Air drag on an archery arrow: Okawa 2012 study

The group measured air resistance on an Easton A/C/E arrow in two ways, both without supports disturbing the flow: in JAXA's 60 centimetre magnetic suspension wind tunnel, where the arrow is held up by the field of a set of coils, and in free flight, filmed by two high-speed cameras.

In the tunnel, across the whole range tested (Reynolds number from 4.0 thousand to 1.5 ten-thousand) the boundary layer on the shaft stayed laminar: the bare shaft gave the theoretical value for a laminar cylinder, and with vanes the drag coefficient rose by about 0.8, the friction on the vanes, reaching around 1.5 for Reynolds between 1.0 and 1.5 ten-thousand.

In free flight the same arrow gave 1.5 at Reynolds 1.0 ten-thousand, but about 2.5 above 1.3, and at two points, around 1.13 and 1.8 ten-thousand, the coefficient took two distinct values, one laminar and one turbulent. The explanation comes from the tunnel: at Reynolds 1.0 ten-thousand, past about six tenths of a degree of angle of attack the coefficient rises, with either point fitted. Arrows shot by a competitor (57 metres per second, Reynolds 2.0 ten-thousand) and by the shooting robot both gave about 2.6, the turbulent value, because a shot arrow flies while flexing.

With GAS Pro vanes in place of Spin-Wings, the boundary layer stayed laminar across the whole range. Between Reynolds 1.7 and 1.8 ten-thousand the histogram of values has two peaks with the Spin-Wings and only one with the GAS Pro, and the coefficient there is about 1.60 against 1.45, because the GAS Pro have less surface area.

The shaft tested is an A/C/E, with the barrelled profile whose diameter is greatest towards the centre; the Reynolds number uses the mean shaft diameter as its reference length. There were two points: the commercial bullet point, widely used in competition, and a streamlined one machined from brass. There were two vane types, Spin-Wing and GAS Pro, fitted in threes.

In the wind tunnel the arrow is suspended by the magnetic field of a set of coils whose current is regulated to cancel its weight and balance the aerodynamic forces: the force is computed back from the current. The tunnel's maximum speed is 45 metres per second, with a variation of 0.06 per cent at 30 metres per second and a turbulence intensity below 0.1 per cent at all speeds. The angle of attack was varied from minus three to plus three degrees, with drag, lift and pitching moment measured.

The free-flight tests were carried out in the indoor facility of the national institute of sports sciences and at the university archery range; the drag coefficient is derived from the arrow's deceleration between the two camera stations and, alternatively, from its attitude. To cover a wide range of speeds the authors built a compressed-air launcher: with the compressor, up to 4.3 megapascals, the initial velocity can be set up to about 60 metres per second, and with a hand pump, up to 15 megapascals, it reaches about 75. They also used a robot that draws the string and releases the arrow, and arrows shot by a competitor.

A single shaft model, and the number of shots is not reported for any configuration: the figures show individual points and histograms without saying how many, and the only uncertainty stated is a relative error of 2.9 per cent on one measurement. In the flight tests the angle of attack could not be brought to zero: the authors estimate it stayed within about a degree from the footage, but the transition happens past six tenths of a degree, that is, inside that margin. It is the same order of magnitude as the effect being measured.

The two experiments do not overlap exactly where it would matter: in the tunnel the boundary layer never became turbulent, so the comparison between tunnel and free flight covers only the laminar branch. The abstract puts the transition above about 1.2 ten-thousand, the body of the paper above 1.3.

The arrow's flexing was not measured: that a flexing arrow keeps the boundary layer turbulent is inferred from the coefficients of the competitor's and the robot's arrows, not from a measurement of the flex. The conclusion that damping the flex or redesigning point and vanes would halve the coefficient is an extrapolation, and in the same paragraph the authors add the opposite thought, that the flex may also serve to keep the boundary layer turbulent; they leave it as a possibility.

There is nothing on wind drift, on the trajectory over 70 metres or on scores, and nothing either on the spin the vanes give the arrow, which is the other reason they are chosen. It is four pages of symposium proceedings, in Japanese, with an English abstract.

This is the clearest measurement of what the air does to a competition arrow, and the figure to hold on to is the jump: the same shaft can fly with a drag coefficient of around 1.5 or around 2.6, depending on whether the air on the shaft stays laminar or turns turbulent, and what trips the change is a few tenths of a degree of angle of attack. A shot arrow flexes, so it usually sits on the turbulent branch: the competitor's arrows and the robot's all sit around 2.6.

The comparison between the two vanes is the part that touches a choice archers make. With the GAS Pro the coefficient stayed on the laminar branch across the whole range tested, and the authors read this as a stronger stabilising effect that keeps the angle of attack small. Note how that should be read, though: in the same paper the authors remark that a coefficient which keeps jumping between two values is worse for accuracy than one that stays put, and this is an argument against treating the lower number as preferable in itself.

What is missing is everything an archer looks at: no trajectory, no wind drift, no group, no score, and nothing on the spin the vanes give the arrow. An arrow with less drag arrives sooner and drifts less in the same wind, but how much of that survives over 70 metres is not calculated here.