Boundary layer along the arrow: Hasegawa 2018 study

This fluid dynamics paper calculates whether the thin layer of air flowing along the arrow shaft, the boundary layer, stays smooth (laminar) or tends to become turbulent. The question matters in practice: the same group had measured in a wind tunnel and in flight that an arrow aligned with the airflow has very different drag from one inclined by about a degree, because in the second case the boundary layer becomes turbulent. The calculation covers a bare shaft with two points, a streamlined one and a blunter competition point (a 'bulge' in the English abstract), at Reynolds numbers between 10,000 and 20,000; the highest value corresponds to a recurve arrow at about 60 m/s.

The disturbances that grow fastest are those that deform the boundary layer with two waves around the shaft and, just behind, the bending mode with a single wave; the symmetrical disturbance, which dominates on flat surfaces, never grows. The shape of the point matters only along the front part of the shaft, where the flow is slightly less stable with the streamlined point; from about 70 radii back the difference disappears. The amplification factor of the disturbances, N, stays below 2.5 even at a Reynolds number of 20,000, far from the value of about 9.81 at which, using the flat-plate formula and the turbulence level of the wind tunnel the group used, transition would begin. With the arrow aligned, then, the boundary layer can stay laminar, as it did in the wind tunnel tests.

The shaft is treated as a cylinder 120 diameters long, with the point at the front. The authors first calculate the base flow around an arrow aligned with the airflow, solving the Navier-Stokes equations for flow symmetrical about the axis on a grid of 1,201 by 301 points; they then study how small disturbances added to that flow grow, using the parallel flow approximation and a 64-point Chebyshev spectral method, checked by reproducing known results for a flat plate. Disturbances with zero, one and two waves around the shaft are considered.

Growth along the shaft is summarised in the N factor of the eN method, a semi-empirical criterion for predicting transition to turbulence. As a benchmark the authors use a formula developed for flat plates, which links the critical value of N to the turbulence of the incoming air (0.05% in the magnetic suspension wind tunnel where the group had made its measurements), and they state themselves that applying it to a cylinder is not justified: it is an indicative reference.

This is a calculation presented at the annual congress of the Japan Society of Mechanical Engineers: five pages in Japanese, the last of them references only, not peer reviewed. The model arrow has no fletching, does not spin, does not flex and flies perfectly aligned with the airflow, whereas a real arrow leaves the bow flexing, at a small angle of attack, and spins because of its fletching; and a small angle of attack is exactly the condition that, according to the same group's measurements, makes the boundary layer turbulent. The authors leave these cases to future work.

The comparison with the critical value of N rests on a flat-plate formula and on the very low turbulence of a wind tunnel; outdoors the air is far more disturbed, and the margin between 2.5 and 9.81 could shrink considerably. Just behind the competition point, where the flow separates and reverses, the calculation shows strong instability (N of around 4.5 on a rough estimate), but the approximation used does not hold there, and the authors defer to a more complete analysis.

For an archer this is one piece of a larger body of work on air resistance. An arrow's drag can change a great deal depending on whether the boundary layer stays laminar or becomes turbulent, and this calculation says that with the arrow well aligned it tends to stay laminar along the shaft, and that the shape of the point affects the flow only over the first stretch.

It does not say which point to choose, since the calculated differences are small and confined to the front of the shaft, and it says nothing about fletching, spin or wind. The part that actually concerns flight, how much an arrow slows down when it is even slightly inclined, lies in the same group's wind tunnel and flight measurements, not in this paper.