Wind and arrow trajectory: Ortiz 2018 study

A numerical simulation of the flight of an Easton A/C/E arrow treated as a rigid body, run to estimate how far background wind pushes it off the centre of a target at 70 metres. The arrow weighs 14 grams, is 626 mm long with an outer radius of 2.60 mm, and leaves at 57.3 metres per second from a height of 1.5 metres towards a centre set 1.3 metres above the ground.

With a 3 metre per second side wind the angle of attack oscillates up to a maximum of 3.0 degrees and the arrow lands 0.36 metres from the centre; with a 1 metre per second side wind the deviation falls to 0.11 metres. A wind of the same strength, 3 metres per second, but head-on or from behind moves it far less: 0.10 metres into the wind and 0.09 metres with it, and in those two cases the maximum angle of attack stays below one degree (0.45 degrees head-on, 0.54 from behind). Side wind, in short, counts for much more than wind in line with the shot.

The other result concerns the launching conditions. The authors compare what they call seemingly ideal conditions, that is zero angle of attack and zero angular velocity at release, which is what archers and coaches usually take to be ideal, with what they call truly ideal conditions, in which the arrow leaves already aligned with the relative airflow. In the second case the angle of attack stays two orders of magnitude smaller, the boundary layer may remain laminar instead of turning turbulent, the drag coefficient falls from 2.69 to 1.50 and the lateral deviation comes out markedly smaller for the same wind. The oscillation of the trajectory shows up mostly over the first twenty metres, because the amplitude of the angle of attack decays over the flight, at a rate of 0.87 per second, through the stabilising effect of the vanes.

The equations of motion are those of a rigid body subject to gravity, drag, lift and a pitching moment, solved with a fourth-order Runge-Kutta method at a time step of half a millisecond. The wind is a three-component vector and is assumed uniform over the whole trajectory, that is identical at every point of the range. The angle of attack is defined as the angle between the arrow's axis and its velocity relative to the air.

The aerodynamic coefficients are not taken from the literature but measured by the same group in earlier work: in the wind tunnel with JAXA's 60 cm magnetic suspension and balance system, in which the model hangs without any support, and in free flight tests recorded with high-speed cameras. From these come the lift coefficient proportional to the angle of attack, the pitching moment coefficient, the spin parameter and the laminar value of drag; the free flight tests give the mean initial velocity and the turbulent value. The two drag values used are 2.69 in the turbulent state and 1.50 in the laminar one; air density is 1.20 kg per cubic metre.

Which of the two applies depends on the angle of attack: the group had already identified a transition zone between 0.40 and 0.60 degrees for an A/C/E shaft at the reference Reynolds number, so above 0.40 degrees the turbulent coefficients are used and well below it the laminar ones.

One parameter was not measured: the axial component of the torque, for which no wind tunnel data were available. The authors had to introduce the assumption that the arrow's rotation rate about its own axis is independent of the angle of attack and proportional to the relative velocity, an assumption they checked during the tunnel tests and which holds only for angles of attack within 3.0 degrees.

The launching angles were adjusted so that with no wind the arrow hits the centre; the vertical and horizontal deviations are then measured from that point, and the radial distance is their combination. Shaft flex is not in the model: the arrow is rigid.

The arrow is rigid: no flexural oscillation, and so no archer's paradox, no pressure button, nothing of the behaviour over the first few metres that other models describe. The authors state this and explain that the simplification serves to isolate the phenomenon they are after, the transition of the boundary layer from laminar to turbulent.

The wind is uniform over all 70 metres and constant in time. A real field is not like that: gusts, the gradient near the ground and shifts in direction are exactly what makes life hard for an archer, and none of them is here. Values of 1 and 3 metres per second are light winds; the work does not reach the strengths seen at windy competitions.

One parameter is assumed rather than measured: the arrow's rotation about its own axis, for want of data on the axial torque. The assumption is stated to hold within 3.0 degrees of angle of attack, which is exactly the maximum reached in the simulation with a 3 metre per second side wind: the most extreme case studied sits on the edge of its own range of validity.

The results apply to one shaft only, the A/C/E with that point and those vanes, at that initial speed. The comparison between seemingly ideal and truly ideal launching conditions is a comparison between two simulations, not between two archers: whether an archer can send the arrow off already aligned with the relative flow, and how repeatably, is not measured here, and the authors write that the task appears very difficult. There is no real target and no score anywhere in the work: it is a short conference paper, with the experimental grounding of the coefficients left to the group's earlier articles.

The figure to take away is the ratio between the two winds. At 3 metres per second, which is a light breeze, a crosswind carries the arrow 36 centimetres off the centre, while the same wind head-on or from behind moves it 9 or 10 centimetres. On a 122 centimetre face that is the difference between losing a couple of rings and losing almost nothing. At 1 metre per second of side wind the deviation falls to 11 centimetres, less than a third, which gives a sense of how fast the problem grows with wind strength.

The second part is more interesting and less comfortable. The simulation says that two identical arrows, launched into the same wind, end up in different places depending on how they were oriented relative to the airflow at the instant they left the bow. An arrow that leaves already aligned holds a very small angle of attack, its boundary layer may remain laminar, drag nearly halves and the drift falls. An arrow that leaves with zero angle of attack relative to its own direction of travel, which is the condition usually taken to be perfect, is not aligned with the flow when there is a crosswind, and it oscillates.

This is not a tuning instruction: the authors do not say how such a launch is achieved, and they observe that it appears very difficult. What remains is an indication of where to look when, in a steady wind, two apparently identical arrows separate from one another: the model suggests the difference may arise in the first few thousandths of a second rather than along the flight.

All of this applies to an A/C/E at 57.3 metres per second, in a uniform wind with no gusts. A real competition field is another matter.