The drag that falls as it speeds up

Drag goes as the square of speed. Everybody knows this, it is right nearly everywhere, and it is built into every intuition about how things move through air.

There is one place where it is not merely inaccurate but backwards. Over a narrow band of speeds, a smooth sphere pushed harder experiences less force, not more. The drag falls as the speed rises, by nearly thirty per cent, and the reason is that the boundary layer on the ball changes character partway through.

What is borrowed here, stated before anything is built on it #

This essay is the first in the field to rest on a correlation for its main quantity, and the terms should be clear before any argument is made on top of it.

This site computes no turbulent boundary layer at any Reynolds number. The reason is arithmetic rather than effort: resolving one needs a grid whose cost nobody can pay. So the separation point on a ball — which is the thing that decides everything below — cannot be predicted here, and any figure claiming otherwise would be inventing.

What is used instead is Morrison’s correlation for the drag of a smooth sphere: an algebraic fit, valid from Stokes’ law at the bottom to $\mathrm{Re} = 10^6$ at the top, which reproduces the plateau in the middle and the crisis near $\mathrm{Re} = 2.6\times10^5$. It is a fit. No part of it is derived and no assertion in this essay tests it.

What is computed, and what the assertions are about, is what follows given that curve: the band in which the drag force falls, located by bisection on the logarithmic slope; the trajectories, whose energy budgets close to a part in a million; and the comparison between the two balls.

The roughened curve is a second, weaker kind of statement. It is the smooth correlation shifted along the Reynolds axis by a stated factor, which is the simplest thing that captures what roughness does — trip the layer early, bring the crisis forward. It is a model of an effect and not a measurement of a golf ball, and it gets one thing wrong that the essay’s last section returns to.

The crisis, and what changes at it #

Below the crisis, the boundary layer on the front of a sphere is laminar. It climbs the pressure rise on the back half, and a laminar layer climbing a pressure rise gives up early — at about eighty degrees from the front stagnation point, barely past the equator. The wake behind is as wide as the ball, the base pressure in it is low, and almost all the drag is pressure drag.

Above the crisis, the layer goes turbulent before it gets there. A turbulent layer carries much more momentum near the wall, because turbulence mixes fast fluid down into the slow region, so it can climb a much steeper pressure rise before separating — around a hundred and twenty degrees, well round the back. The wake is narrower, the base pressure higher, and the pressure drag much smaller.

The trade is the essay’s refutation in one line:

• Friction goes up. A turbulent boundary layer has several times the skin friction of a laminar one at the same Reynolds number, which is computed elsewhere on this site for a flat plate and rises further with Reynolds number.
• Pressure drag goes down, by more. On a bluff body, pressure drag is the dominant term, and narrowing the wake cuts it dramatically.

Total drag coefficient falls from about 0.5 to about 0.15. The smaller of the two contributions went up; the larger one collapsed.

It is worth pausing on why the pressure term is so large on a bluff body in the first place, because the answer is the site’s founding paradox. Ideal flow predicts no drag at all: the pressure recovers perfectly on the back of a sphere, the fore-and-aft distribution is symmetric, and the integral is exactly zero. Every bit of the pressure drag a real sphere has is the failure of that recovery, and the failure happens because the flow separates. So the pressure drag is not a separate mechanism from separation — it is separation, measured as a force, and anything that delays separation attacks it directly.

That also explains the size of the effect. Moving the separation point from eighty degrees to a hundred and twenty does not shave a percentage off the drag; it recovers a large part of the pressure on the back of the ball that ideal flow said should have been there all along.

The force, not the coefficient #

A falling coefficient is unremarkable — almost every drag coefficient falls with Reynolds number somewhere. What is remarkable is that here the force falls, and that requires something much stronger.

Drag force is $\tfrac{1}{2}\rho U^2 A\,C_D$, and at fixed size and fluid $U \propto \mathrm{Re}$, so the force goes as $\mathrm{Re}^2 C_D$. For that to fall, $C_D$ must fall faster than $\mathrm{Re}^{-2}$: the logarithmic slope $\mathrm{d}\ln C_D/\mathrm{d}\ln\mathrm{Re}$ must be steeper than $-2$.

The band comes out at $\mathrm{Re} = 2.40\times10^5$ to $3.53\times10^5$, with the force falling by 29.2 per cent across it. In air, for a 70 mm ball, that is roughly 51 to 76 metres per second: a sphere at 76 m/s is being pushed backwards less hard than the same sphere at 51.

Both halves of that are asserted. A curve reported as having a crisis must actually have a band where the force falls; and — the more useful half — a curve with no such band must not be reported as having one. That second refusal exists because the failure is symmetric: a correlation with the crisis term dropped still falls with Reynolds number everywhere, and would pass any test that merely asked whether $C_D$ decreases.

The flight, integrated #

A ball in flight is a Reynolds number sweep. It launches fast, slows, and its drag coefficient walks along the curve while it does so — which is why a trajectory, rather than a coefficient, is the figure that settles the argument.

Two assertions run on every trajectory, and both are about the integrator rather than the physics — which is the right place for them, because the comparison between two balls is only worth anything if the arithmetic is not leaking.

The Magnus force does no work. A force perpendicular to the velocity cannot, and the check evaluates the lift acceleration through the same code path that moves the ball and dots it with the velocity. It comes out at $10^{-17}$ joules. The failure it catches is a sign or an index the wrong way round in the two components, which produces a perfectly plausible trajectory — and a sign error of exactly that kind has been caught by measurement rather than by looking four times in other fields on this site.

The energy budget closes. The kinetic and potential energy lost must equal the work done against drag, integrated along the path. It comes out at $3\times10^{-9}$ of the launch energy. An integrator leaking a per cent would change the range by metres, which is the size of the effect being argued about.

The second figure is the more instructive one. Roughness is not a free improvement; it is a transposition. It moves the crisis to a lower speed, which is worth a great deal if the ball flies near the shifted crisis and worth nothing if it flies far below it. A rough ball thrown slowly is simply a rough ball, with all the extra friction and none of the benefit.

Where the drag actually is #

The split between the two contributions is worth seeing directly, because the folk explanation is entirely about the smaller one.

That single distinction — streamlined or bluff — decides which way the trade goes, and it is why the dimple argument does not generalise. An aircraft wing is streamlined; its drag is mostly friction; a tripped boundary layer is a straightforward loss, and enormous effort goes into keeping laminar flow on one. A golf ball is bluff; its drag is mostly pressure; tripping the layer is worth a factor of three. The same modification, on two shapes, in opposite directions.

The same crisis, on everything bluff #

A sphere is not special. Every bluff body has a drag crisis, at a Reynolds number set by its own geometry, and the mechanism is identical each time: the boundary layer goes turbulent before it has to climb the pressure rise, separation moves aft, the wake narrows.

A circular cylinder has one at about the same Reynolds number as a sphere, and it is the reason the site’s regime ladder puts a threshold at 2·10⁵ — that is where a cylinder’s drag coefficient falls from about 1.2 to about 0.3. The cylinder case is the one this site can draw the flow of at low Reynolds numbers, and it is a very different picture at each of them.

The consequences turn up wherever a bluff body moves fast enough:

• Cycling. A rider’s limbs and body are bluff, and much of the speed range of a racing cyclist sits near the transition for a cylinder of a limb’s diameter. Textured skinsuit fabric on the arms is a trip wire, and it is legal in some codes and banned in others.
• Ski jumping and speed skating. Same argument, same fabrics, same regulatory arguments.
• Chimneys, cables and bridge decks. Here the crisis is a hazard rather than a benefit: a structure that sits near its own critical Reynolds number has a drag that changes sharply with wind speed, and one side of a circular member can be in a different regime from the other. Helical strakes on a chimney exist to force the issue.

None of these is a different physics. They are the same curve read at a different Reynolds number, which is the site’s oldest claim — the shape is not the question, the ratio is — applied somewhere with money on it.

The crisis is not a property of the ball #

Everything above treats the critical Reynolds number as a number belonging to a sphere, the way its diameter does. It is not. It belongs to the sphere and the stream it is in, and the second half of that pairing moves it further than roughness does.

The reason follows from the mechanism. The crisis happens when the boundary layer goes turbulent before it reaches the separation point, and transition is triggered by disturbances — which may come from the surface, as a dimple or a trip wire, or may arrive already present in the oncoming air. A stream that is itself turbulent supplies them for free, so the layer transitions earlier, and the whole curve shifts left exactly as a roughened ball’s does. From the drag curve alone the two causes are indistinguishable.

The size of it is large. A sphere in genuinely quiet air — free flight, or a tunnel built for low turbulence — reaches the post-crisis state at about $3.9\times10^5$. The same sphere in a stream carrying a per cent or two of turbulence reaches it below $2\times10^5$, and in a deliberately stirred one lower still. That is a factor approaching two, which is a third of the six-fold shift this essay uses to represent a golf ball’s dimples, produced by nothing being done to the ball at all.

So it was turned into an instrument. Through the 1920s and 1930s a standard sphere was the accepted way of measuring how quiet a wind tunnel was: run it up, find the Reynolds number at which its drag coefficient passed a fixed value, and divide the free-air figure by the one the tunnel gave. The ratio was the tunnel’s turbulence factor, and every drag measurement made in that tunnel was corrected by it. Tunnels of the period ran from about 1.1 for the best to nearly 3 for the worst, and a model tested in one was effectively being tested at a higher Reynolds number than its speed implied.

That is a slightly vertiginous piece of history, and it has a direct bearing on this essay’s borrowing. The sphere’s own drag curve was for two decades the calibration standard for the instrument used to measure drag curves — so the correlation the whole argument rests on is a fit to measurements each of which carries its own stream’s disturbance level inside it. The band located above by bisection, $2.40$ to $3.53\times10^5$, is therefore a property of a dataset rather than of spheres, and quoting it to three figures says more about the fit than about the physics.

None of that touches the argument. The falling force is real, the mechanism is real, and the trade between friction and pressure runs the same way whatever triggers the transition. What moves is the speed at which a particular ball, on a particular day, meets it.

Where the model stops #

The shift is a model, and it is wrong in a knowable way. Shifting the smooth correlation predicts that a tripped ball reaches the smooth ball’s post-crisis coefficient of about 0.13. A real golf ball sits nearer 0.25. The difference is that dimples do not only trip the layer — they are themselves roughness elements with their own form drag, and the fully-rough asymptote of a sphere is higher than the smooth-sphere post-crisis value. The shift model captures the transition and not the penalty, and the trajectory figures therefore overstate the tripped ball’s range.

That is stated rather than corrected because correcting it would mean fitting a second parameter to golf-ball data, which would make the figure a curve fit rather than an argument.

No spin effect on drag. Spin changes the separation pattern and therefore the drag, not only the lift. The model here has spin entering the lift alone.

The lift model is linear and stated. $C_L = k\,\omega R/V$ with $k$ chosen so that a driven ball comes out near the measured lift coefficient. Real lift coefficients saturate at high spin. This matters for the apex of the trajectory and not for the drag argument.

And nothing here is a solution. The separation angles, the base pressure, the correlation and the shift are all borrowed or stated. What this site contributes is the arithmetic around them, and the distinction is worth keeping sharp: the crisis is a measurement, and the falling force is a calculation.

Who found it, and when #

Gustave Eiffel measured the drag crisis in 1912, dropping spheres down the tower that carries his name and later in the wind tunnel he built at Auteuil. Prandtl explained it almost immediately, and in 1914 demonstrated it in the neatest way anybody has: he fitted a trip wire — a thin ring of wire around the front of a sphere — and photographed the wake narrowing. A sphere with a wire on it had measurably less drag than the same sphere without.

That experiment is the whole essay in one object. Something was added to the surface, the friction went up, and the total drag went down, because the added thing changed where the flow let go.

The dimple came from the other direction entirely. Golfers in the 1840s noticed that old scuffed gutta-percha balls flew further than new smooth ones, which is an observation seventy years ahead of any explanation, and the deliberately patterned ball dates from the 1890s. It is one of the clearest cases in engineering of a practice that worked for half a century before anybody could say why, and of an explanation that, once available, turned out to be the opposite of the obvious one.

Where the ladder goes #

This rung is about a symmetric ball, where moving the separation point changes the drag. The next is about an asymmetric one, where moving it on one side only changes the direction.

A cricket ball swings without spinning at all. The seam trips the boundary layer on one side, that side separates later, the pressure distribution is no longer symmetric, and the ball goes sideways. The calculation of the side force is the same pressure integral this essay’s mechanism implies — performed both by quadrature and in closed form — and it has a property that makes it unusually instructive: the sign of the answer depends on one borrowed number, and the site’s own inviscid solver supplies the value that gets it wrong.