Boundary layer and arrow attitude: Matsumoto 2015

The study brings three tools to bear on the same arrow: a magnetic suspension wind tunnel, free-flight tests with a compressed-air launcher, and a numerical computation of the rigid-body motion. The arrow is an Easton A/C/E shaft with an ACE bullet point of 3-36 grains, tested with two vanes widely used in competition, Spin Wing and Gas Pro.

In the tunnel, with incidence varied between minus three and plus three degrees, the lift coefficient and the pitching moment coefficient grow in proportion to incidence, and the slope of both depends little on the Reynolds number. The oscillation's own damping comes out negative, that is the oscillation amplifies, but by an amount of the order of 10 to the minus six, negligible in the computation. The tunnel measurements make the Spin Wings more stabilising than the Gas Pro vanes, against what an earlier flight test by the same group had found.

In the flight tests, with Spin Wings and no conditions imposed on the launch, the drag coefficient splits into two values between Reynolds numbers of 1.3 and 1.8 times 10 to the fourth: about 1.5 with a laminar boundary layer and about 2.6 with a turbulent one. With the Gas Pro vanes, at the two Reynolds numbers retested, every measurement gives the turbulent value, and the laminar condition found previously does not reproduce.

The numerical computation places the transition at around a maximum incidence of 0.45 degrees at a Reynolds number of 1.77 times 10 to the fourth, lower than the 0.6 degrees a static tunnel measurement had given at 1.1 times 10 to the fourth: the faster the arrow, the less tilt it takes to trigger turbulence. At 1.5 times 10 to the fourth the Spin Wings keep incidence smaller and make it oscillate faster than the Gas Pro vanes, which in turn spin the arrow faster (spin parameter 0.039 against 0.029). The computation adds two things: the higher the initial velocity, the smaller the incidence in flight, and with an initial angular velocity equal to minus g times the cosine of the elevation angle divided by the velocity the arrow flies at almost no incidence; a departure of about 0.2 radians per second from that value takes maximum incidence above 0.45 degrees. Finally, computing the vertical deviation at the target at 70 metres, the deviation falls when the slope of the pitching moment is high and that of the lift is low: the vane that suits accuracy generates little lift and a large restoring moment.

The arrow tested is an Easton A/C/E shaft, barrel-shaped and thickest in the middle: length 0.626 metres, mass 14.3 grams, mean diameter 5.24 millimetres, slenderness ratio 119. The point is a 3-36 grain ACE screw-in, the vanes are Range-O-Matic Spin Wings and D.S.D Gas Pro. The Reynolds number is computed on the diameter.

The tunnel tests were run with JAXA's 60-centimetre magnetic suspension and balance system, which avoids any mechanical support: neodymium magnets are inserted in the arrow and external coils hold it suspended. The electromagnet currents are controlled to cancel the weight and balance the aerodynamic forces, and the forces are derived from the currents. The tunnel reaches 45 metres per second, with a fluctuation of 0.08% at 30 metres per second and a turbulence intensity below 0.1% at all speeds. Incidence in the vertical plane was varied quasi-statically between minus three and plus three degrees to measure lift and pitching moment; to assess the damping, the arrow was set oscillating by varying the current periodically, with an amplitude of 0.5 degrees and a frequency of 5 Hz at 16 metres per second for both vanes.

The flight tests were carried out in the indoor facility of the Japan Institute of Sports Sciences, with a compressed-air launcher chosen precisely so as not to produce flexing of the shaft: the arrow leaves without the archer's paradox oscillation. The recording is by high-speed cameras: two serve to obtain the initial conditions, including the shaft's angular velocities, and two to measure the decay of velocity, from which the drag coefficient is derived.

The computation integrates the rigid-body equations of motion over time, with lift and pitching moment taken from the tunnel measurements and approximated, for small incidences, as proportional to the sine of the incidence. Since the torque about the arrow's axis was not measured, the computation assumes that the spin parameter stays constant even when an incidence appears. For both vanes the lift and moment values measured at a Reynolds number of 7.7 times 10 to the third were used, neglecting the dependence on Reynolds number, and the damping term was omitted.

There is no archer in the flight tests. The arrow is fired by a compressed-air launcher, chosen precisely to eliminate flexing of the shaft: that is the right choice for isolating the aerodynamics, but it means the archer's action, the fingers leaving the string and the archer's paradox do not enter the measurements. How far the initial conditions reconstructed in the laboratory resemble those of an arrow shot from a bow lies outside the study.

The computation rests on two declared simplifications. The torque about the arrow's axis was not measured, so the spin parameter is assumed to stay constant even once incidence appears: it is the assumption on which the whole section about the ideal initial spin depends. And the lift and moment slopes were taken at a Reynolds number of 7.7 times 10 to the third and used across the whole range, that is at speeds well away from competition ones. The tunnel measurements show a small dependence on Reynolds number, but not a null one.

There is then a finding the authors cannot reconcile, and they say so. With the Gas Pro vanes, an earlier flight test by the same group had found a laminar boundary layer up to a Reynolds number of 2.3 times 10 to the fourth; here the tests redone at 1.5 and 1.77 times 10 to the fourth all give the turbulent value, and the tunnel says the Gas Pro vanes stabilise less than the Spin Wings. The explanation proposed, that in that test the arrow flew with an almost ideal initial spin, is plausible but remains an interpretation after the fact: two sets of measurements on the same vane disagree, and the study does not settle it.

Nor is the incidence threshold a firm number. Here it is given as about 0.45 degrees at a Reynolds number of 1.77 times 10 to the fourth; the same group's 2016 paper presents it as a transition band between 0.4 and 0.6 degrees at 1.75 times 10 to the fourth, and in its English summary as about 0.5 degrees at 1.8 times 10 to the fourth. These are compatible values but not the same value, and they come from computation rather than from a direct measurement of the boundary layer.

Finally the editorial context. This is a four-page Japanese conference paper, with the results almost all in graphs and no tabulated figures beyond the arrow's specifications and the excitation parameters. There are no stated repetitions, no measurement uncertainties on the coefficients and no statistical analysis. The number of arrows tested and of shots per condition is not reported. The conclusion about the ideal vane, little lift and a large restoring moment, comes from a computation of vertical deviation at 70 metres rather than from arrows shot at 70 metres. The same programme of tests is also published in the 2016 paper: the two articles are not independent confirmations of each other.

The point that changes how one thinks about arrow flight is how little it takes. Half a degree of tilt relative to the air, less than can be seen by eye or controlled consciously, decides whether the layer of air around the shaft stays laminar or turns turbulent, and with turbulence the drag goes from about 1.5 to about 2.6, nearly doubling. It is not a gradual difference: the arrow sits in one regime or the other.

From this comes a practical consequence that bears on tuning. A Reynolds number of 1.3 to 1.8 times 10 to the fourth on an A/C/E shaft corresponds, converting with the mean diameter of 5.24 millimetres and air at room temperature, to a speed between about 38 and 52 metres per second (that is a conversion, not a figure from the paper). This is the range a recurve arrow crosses as it slows towards 70 metres: the same arrow can leave in one regime and arrive in another, and a launch even slightly tilted takes it across sooner.

The second useful point is that the spin the arrow has as it leaves matters, not only the tilt. The computation indicates an ideal initial angular velocity, with which the arrow flies at almost no incidence, and shows that departing by about 0.2 radians per second is already enough to take maximum incidence above the threshold. It is a measure of how sensitive the release is: there is no wide margin.

On vanes, the comparison should be taken with care, but the direction is clear in this study's data: the Spin Wings keep incidence smaller and damp it faster than the Gas Pro vanes, which in exchange spin the arrow more. It is worth remembering that spinning an arrow and stabilising it are not the same thing. The criterion the computation suggests for accuracy at 70 metres is that a good vane should generate little lift and a strong restoring moment, that is straighten the arrow without pushing it sideways. The same group, in an earlier test, had found the opposite for the Gas Pro vanes, and the contradiction is unresolved: this is not a basis on which to choose your own vanes.

One last point for anyone reading aerodynamics studies: here the arrow is fired by a compressed-air launcher, which does not make it flex. Everything that happens between the fingers and the pressure button, that is the archer's paradox, is deliberately excluded.