Along the shaft the airflow destabilises only under non-symmetric disturbances, and the disturbance always travels towards the nock

The summary today

The answer comes from a single calculation, from 2018, by a Japanese group working on a bare shaft without fletching at recurve speeds: around 60 metres per second, with a Reynolds number between ten and twenty thousand based on the diameter and a shaft about a hundred and twenty diameters long. In the layer of air next to the shaft only the non-symmetric disturbances grow: the two-lobed one, already in the front part, and the bending one, which keeps growing all the way to the nock, while the symmetric disturbance never grows. At lower speeds only the bending mode remains. The group velocity is always positive, so disturbances travel only towards the tail and whatever happens behind cannot destabilise the part of the arrow further forward. The amplification stays small all the same, with an N factor below 2.5 even at the highest speed, and the shape of the point matters only at the front: after about seventy radii the difference between a tapered point and a bulge point disappears. There is an exception right there, though: behind the bulge point, where the flow separates, all disturbances grow over about three radii, the symmetric one included, with a rough estimate of N around 4.5, in a region where the authors themselves say their method is not suited. It is all calculation, on a shaft without fletching, which means without the very part of the arrow that disturbs the air most at the tail; the authors describe it as consistent with their wind tunnel tests, where the boundary layer stayed laminar, but no measurement has checked the individual disturbances. On the state of the flow, an Australian water channel experiment from 2011 actually points elsewhere: with the standard bullet point the air turns turbulent at about 81 millimetres along the shaft for an arrow at 85 metres per second, and behind the short bulge point it stays turbulent along the whole shaft.

Also phrased as: "arrow boundary layer"; "airflow along the shaft"; "turbulence on the shaft"; "laminar flow on the arrow"; "flow instability on the shaft"

Evidence

• 2011
  Context Equipment tests Published later

  A 15.2 to 1 scale model in a water channel, with dye and particle image velocimetry: with the standard bullet point the flow stays laminar over the point and the first part of the shaft and turns turbulent at a Reynolds number of about 450,000, which for an arrow at 85 metres per second means about 81 millimetres along the shaft; with the short bulge point the flow separates behind the bulge and stays turbulent along the whole shaft at the usual recurve and compound speeds. At 3 degrees of angle of attack the transition moves by about 0.1 metres on the model, forward on one side and back on the other, and on average does not change. It says where the air turns turbulent, not which disturbance destabilises it.

  Air flow around the point of an arrow · James L Park, Michael R Hodge, Salam Al-Mulla, Michael Sherry, John Sheridan

• 2018
  For Model Published later

  Linear stability analysis on a model shaft without fletching, with a diameter-based Reynolds number between 1.0 and 2.0 times 10 to the fourth, about 60 metres per second: only the non-symmetric modes grow, the two-lobed one and the bending one, while the symmetric mode never grows.

  Linear stability of the boundary layer flow along an archery arrow · Toshinari Hasegawa, Atsushi Serino, Takeshi Miyazaki, Naoya Takahashi, Reima Iwatsu

• 2018
  For Model Published later

  The two instabilities arise in different places: the two-lobed mode already grows in the front part of the shaft, while the bending mode has a growth rate that keeps increasing to the rear end.

  Linear stability of the boundary layer flow along an archery arrow · Toshinari Hasegawa, Atsushi Serino, Takeshi Miyazaki, Naoya Takahashi, Reima Iwatsu

• 2018
  For Model Published later

  The group velocity is positive in every case calculated, for both modes and across the whole Reynolds range: disturbances propagate with the flow, towards the nock, and the stable region further forward receives no influence from the one behind.

  Linear stability of the boundary layer flow along an archery arrow · Toshinari Hasegawa, Atsushi Serino, Takeshi Miyazaki, Naoya Takahashi, Reima Iwatsu

• 2018
  Context Model Published later

  At lower speeds only bending remains: the two-lobed mode does not grow below a Reynolds number of 1.2 times 10 to the fourth, while the bending mode still grows at 1.0 times 10 to the fourth in the rearmost region. The amplification factor N always stays below 2.5, even at the highest Reynolds number.

  Linear stability of the boundary layer flow along an archery arrow · Toshinari Hasegawa, Atsushi Serino, Takeshi Miyazaki, Naoya Takahashi, Reima Iwatsu

• 2018
  Context Model Published later

  The shape of the point matters only at the front: with the tapered point the two-lobed mode grows slightly more, but beyond about 70 radii from the junction the difference from the bulge point almost disappears; behind the bulge point, where the flow separates near the junction with the shaft, there is instead a region about 3 radii long in which all disturbances grow strongly, the symmetric one included, with a rough estimate of N around 4.5; with the tapered point that region is stable, and the authors warn that their method is not justified there.

  Linear stability of the boundary layer flow along an archery arrow · Toshinari Hasegawa, Atsushi Serino, Takeshi Miyazaki, Naoya Takahashi, Reima Iwatsu

• 2018
  Context Model Published later

  The recurve arrow is a very long, slender body flying at a low Reynolds number: a typical launch speed of around 60 metres per second, a diameter-based Reynolds number of about 2 times 10 to the fourth, and a shaft modelled as a cylinder 120 times as long as its diameter.

  Linear stability of the boundary layer flow along an archery arrow · Toshinari Hasegawa, Atsushi Serino, Takeshi Miyazaki, Naoya Takahashi, Reima Iwatsu

Status history

1. 1990 Not studied In the registry's bibliography nobody had yet studied how the layer of air flowing along an arrow shaft becomes unstable: what was measured on arrows in flight was drag and trajectory, not the behaviour of the air along a long, slender shaft.
2. 2011 Not studied An experiment in a water channel on a scale model shows where the air along the shaft stops being laminar: with the standard bullet point the transition comes at a Reynolds number of about 450,000, about 81 millimetres along the shaft for an arrow at 85 metres per second, while behind the short bulge point the flow separates and stays turbulent along the whole shaft. It says where turbulence begins, not which disturbances make it grow or in which direction they travel: the question in the claim is still unstudied.
3. 2018 Transferred The only piece of work arrives, a linear stability analysis of a bare shaft at recurve speeds, with a Reynolds number of around twenty thousand based on the diameter: only the non-symmetric disturbances grow, the two-lobed one already in the front part and the bending one all the way to the nock, while the symmetric one never grows, and at lower speeds only bending remains. The group velocity is always positive, so what happens at the tail cannot disturb the part of the arrow further forward. The exception is the short region behind the bulge point, where all disturbances grow, but there the method itself does not hold. It is all calculation, on a shaft without fletching and with no accompanying measurement: with models alone the status is transferred.

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