Arrow aerodynamics in recurve shooting: Shen 2025 study

No archers and no arrows shot: this is a computational study, in two steps. First the authors simulated the air around an Easton X10 arrow of 19.7 grams and 62.5 centimetres with computational fluid dynamics, to find where the aerodynamic pressure centre falls, the point at which the resultant of the air forces can be taken to act. It sits towards the tail, at around 48.5 centimetres from the point, and it is not fixed: it shifts by a few millimetres as the angle of attack changes, the angle between the arrow's axis and the direction in which it is actually moving. From those simulations they derived two empirical equations describing that shift.

In the second step they inserted those equations into a rigid-body trajectory model and flew the arrow, in the computer, over 70 metres at 57 metres per second, with initial angles of attack from minus 3 to plus 3 degrees and uniform wind from 1 to 3 metres per second. The comparison is with the same model holding the pressure centre fixed at 48.4 centimetres, as was previously done.

The most readable result concerns wind direction. A crosswind moves the arrow far more than a headwind or tailwind: at 3 metres per second across, the computed deviation is 0.258 metres, while at 3 metres per second against or behind it sits between 0.078 and 0.085, about a third as much. Between 1 and 3 metres per second of crosswind the deviation goes from 0.091 to 0.258 metres. The starting conditions matter just as much, though: with the same 3 metres per second crosswind, an arrow leaving with a 3 degree angle of attack drifts 0.27 metres, one leaving with zero angle 0.125.

Two properties of the arrow pull the same way. The heavier the arrow, the less it drifts: at 1 and 3 metres per second of crosswind a 14.2 gram arrow deviated by 0.12 and 0.322 metres, a 29.1 gram one by 0.078 and 0.229. And the further forward the balance point, the less it drifts: with the centre of mass 18.5 centimetres from the point and a 3 metre per second crosswind the deviation is 0.116 metres, and moving the balance back to 20 centimetres increases it by 0.03 metres. The authors add, however, that a more forward balance point makes the trajectory drop more at the end, at medium and long ranges, and calls for a different launch angle.

The comparison between the two models gives the figure that appears in the paper's abstract: letting the pressure centre move, the predicted deviations are 13 to 17% smaller under crosswind and 6 to 9% smaller under head or tailwind, which the authors summarise as an improvement in accuracy of 15% and 8%.

The fluid dynamics part solves the mass and momentum conservation equations with a Reynolds-averaged k-epsilon turbulence model. The geometry is a complete Easton X10: a shaft of carbon fibre and aluminium composite, a tungsten steel point, three thin plastic vanes, with a mass of 0.0197 kilograms, a length of 0.625 metres, a shaft radius of 2.4 millimetres and a vane thickness of 0.1 millimetres. The inlet velocity is 57 metres per second, which the authors give as the typical initial velocity of a top archer at 70 metres. The computational mesh is unstructured tetrahedra, with three refined layers around the arrow's surface and greater density near the point and the nock; grid independence was checked by comparing surface pressure across four configurations at three different angles of attack.

From the pressure field the authors derive the pressure coefficient point by point, the pressure vector and, integrating over the surface, the position of the pressure centre. In the vertical and transverse directions this position is held fixed, so the only variation computed is the one along the arrow's axis as the angle of attack changes. The turbulence model was verified by reducing the arrow's velocity to 23 metres per second, a Reynolds number of 7.7 thousand, and comparing the lift coefficient and the pitching moment coefficient against already published wind tunnel data: a mean error of 6.44% and a maximum of 9.31% on lift, a mean of 4.21% and a maximum of 8.07% on the moment.

The ballistic part is a rigid-body model: the arrow's deformation in flight is neglected, on the grounds, taken from the literature, that significant bending only occurs over the brief stretch just after the arrow leaves the string. The arrow's aerodynamic characteristics are not measured here but taken from a wind tunnel study by Ortiz: the two parameters tied to lift and pitching moment, two separate drag coefficients for laminar and turbulent boundary layers, the spin parameter, the position of the centre of gravity and the two moments of inertia. The equations of motion are written in a ground-fixed frame, with wind as a uniform vector and relative velocity as the difference between the centre of mass velocity and the wind; from these the angle of attack is obtained instant by instant. The whole was programmed in MATLAB.

The conditions explored are: 70 metres distance, 57 metres per second initial velocity, initial angle of attack from minus 3 to plus 3 degrees, crosswind, headwind or tailwind from 1 to 3 metres per second, that is, a light breeze, with zero initial angular velocity in the main tables. Masses were varied from 14.2 to 29.1 grams and the centre of mass position between 18.4 and 20.0 centimetres from the point, keeping the total mass fixed.

No arrow was shot. Every deviation figure on the target is the output of a model, and the model was not compared against measured impacts. The only experimental check concerns two aerodynamic coefficients, lift and pitching moment, compared with wind tunnel data at a velocity reduced to 23 metres per second, less than half the 57 metres per second then used in all the simulations.

The figure the paper puts in its abstract needs reading carefully. The 15% under crosswind and the 8% under head or tailwind are the difference between the deviations predicted by two models, the one with a fixed pressure centre and the one with a moving pressure centre. The authors call it an "improvement in accuracy", but nobody measured which of the two models better predicts where the arrow lands: it is the distance between two predictions, not a verified gain. From the table, moreover, the value runs from 13 to 17% across crosswind speeds and from 6 to 9% under head or tailwind, and the 15% and 8% are roundings.

The model simplifies in ways that matter. The arrow is a rigid body and bending in flight is neglected, justified with a citation rather than a check. The wind is uniform and steady over all 70 metres, while the wind on a competition field changes in gusts, in direction and with height. The pressure centre is computed only along the arrow's axis and held fixed in the other two directions. The basic aerodynamic characteristics, including the two drag coefficients and the spin parameter, do not come from this study but from an earlier wind tunnel work. The main tables are computed with zero initial angular velocity, while a real arrow with helical vanes spins.

The arrow modelled is a single one. Varying the mass from 14.2 to 29.1 grams is a parameter exercise rather than a comparison between existing arrows: an X10 of 62.5 centimetres weighing 14.2 grams is not something that can be bought, and changing the mass without changing stiffness and geometry gives something no real arrow reproduces. The same holds for shifting the balance point with total mass unchanged. Spine is not discussed, and with a rigid arrow it could not enter the model at all.

Then there is the reach of the practical conclusions. The recommendations the authors address to archers, coaches and manufacturers come from the model rather than from tests on a shooting line, and one of them contradicts itself: a more forward balance point reduces lateral drift but, they write, accentuates the drop of the trajectory at the end over medium and long ranges. Finally, the computed deviations are a few tens of centimetres on a 122 centimetre face, so of the order of the scoring rings: a comparison with an archer's real dispersion, which the model does not contain, would be needed to know how much of that deviation would be visible.

What this work lines up is the hierarchy of wind. In the model, at equal speeds, a crosswind moves the arrow about three times as much as a wind in the face or at the back: at 3 metres per second it goes from 0.258 metres of lateral drift to 0.078 to 0.085 metres of deviation under head or tailwind. It is an indication of why a field with a crosswind is a different job from one with wind along the shot, and it comes from a calculation rather than from arrows shot.

The second number concerns the launch rather than the wind. With the same 3 metre per second crosswind, an arrow leaving with a 3 degree angle of attack ends up 0.27 metres from the predicted point, one leaving aligned 0.125: twice as far. In the model, then, how straight the arrow leaves weighs as much as the wind it meets, and in archery terms that concerns how it comes off the string, that is, tuning and release.

On the arrow itself the model says two things that run with what was already expected: heavier drifts less, with a 29.1 gram arrow deviating 0.229 metres in a 3 metre per second crosswind against 0.322 for a 14.2 gram one, and a more forward balance point drifts less, by 0.03 metres when the centre of mass moves from 20 to 18.5 centimetres from the point. These are variations obtained by changing one parameter at a time inside a calculation rather than comparisons between real arrows, and the authors themselves note that a more forward balance makes the trajectory drop more at the end.

What this record does not license is treating the 15% as precision gained. That figure says how far the two versions of the model diverge from one another, not how much better an archer shoots: finding out whether the arrow goes where this model says would take a field, an anemometer and a series of measured impacts, and this study has none of those.