Arrow attitude and airflow: Matsumoto 2016 study

The paper takes up the previous year's programme of tests and concentrates on one question: how the arrow's attitude in flight determines the passage of the boundary layer from laminar to turbulent, in the Reynolds number range between 1.3 and 1.8 times 10 to the fourth where the two conditions coexist. The arrow is the Easton A/C/E shaft with a bullet point, tested with Spin Wing and Gas Pro vanes; the tools are JAXA's magnetic suspension tunnel, free-flight tests with a compressed-air launcher and four cameras at 3000 frames per second, and the numerical computation of the rigid-body motion.

In the tunnel, with incidence varied between minus three and plus three degrees, lift and pitching moment stay proportional to incidence and their slope depends little on Reynolds number: in the computations the values measured at 7.7 times 10 to the third were used for both vanes. The oscillation's own damping comes out negative, that is amplifying, but of the order of 10 to the minus six, and is neglected.

In the flight tests, with Spin Wings and no conditions imposed, the drag coefficient splits into two values: about 2.6 in the turbulent regime and about 1.5 in the laminar one. Reproducing instead the launch conditions that bring maximum incidence towards zero, the coefficient stays at the laminar value. Deliberately forcing non-ideal conditions, with Spin Wings the values split between about 2.6 and about 1.8, an intermediate value the authors read as an arrow alternating between the two regimes during flight; with the Gas Pro vanes the turbulent value is always obtained.

The computation, with a launch at 60 metres per second, shows an oscillation of incidence that damps out and that takes place almost entirely in the vertical plane, and with ideal initial conditions brings incidence practically to zero. There are two thresholds, and they fall as speed rises: at a Reynolds number of 1.75 times 10 to the fourth the boundary layer stays laminar with maximum incidence below 0.4 degrees and turns turbulent above 0.6, with a transition band in between; at 1.55 times 10 to the fourth the same thresholds are 0.45 and 0.65 degrees.

The arrow is an Easton A/C/E shaft with a 3-36 grain ACE screw-in bullet point and two types of vane, Range-O-Matic Spin Wings and D.S.D Gas Pro. The Reynolds number is computed on the shaft diameter.

The static measurements of lift and pitching moment were made in the wind tunnel with JAXA's 60-centimetre magnetic suspension and balance system, which has no mechanical supports: neodymium magnets inserted in the arrow, external coils, and the forces derived from the currents needed to cancel the weight and balance the aerodynamic forces. 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%. Incidence in the vertical plane was varied quasi-statically between minus three and plus three degrees. Damping was estimated by setting the arrow oscillating with a periodic variation of the current, for both vanes.

The flight tests took place in the indoor facility of the Japan Institute of Sports Sciences, with a compressed-air launcher that produces no flexing of the shaft. Four high-speed cameras recorded at 3000 frames per second: two for the initial conditions fed to the computation, including incidence and spin angular velocity at launch, two for the decay of velocity, from which the drag coefficient is derived. The tests were repeated with different launch conditions: with no conditions imposed, reproducing the ideal attitude as closely as possible, and deliberately forcing a non-ideal attitude; the results are presented as the dependence of the drag coefficient on Reynolds number and as histograms of the coefficient for each condition.

The computation integrates the rigid-body equations of motion, with lift and pitching moment approximated as proportional to the sine of the incidence for small incidences. Since the torque about the arrow's axis cannot be measured with this apparatus, the spin parameter is assumed to stay constant; the damping term is omitted because it came out negligible. The variables and the geometry of the angles are defined in a table and a figure in the paper.

The incidence thresholds are the main result, and they come from computation rather than from a measurement of the boundary layer. What is measured directly is the drag coefficient in flight, that is the effect; the link with maximum incidence runs through the integration of the equations of motion with initial conditions read off the footage and coefficients taken from the tunnel. If the computation's simplifications do not hold, the thresholds move.

Two of those simplifications are declared. The torque about the arrow's axis cannot be measured with this apparatus, so the spin parameter is assumed to stay constant even once an incidence appears. And the lift and moment slopes are those measured at a Reynolds number of 7.7 times 10 to the third, used across the whole range: the Reynolds numbers in question, 1.55 and 1.75 times 10 to the fourth, are more than double. The measured dependence on speed is small, but the computation sets it to zero.

The same number changes from one place to another. The English summary gives the transition above about 0.5 degrees at a Reynolds number of 1.8 times 10 to the fourth; the results give a band between 0.4 and 0.6 degrees at 1.75 times 10 to the fourth and between 0.45 and 0.65 at 1.55; the previous year's paper by the same group gave about 0.45 degrees at 1.77 times 10 to the fourth. These are values of the same order but not the same value, and they all come from the same computational scaffolding.

The programme of tests, moreover, is the 2015 one: this paper extends it rather than verifying it. No other group has repeated these measurements, and the disagreement the 2015 paper could not settle, over the behaviour of the Gas Pro vanes, stays open here too, where those vanes always give the turbulent value.

There is no archer in the flight tests, and that is deliberate: the compressed-air launcher exists precisely to remove the flexing of the shaft. It is the right choice for isolating the aerodynamics, and at the same time the reason these numbers cannot be carried straight over to an arrow shot from a bow, where the launch includes the archer's paradox and contact with the pressure button.

The format remains: four pages of conference proceedings in Japanese, with the results in graphs and histograms. There are no tables of values beyond those of the variables and parameters, the number of arrows and shots per condition is not reported, and there are no measurement uncertainties on the coefficients and no statistical analysis of the histograms. The paper's English title carries a mistake, «archery allow» for «archery arrow».

This paper puts a number on the jump in drag and one on the threshold that triggers it. The drag coefficient goes from about 1.5 to about 2.6 when the layer of air around the shaft turns turbulent, a rise of about 70%, and the threshold is between four and six tenths of a degree of tilt relative to the air. A tilt that small cannot be seen or controlled: it does, however, decide which of the two regimes the arrow flies in.

The second thing to learn is that the threshold falls as the arrow goes faster. At a Reynolds number of 1.75 times 10 to the fourth less than half a degree is enough, at 1.55 times 10 to the fourth slightly more. Converting with the A/C/E shaft's mean diameter, 5.24 millimetres, and air at room temperature, those two numbers correspond to about 50 and 45 metres per second (the conversion is mine, not the paper's). Read that way, it says the most delicate part of the flight is the early part, where the arrow is fast and the tolerance on attitude is at its smallest.

The third result is the most interesting for an archer, and it comes from the tests with different launch conditions. When the authors reproduce the ideal attitude, the arrow stays in the laminar regime for the whole flight. When they force a non-ideal attitude, the drag coefficient also settles at an intermediate value, about 1.8, which means an arrow passing from one regime to the other while in flight. So this is not a matter of a few hundredths of drag: a clean launch and a dirty one put the arrow into two different aerodynamic histories, and the dirty one can change it halfway.

Note where these numbers stop, though. The arrows are fired by a compressed-air launcher that does not make them flex, so the whole effect of the fingers leaving the string and of the archer's paradox is excluded. On vanes, the Gas Pro here always give the turbulent regime, but the same group in an earlier test had obtained the opposite, and the contradiction is unresolved: this is not a basis on which to choose.