Arrow in flight and uncertain drag: Miyazaki 2017

The study measures air resistance on an Easton A/C/E arrow and shows that, at recurve speeds, it can take two very different values depending on how the arrow flies. The layer of air next to the shaft can stay laminar or turn turbulent, and in the second case the drag coefficient doubles. In the wind tunnel, with the arrow perfectly aligned, the coefficient stayed at about 1.5 up to a Reynolds number of 15,000, the highest available (45 m/s); tilting it by 0.75 degrees was enough for it to jump from 1.6 to 3.1 at a Reynolds number of about 11,000. About 0.75 of that coefficient comes from the vanes, whichever type was fitted.

In launches from a compressed-air machine, between Reynolds numbers of 12,000 and 18,000 the measured drag was widely scattered and often split into two groups, one laminar and one turbulent: at 13,000, over 137 launches, the values clustered around 1.4 and 3.1. At 17,500 the coefficient stayed laminar (about 1.5) only when the arrow's largest angle to the airflow during flight, worked out by simulation, was below 0.4 degrees, and became turbulent (about 2.7) above 0.6. The larger Spin Wing vanes steadied the arrow more than the Gas Pro vanes, with lift and moment higher roughly in proportion to area (ratios of 0.74 and 0.69, against an area ratio of 0.75); in the simulation the arrow oscillated at 3.95 Hz against 3.28 and settled sooner.

Shot by an expert female archer from her recurve at 58 m/s, the arrow flexed with an amplitude of 6.9 mm at the point and a frequency of 87 Hz. The boundary layer was turbulent every time, with a coefficient of 2.68 with Spin Wings and 2.63 with Gas Pros and a spread of about 3%.

The arrow was an A/C/E shaft, carbon over an aluminium core and barrelled, 625 mm long with a mean diameter of 5.25 mm, fitted with a 5.83 mm bulge point and two types of vane: Spin Wing, 12.9 cm², and Gas Pro, 9.7 cm². It weighed 14.3 and 14.4 grams. In the machine tests it had no nock.

In the wind tunnel (JAXA, 60 cm) the arrow was held in place by magnetic fields, with no supports to disturb the flow: ten small neodymium magnets inside the shaft and a set of coils kept it in position, and lift, drag and moment were worked out from the currents needed. The top speed was 45 m/s, the Reynolds number from 4,000 to 15,000, and the angles below 5 degrees.

The free-flight tests took place at the indoor range of the Japan Institute of Sports Sciences, with the air conditioning switched off. A machine launched the arrow with compressed air, so it did not flex, and five high-speed cameras at 3,000 frames per second filmed it: two near the launch for initial speed, rotation and angle, one at 9 metres and one 55 metres further on to derive drag from the loss of speed, and one from above. Different nozzles (straight, cut at 80 and 60 degrees, and slotted) were used to vary the initial rotation. The angle during flight was not measured but calculated, by integrating the equations of a rigid body with the tunnel coefficients and the initial conditions from the video, for a target at 70 metres. Finally an expert archer shot 36 arrows with each vane type from her recurve.

Almost all of the work concerns an arrow launched by a compressed-air machine, which flies as a rigid body, without the flex of a shot from a bow and without a nock; the angles in flight, on which the main finding rests, come from a model rather than a measurement. The tunnel tops out at 45 m/s, below the roughly 60 m/s of a recurve, and with Gas Pro vanes the arrow could not be held steady above a Reynolds number of 9,000 because it spun too fast for the control system. There was one shaft, one point and two vane types.

The part with the bow rests on a single archer, 36 arrows per vane type, at one speed. The authors' final conclusion, that flex makes the boundary layer turbulent right from the point and that this improves accuracy and repeatability, is an interpretation: the repeatability measured is that of the drag coefficient, and groups on the target were not recorded. The archive PDF is the accepted manuscript, not the journal's typeset version.

For recurve archers the most concrete finding is the last one: with an arrow shot from the bow, which flexes on every shot, the boundary layer was always turbulent and drag was very consistent from arrow to arrow, with a spread of about 3%. The uncertainty between laminar and turbulent flow, which makes drag unpredictable, concerns arrows that fly almost perfectly aligned and without flex, like those from the machine. For real shooting the study suggests it is not an everyday problem, although it shows this with a single archer.

The vane comparison confirms that a larger vane straightens the arrow faster, in proportion to its surface, and that, in the machine launches, the arrow's initial rotation about its centre of mass mattered a great deal for how it flew afterwards. The study measures neither scores nor groups and cannot say which vane or point earns more points.