Arrow drag and axial rotation: Park 2011 study

Park puts forward a way of calculating air resistance on an arrow by adding up four parts (point, nock, shaft and fletching) and tests it by measuring the speed of three very different arrows out to 90 metres. For a 23.5-gram Protour 470 launched at 78.6 m/s from a compound, the calculated initial drag was 0.94 times the arrow's weight: 73% came from the shaft, 11% from the nock, 12% from the vanes (half from skin friction on their surfaces, half from their frontal thickness) and 4% from the point. For a LightSpeed 400, light and fat (7.44 mm), drag at 90.8 m/s was 1.92 times its weight; for an Axis FMJ 400, heavy and fat, 1.13 times at 81.3 m/s. The calculated speeds tracked the measured ones better than Liston's method did, while Rheingans' historical method gave a drag of about half.

On spin, with vanes set at about 0.45 degrees the Protour settles at one full turn every 2.12 metres, reaches that rate in about 10 metres and loses 0.46 m/s in spinning up. In a shooting-machine test at 30 metres, angling the vanes by about 0.25 degrees more than halved group size compared with straight vanes. In a simulated comparison of arrows as close as possible to 400 spine (LightSpeed 400, ACE 400, Protour 420) from the same compound, the LightSpeed was fastest off the bow but beyond 30 metres was overtaken by the ACE and, from about 50 metres, by the Protour as well.

Each part of the arrow gets its own coefficient from the classic aerodynamics literature: 0.1 for the point and 0.3 for the nock; for the shaft, turbulent skin friction calculated point by point along its length, which allows for tapered and barrelled shafts, with a correction for surface curvature; for the vanes, turbulent friction on both sides plus a coefficient of 0.2 for the leading edge, a value chosen by the author pending a direct measurement. Spin is calculated from the lift of the vanes and the arrow's inertia, and compared with the rotation angle measured by shooting from a machine into a boss that stops the arrow at once, from 1 to 15 metres in 1-metre steps, reading the angle by eye to the nearest 5 degrees.

Speeds were measured with an Easton chronograph out to 90 metres, always from the machine and on still days, because the arrow had to pass through an area about 50 mm across above the sensors; accuracy is about 0.3 m/s, which means about 3% on the drag estimate. The Protour was launched from an Archery Research AR34 single-cam bow, the other two from a more powerful Hoyt Carbon Matrix. Drag is given as the ratio of force to arrow weight, so that arrows of different mass can be compared.

Three arrows, two compound bows, no recurve. Some coefficients are estimates: the vane-edge value is the author's choice and the nock value is reasoned rather than measured, and interference between shaft and vanes is taken as zero. Angle of attack and arrow flex are ignored in the drag calculation, on the basis of classic data on cylinders. The number of arrows measured at each distance and the spread of the readings are not reported, and agreement between calculation and measurement is shown only in graphs.

The group test with straight and angled vanes is described in two sentences, with no arrow count and no values. For the Axis FMJ the table gives a mass of 0.273 kg, clearly a typo for about 27 grams, and one figure caption calls it an Axis FMJ 500 while the text and table say 400. The view that thin, heavy arrows drift less in wind rests on drift being roughly proportional to drag and on the practical difficulty of getting fat arrows to pass over the chronograph: drift itself was not measured.

The study puts numbers on something outdoor archers sense: almost all the drag on a competition arrow comes from friction along the shaft, so its surface area is what matters most. Diameter and mass decide how much speed is left at long range, and a light, fat arrow that is fastest off the bow can be the slowest beyond 30 metres. For compound field archery, where speed at unmarked distances counts, the author favours thin, light arrows such as the ACE; for target archery, where wind counts, thin and heavy ones.

Vanes add little to total drag but govern stability, and setting them at an angle helps: with straight vanes the machine's groups were more than twice the size. The author thinks a smaller angle than the one tested would do, with one full turn about every 5 metres. Nocks wider than the shaft and points with an abrupt step also add drag. These are pointers drawn from a model tested on three arrows and from machine shooting, not evidence about an archer's score.