up in the lift of a wing is the kind of thing this subject does regularly, and it is worth not becoming used to.

The coefficient, and why lift is quoted that way #

The vertical axis is not lift but lift coefficient, and the substitution is worth explaining because it is what makes the curve a property of the shape rather than of the day.

Lift depends on air density, on the square of the speed, and on the wing area, none of which say anything about the wing’s design. Divide them out —

$$C_L = \frac{L}{\tfrac12 \rho U^2 S}$$

— and what is left describes the section alone. The same curve then applies at sea level and at altitude, at ninety knots and at four hundred, to a model and to the full-size aircraft.

That is why aerodynamic data is almost always presented in coefficients, and why the curve has no units on its vertical axis. It is also the first appearance of the habit that dominates the whole subject: find the dimensionless group, and the answer stops depending on the particulars.

What the slope means in the cockpit #

The slope is not an abstraction. It is how hard the aircraft is to fly.

A high slope means a small change in attitude produces a large change in lift. That makes the aircraft responsive in pitch and also makes it sensitive to gusts, since a vertical gust is a change of incidence the pilot did not ask for. A glider in rough air is being shaken by exactly this.

A low slope — a highly swept or very low-aspect-ratio wing, say — gives a docile response and requires large attitude changes to manoeuvre, which is why a delta-winged aircraft approaches at such a conspicuous nose-up angle.

So a number that fell out of the circumference of a circle ends up setting how an aeroplane feels to its pilot, and that chain has no empirical step in it anywhere.

Where the straight line comes from, again #

It is worth restating the causal chain in one place, because the result is so clean that it can look like a fit to data.

The sharp trailing edge forbids all but one circulation. That circulation is $-4\pi a U \sin(\alpha

• \beta)$, in which $a$ and $\beta$ are pure geometry. Kutta–Joukowski converts circulation to lift in one multiplication. Dividing by $\tfrac12 \rho U^2 S$ removes the speed and the density.

What is left is $C_L \propto \sin(\alpha + \beta)$, and near zero a sine is its own argument.

At no point in that chain does an experiment appear. The straightness, the offset and the slope are predictions, and they can be checked against measurement afterwards — which is the right order, and the opposite of how the subject developed historically.

What thickness does #

The nine percent by which the computed slope exceeds $2\pi$ deserves an explanation, since it is the one departure from the thin-aerofoil result.

A thick section displaces the oncoming flow more than a thin one, so the flow that arrives at the surface has already been accelerated slightly. The effective incidence the section experiences is a little larger than the geometric one, and the lift responds accordingly.

The correction is roughly $2\pi(1 + 0.77\,t/c)$ for a section of thickness ratio $t/c$. For the ten percent section drawn here that gives 6.77 against a computed 6.84 — agreement to about one percent, from a formula the solver knows nothing about.

That is a useful kind of check. The solver and the analytical correction are independent, and their agreement is evidence that neither has a mistake in it of the kind that would show up in the slope.

What the solver computed #

Every point on the curve is a separate solve. lib/flow.js constructs the Joukowski section at the given incidence, applies the Kutta condition to get the circulation, and converts it to a lift coefficient against the mapped chord. Forty-one points, forty-one flows.

The slope quoted on the figure is measured from the curve rather than taken from the formula: two solutions a twentieth of a radian apart, differenced. It comes to 6.84.

The lift at each point is cross-checked against an entirely separate calculation — integrating the surface pressure and resolving perpendicular to the stream, which never mentions circulation. The two agree to within 0.8 percent at zero incidence and 3.6 percent at twelve degrees, the growing gap being the discretisation of the surface integral as the pressure peaks sharpen.

That cross-check exists because it caught the site’s worst bug: a circulation sign error that had the section generating lift downwards, with every other check passing. A lift curve with the wrong sign is still a straight line with a plausible slope.

What the curve does not have on it #

The most important feature of a real lift curve is missing here, and its absence is the honest limit of the model.

Real wings stop. Somewhere around fifteen degrees the curve bends over, peaks, and falls — sometimes gently, sometimes catastrophically. That is the stall, and there is no hint of it in anything above. This model will happily report the lift at forty degrees, and the number will be wrong.

The reason is that stall is a viscous phenomenon. The flow over the upper surface is being asked to climb an ever steeper pressure hill towards the trailing edge, and past a certain incidence the boundary layer cannot manage it and lets go of the surface. Once that happens the Kutta condition has nothing to enforce, the circulation collapses, and so does the lift.

The ideal theory has no boundary layer, so it cannot see any of that coming.

Reading the curve as a designer would #

Four numbers matter, and the curve gives three of them.

The slope decides how sharply lift responds to attitude — how twitchy the aircraft is in pitch, and how much a gust changes the load.

The zero-lift angle sets the attitude at which the wing does nothing, and therefore where the fuselage sits in cruise.

The lift at a given angle sets the wing area needed for a given weight.

The maximum, which this model does not have, sets the stall speed — and therefore the length of the runway, the approach speed, and a good deal of the certification. That the exact theory supplies three of the four and is silent on the fourth is a fair summary of its usefulness.

The curve is the same for everyone #

One more property of the straight line is worth drawing out, because it is the reason aerodynamic data is worth publishing at all.

The slope near $2\pi$ is not a property of any particular aerofoil. Every thin section has it — symmetric, cambered, thick, thin, ancient or modern. Sections differ in their zero-lift angle, in their maximum lift, in how gracefully they stall and in how much drag they make, but they do not differ much in slope.

That is a strong statement and it is easy to test: compute the slope for a symmetric section and a heavily cambered one of the same thickness and the two agree to about a percent, with the difference attributable to thickness rather than camber.

So a designer choosing between sections is not choosing a lift curve slope. That number is fixed by the physics. What is being chosen is where the line sits, where it stops, and what the drag is doing while all this happens.

Angle of attack is not pitch attitude #

A distinction that causes real accidents, and which the curve makes precise.

Angle of attack is the angle between the wing and the air arriving at it. Pitch attitude is the angle between the aircraft and the horizon. They are equal only in level flight at constant speed.

An aircraft descending steeply is meeting air that is coming from below, so its angle of attack is larger than its attitude suggests. One in a steep climb at low speed can have a smaller angle of attack than its nose-high attitude implies.

The lift curve is a function of angle of attack and of nothing else. It does not know about the horizon. That is why aircraft carry angle-of-attack indicators rather than relying on attitude, and why a wing can stall in any attitude at all, including nose-down.

The curve against measurement #

Since every point here was computed, the obvious question is how it compares with a wind tunnel.

Well, in the middle, and not at the ends. Between about minus four and plus ten degrees, a measured lift curve for a section of this kind lies within a few percent of the computed one — the slope is right, the zero-lift angle is right, and the straightness is real.

Below that range the measured curve begins to bend as the flow separates from the lower surface. Above it, the measured curve peaks and falls where the computed one goes on rising for ever, and the gap grows to everything.

So the honest summary is that this model predicts the part of the curve an aircraft cruises on and none of the part that kills people. Which is a fair description of ideal flow generally: excellent where nothing has gone wrong, silent about going wrong.

Where the model stops #

No stall, as above, and it is the number a designer most needs.

Two-dimensional. A real wing has ends; the trailing vortices there add induced drag and reduce the effective incidence, which lowers the slope. A wing of aspect ratio six has a slope perhaps three-quarters of the section’s.

Incompressible. Above about Mach 0.3 the slope starts to change, and near Mach 1 it changes a great deal.

Thickness handled, roughness not. The nine-percent slope increase from thickness is predicted; the effect of surface finish, which can be substantial near the stall, is entirely outside this model.

Who worked it out, and when #

The lift curve was measured before it was explained, as usual in this subject. Otto Lilienthal published tables of lift against angle in 1889 from measurements on a whirling arm, and the Wright brothers rebuilt their own after concluding — correctly — that Lilienthal’s figures were off.

The theory arrived afterwards: Kutta in 1902, Zhukovsky in 1906, and Max Munk’s thin-aerofoil theory around 1920, which is where the clean $2\pi$ statement and the treatment of general camber lines come from. Munk was working at the Langley laboratory as a German national shortly after the war, under restrictions that included being escorted between buildings.

Why the coefficient, once more #

There is a reason to labour the coefficient, and it is the same reason the Reynolds number matters: a quantity with units is a quantity that depends on circumstances, and a quantity without them is a property of the design.

Lift in newtons describes a particular aircraft on a particular day at a particular altitude and speed. Lift coefficient describes the wing.

That is why aerodynamic data has been exchanged in coefficients for a century, why a model in a tunnel can stand in for an aircraft, and why the curve in this essay can be published without saying what it is a wing of.

The habit is worth carrying: when a result seems to depend on too many things, look for the combination in which the units cancel, and expect the physics to live there. It works for lift, for drag, and for the boundary layer’s thickness alike.

The ladder from here #

Next rungs: thin-aerofoil theory in general, where the camber line is arbitrary rather than circular; the stall and what precedes it; finite wings and the slope reduction that comes with having ends; and the aerodynamic centre, which is the point the lift can be taken to act through at any incidence.

Then across to the separation that ends the straight line, and to the thin layer where it happens.