Air drag on an archery arrow: Miyazaki 2013 study

The authors measured the drag on an Easton A/C/E arrow by two methods that need no mechanical support, since any support would distort the flow around a body this thin. The first is JAXA's wind tunnel, where the arrow is held in the air by magnetic fields and the forces are read from the currents in the coils; the second is free flight, filmed by two high-speed cameras.

In the tunnel, up to 45 metres per second, the boundary layer on the shaft stays laminar across the whole range tested and the drag coefficient settles around 1.5 above a Reynolds number of 1.0 times ten to the fourth. The bare shaft, with no vanes, sits about 0.15 off the theoretical value for a cylinder, a difference the authors attribute to base drag; fitting the vanes raises the coefficient by about 0.75, through friction on the vanes themselves. The point shape makes no difference, and the arrow, although spinning, keeps its boundary layer laminar.

In free flight the picture changes. Below 1.2 times ten to the fourth the two methods agree and the coefficient is about 1.45. Above 1.3 times ten to the fourth, with the bullet point, the boundary layer becomes turbulent and the coefficient rises to about 2.6; at 2.4 times ten to the fourth the transition is complete. With the streamlined point it does not complete even there: two distinct values are found, about 1.6 for the laminar regime and about 2.6 for the turbulent one, and the distributions show two peaks, as though there were two kinds of shot. The authors doubled the number of shots thinking they had too few data, and the two peaks stayed.

The explanation comes from an extra test in the tunnel. Varying the angle of attack, the coefficient stays around 1.5 as long as the angle is below 0.6 degrees, and rises abruptly beyond one degree, even at 1.0 times ten to the fourth. In flight the attitude is never as perfect as in the tunnel, and that one degree is enough to change the nature of the flow.

The arrow shot by an expert female national-team archer, at 58 metres per second, gave a coefficient of about 2.65, the turbulent value, with a relative error under 3 per cent and a single distribution, with no double peak. The authors hold that the oscillation of an arrow leaving a bow introduces asymmetric disturbances that make the boundary layer turbulent on every shot. They close by estimating the ratio of mass to drag at 60 metres per second: about 0.50 with a laminar boundary layer and 0.86 with a turbulent one, a gap that corresponds to about 15 centimetres of height on a target 60 metres away.

The arrow is an Easton A/C/E series 1000 C.6, a barrel-shaped shaft with a mean external diameter of 5.25 millimetres and a length of 625 millimetres, shortened by cutting from the front; with point and vanes it weighs 14.3 grams. The vanes are one-and-three-quarter-inch Spin Wings from Range-O-Matic. Two points were tested: the standard bullet point, an ACE Screw Point of 36 grains with a maximum diameter of 5.83 millimetres, wider than the shaft at the join, and a hand-made brass streamlined point with a maximum diameter of 5.30 millimetres reached at its base.

In the tunnel the arrow is suspended by JAXA's 60 cm magnetic system, with ten coils around the test section and ten cylindrical neodymium magnets of 4 millimetres diameter and 30 millimetres length inserted inside the shaft. The currents are adjusted so that the arrow stays still in the chosen position and attitude, and from those currents the drag, lift and pitching moment are calculated. The control circuit responds at 245 hertz, the data are averaged over 8 seconds, the maximum wind speed is 45 metres per second and the fluctuation at 30 metres per second is 0.06 per cent.

The flight tests were run in the 60-metre indoor range of the Japan Institute of Sports Sciences. Two Phantom V310 cameras, at 1,280 by 800 pixels and 2,800 frames per second, film from the side: the first 13 metres from the launching point, the second 2 metres from the target, that is, 45 metres further on, both set 3 metres off the shooting line. The images give the horizontal and vertical components of the velocity at the two points, and from these the drag coefficient, by two independent procedures: the first uses only the ratio of the horizontal velocities, the second adds the change in the arrow's attitude. Only shots whose two values agreed within 3 per cent were kept.

Since an archer launches an arrow at a single speed, the authors built a compressed-air launcher: an electric compressor reaches 4.3 megapascals and 62 metres per second, and a hand pump of the kind used for air rifles allows 7 megapascals and 75 metres per second. The relative error on launch velocity is under 0.3 per cent up to 62 metres per second and rises to 0.8 per cent above it. For comparison, the expert archer's repeatability is 0.6 per cent at 58 metres per second. About 30 shots were recorded at each Reynolds number, doubled where the scatter was large. The arrow launched by the machine translates and rotates as a rigid body and does not oscillate along its axis, unlike the one shot from the bow; the archer, an experienced recurve shooter, shot 36 arrows.

An empirical correction allows for the fact that the scale in the image changes with the distance between arrow and camera: the archer was asked to scatter her shots horizontally, the relationship between deviation on the target and apparent velocity was established, and every data point was corrected along that line.

All the measurements concern a single shaft, the A/C/E, with a single set of vanes: other diameters, other shapes and other vanes were not tested, and this work says nothing about how they behave.

The tunnel reaches 45 metres per second while an archer shoots at about 60: the range where bow and measurement overlap is therefore covered only by free flight, and the link between the two methods rests on their agreement at low speed. In flight, moreover, the coefficient is calculated on the assumption that there is no lift, that is, a zero angle of attack; the authors themselves measure in the tunnel a lift proportional to the angle with a slope of about 0.69 per degree, and argue that in flight the direction of the lift rotates with the arrow's precession and averages to nothing. It is a reasoned explanation rather than a direct measurement.

The angle of attack in flight was not measured precisely: the authors estimate it at under one degree by looking at the images. The oscillation of an arrow leaving the bow is given as the cause of the turbulence without anyone having quantified it, and is presented as the authors' suspicion.

The critical Reynolds number is not determined: in the transition zone the distributions have two peaks, the choice of where to separate them is declared arbitrary, and the authors say so. An inconsistency also remains between the two methods between 1.2 and 1.5 times ten to the fourth, where the tunnel stays laminar and flight produces turbulent shots as well.

Some values change between abstract and text: the abstract puts the transition with the bullet point above 1.2 times ten to the fourth and the conclusions above 1.3. Park's formula, which the authors cite, predicts 3.14 for the archer's arrow against the 2.65 measured, and the gap is left unexplained.

Finally, the link with scoring is a theoretical estimate: the 15-centimetre difference in height at 60 metres comes from the mass-to-drag ratio, not from arrows counted on a target. One archer shot, and nobody measured her group.

The most useful thing to come out of this work is that an arrow shot from a bow and the same arrow launched by a machine do not have the same drag. The difference lies not in the material or the speed but in the oscillation: an arrow leaving a bow flexes, and that flexing makes the flow along the shaft turbulent every time, while an arrow launched straight can stay in the laminar regime. Between the two regimes the drag coefficient goes from about 1.5 to about 2.6.

The second point concerns repeatability, and it runs against intuition. In the machine shots, in the middle of the speed range, the same arrow sometimes gave the laminar value and sometimes the turbulent one, with double-peaked distributions. In the archer's thirty-six shots this never happened: they were all turbulent, all alike. On the authors' reading, the flexing of the arrow, usually described as a problem to be managed, here keeps the aerodynamic condition constant from shot to shot. In their calculation, a laminar shot and a turbulent one arrive at heights about 15 centimetres apart over 60 metres, so always having the same regime matters more than having any particular one.

A third point, for anyone looking at numbers for vanes: fitting them raises the drag coefficient by about 0.75 compared with a bare shaft. That is the aerodynamic price of the stability they provide.

What this study does not allow is choosing a point. The two shapes tested differ in the laboratory over when the change of regime happens, but in the archer's shots the regime was turbulent anyway, and nobody measured groups or scores with one or the other.