The two drags a wing pays

Drag is not one thing. A wing moving through air is charged twice, by two mechanisms that have nothing in common, and the two bills behave in opposite ways as the speed changes.

That opposition is the whole of subsonic aircraft performance. It is why there is a speed that gives the best range, a different one that gives the longest endurance, and a third that gives the flattest glide; and it is why all three exist at all rather than the answer simply being as fast as possible.

The first bill: having a surface #

A wing has area, the air sticks to it, and dragging a surface through a fluid costs something. That is skin friction, and it is the drag the boundary layer accounts for.

Its defining feature is that it does not care about lift. A wing gliding at its best angle and a wing diving vertically have the same wetted area and very nearly the same friction. So on a plot against lift coefficient it is a horizontal line.

What it does care about is the Reynolds number, and it falls as the inverse square root of it.

For the wing this essay costs out — chord Reynolds number of a million, laminar — the friction drag coefficient is 0.002 66. That number comes from the Blasius wall slope, doubled to account for both surfaces, and it is a lower bound rather than an estimate: real layers go turbulent, and a turbulent layer has several times the friction of a laminar one at the same Reynolds number.

The second bill: having ends #

The other charge is induced drag, and it exists in a fluid with no viscosity at all.

A wing of finite span sheds a sheet of vorticity, the sheet induces a downwash at the wing, the downwash tilts the local lift vector backwards, and the backward component is drag. It has nothing to do with friction and would be there if the air were perfectly slippery.

Its defining feature is that it is all about lift. It goes as the square of the lift coefficient:

$$C_{D_i} = \frac{C_L^2}{\pi A\!R\, e}$$

so a wing making no lift has none of it, and a wing near its stalling angle has a great deal.

There is one more thing worth reading off the exponent. Because induced drag goes as CL², and CL goes as the inverse square of speed at fixed weight, induced drag goes as the fourth power of one over speed. Halving the speed multiplies it by sixteen. Nothing else in ordinary aerodynamics has that sensitivity, and it is why an aircraft flown slowly enough eventually cannot maintain height whatever the engine is doing.

Why the two go opposite ways with speed #

The two bills are drawn against lift coefficient above, and lift coefficient is a proxy for speed in the only way that matters. An aircraft in steady flight must produce lift equal to its weight, so as it flies more slowly the lift coefficient must rise to compensate — as the inverse square of speed.

So reading the figure from right to left is reading the aircraft from slow to fast. At the left-hand edge it is going quickly, making little lift coefficient, and paying almost entirely friction. At the right-hand edge it is going slowly, close to the stall, and paying almost entirely induced drag.

That reversal is the reason drag against airspeed is a U-shaped curve rather than a rising one, and the reason every aircraft has a speed below which going slower requires more power. Pilots call the region on the wrong side of the minimum the back of the drag curve, and it is where a great many accidents happen, because the control that normally slows the aircraft down now speeds it up.

What the solver computed, and how it was checked #

Both halves of the budget are computed here, from two solvers that share no code.

The friction comes from the Blasius profile, shot by fourth-order Runge–Kutta to the free-stream condition. Its momentum thickness, integrated over the whole profile, is 0.664 114 5, and its wall slope, which is where the march started, is 0.332 057. The two must stand in a ratio of exactly two by von Kármán’s momentum integral, and they do to seven figures. The friction coefficient in this essay is that momentum thickness, doubled for two surfaces and divided by the square root of the Reynolds number.

The induced drag comes from the lifting line, solved by collocating the monoplane equation at eight stations. Its span efficiency is checked against the theorem that nothing beats an elliptic loading, its induced drag is computed twice by unrelated routes that agree to 10⁻¹⁵, and the residual of its own equation, tested deliberately between the collocation points where nothing forces it, is 1.4 × 10⁻¹⁷.

Neither computation knows the other exists. Adding them is the only step in this essay that is not checked, and it is addition.

Where the two are equal #

Set the two expressions equal and solve for the lift coefficient:

$$C_L^{\text{opt}} = \sqrt{C_{D_0}\,\pi A\!R\, e}$$

For an aspect ratio of eight with elliptic loading and the friction figure above, that is CL = 0.258, and the figure marks it.

At that lift coefficient the total drag coefficient is exactly twice the friction coefficient, because the two halves are equal by construction. And the ratio of lift to drag — which is the glide ratio, and the single most useful number about an aircraft — is at its maximum there.

The value that comes out is 48.6 to one. That is a very good glider, and it is too good, for a reason stated below.

Why the same point is the best glide #

It is not obvious that the minimum-drag point should also be the best-glide point, and the reason is worth a line because the two questions sound different.

An aircraft gliding without power descends along the direction of its total aerodynamic force, and the angle it makes with the horizontal is set by the ratio of drag to lift. Minimising the glide angle therefore means maximising L/D, not minimising D.

But lift is fixed at the weight. With lift held constant, minimising D and maximising L/D are the same problem, and they land on the same point. That is why one number does both jobs, and it is why gliders have a single “best glide speed” on the airspeed indicator rather than two.

The same reasoning with a different quantity held constant gives the other speeds. Maximum endurance minimises power rather than force, which puts it slower; maximum range for a jet minimises drag per unit distance, which puts it faster. All three come off the same two curves.

The ratios between them are fixed, which is the neat part. Minimum-power speed is 1/⁴√3 times minimum-drag speed — about 76% of it — and the best-range speed for a jet is ⁴√3 times it, about 132%. Those two factors are the same number inverted, they come from setting a derivative of a two-term expression to zero, and they are the same for every aircraft with a parabolic polar regardless of size, weight or wing shape. A number that survives that much variation is usually a sign that the model underneath it is the right one.

Where the aspect ratio goes #

The optimum lift coefficient contains the aspect ratio under a square root, and the best glide ratio contains it the same way. That means the return on span is real but diminishing.

Doubling the aspect ratio from eight to sixteen improves the best glide ratio by a factor of √2, or about 41%. Doubling it again buys another 41%, by which point the wing is sixty metres across and the structure to hold it is the design problem rather than the aerodynamics.

That trade is visible in what actually gets built. Competition gliders sit at aspect ratios near thirty because they are permitted to be structurally marginal and are not required to carry anything. Airliners sit near nine or ten, because a wing must also hold fuel, carry engines, survive gusts and fit an airport gate. Fighters sit near three, because manoeuvre matters more than cruise efficiency and a short wing is stiffer.

The drag polar #

Plotting lift against total drag rather than each against lift gives the shape every aircraft is characterised by.

The polar’s two features are both readable. The left-hand edge is where all the curves meet, and its position is the friction drag alone. The rate at which each bends away from that edge is the induced term, and a higher aspect ratio bends later.

The best glide ratio is found on this plot by drawing a line from the origin tangent to the curve: the tangent point is the optimum, and the slope of the line is the glide ratio. That construction is why the polar is drawn in this orientation at all.

Why 48.6 is too good #

The number is honest arithmetic on a dishonest input, and saying so is the point of this section.

Three things have been left out. The friction figure is laminar, and no wing of any size holds a laminar layer over its whole chord; a realistic turbulent figure is three or four times larger.

There is no form drag — the pressure drag of a body that is not infinitely thin. Even an attached flow over a section of finite thickness leaves a pressure imbalance front to back, and this site’s solver cannot compute it, for reasons set out below.

And there is no fuselage, tail, undercarriage or interference, which on any real aircraft add up to as much again as the wing.

A real high-performance glider achieves about 60 to 1, which is better than the number above; a real light aircraft manages about 10 to 1, which is much worse. A modern airliner in cruise is close to 18 to 1, and a bird of prey is around 10. The first beats it because sixty metres of span puts the induced term far lower than an aspect ratio of eight can; the second falls short because everything omitted above is present.

The area that should have been in the budget #

The section above blames the optimistic glide ratio on a laminar friction figure and on everything that is not a wing. Both are true, and there is a tidier way to say them together which turns the budget into a rule that works across every flying thing there is.

Friction is charged on wetted area — every surface the air touches — while lift is made by the wing area. The budget above used the wing’s area for both, which is why it came out flattering. Put the right area in each place and the best glide ratio rearranges into

$$\left(\frac{L}{D}\right)_{\max} \;\propto\; \sqrt{\frac{b^2}{S_{\text{wet}}}}$$

where the group under the root is the wetted aspect ratio: the square of the span over the total wetted area, rather than over the wing area. Everything else has collapsed into a constant, which is the equivalent friction coefficient of a whole aircraft — around three parts in a thousand for anything clean, and remarkably similar across types because a well-made surface is a well-made surface.

The rule that comes out is roughly fifteen times the square root of the wetted aspect ratio, and it works everywhere. An airliner sits near 1.3 in that group and achieves about seventeen. A fighter sits near 0.4 and achieves about nine. A high-performance sailplane, which is nearly all wing and carries a fuselage barely wider than a person, reaches ten or more and achieves fifty-odd. One line, three orders of magnitude of aircraft.

And it says where the design effort goes. Span helps as its square and is charged for structurally; wetted area hurts directly and is where a fuselage, a tail and a nacelle spend their entire aerodynamic budget. The wing’s own aspect ratio, which the earlier section made so much of, is only the version of this ratio that an aircraft made of nothing but wing would have.

What the picture cannot show #

The figures plot coefficients rather than forces, and coefficients hide the speed. Every point on the first figure is a different flight speed, and the axis does not say so.

More seriously, they hold the friction coefficient constant while lift changes, which is a small lie. The friction depends on the Reynolds number, and the Reynolds number changes with speed, so the horizontal line ought to be very slightly sloped. Over the speed range of interest the effect is a few per cent and drawing it would suggest a precision the rest of the calculation does not have.

The polar is drawn as a smooth parabola all the way up. Real polars depart from the parabola near the stall, where the flow starts letting go and drag rises much faster than the square law, and they depart at the bottom too. The parabola is the middle of the range and the figure does not mark where it stops applying.

Where the model stops #

The largest omission is deliberate, and it is worth naming rather than hiding: there is no pressure drag in this budget.

Pressure drag is the third kind, it is the dominant kind for anything that is not streamlined, and computing it needs a resolved wake. The route a wind tunnel uses — surveying the momentum deficit downstream — was tried on this site’s grid solver and does not work. Close behind a cylinder at Re = 100 the deficit comes out negative, because the formula assumes a pressure that has recovered and inside the recirculation it has not; four diameters back the coarse grid has diffused the wake away and the deficit is nearly zero. Between stations the answer varies by more than its own size.

The site’s own assertion refuses that wake, and the gate asserts that it refuses it, so no essay here can quietly quote a drag coefficient the machinery rejects. What is drawn in this essay is the part that can be computed exactly, and the part that cannot is absent by decision rather than by oversight.

Who found it, and when #

The two-part decomposition is Prandtl’s, and it arrives with the lifting line between 1918 and 1921. It is the moment aerodynamics acquired an accounting system: before it, drag was one measured quantity; after it, drag was a sum of terms each of which could be attacked separately.

The parabolic polar as a design tool is Oswald’s, in a 1932 NACA report that introduced the efficiency factor still written e and still called the Oswald factor. It is a fudge in the honest sense — a single number absorbing everything that makes a real aircraft’s induced drag worse than the elliptic ideal — and it survives because it works.

The reason the decomposition mattered so quickly is that it told designers which thing to fix. An aircraft dominated by induced drag needs span; one dominated by friction needs area removed or a smoother surface. Before 1921 those were both simply “make it less draggy”, and the two answers frequently pull in opposite directions.

It is worth noticing how directly that shows in what aircraft look like. The biplanes of 1918 have enormous wing area and short span, which is the configuration that minimises structural weight and maximises induced drag; within fifteen years of the theory, monoplanes with twice the aspect ratio and half the area had replaced them entirely. The change was not made possible by a new material or a new engine. It was made possible by knowing which of two numbers to reduce, and by the number that says how much a longer wing is worth.

Where the ladder goes next #

Next rungs on this anchor: form drag and how a streamlined shape trades pressure drag for friction, and where the optimum thickness sits; the turbulent friction law, and the transition point that decides which law applies where; laminar-flow sections, which chase the friction figure in this essay by holding the layer laminar over half the chord and pay for it in sensitivity to dirt; and the compressibility drag rise, which adds a fourth term near Mach one that dwarfs the other three.

Then across to how thick is thin, which supplies the first half of this budget, and to the price of having ends, which supplies the second.