An arrow, unless it is shot vertically upwards, can never fly straight.
Because the arrow vertical velocity is always changing under the effect
of gravity the direction of the air flow (and hence the direction of the total drag force)
is virtually always at an angle to the arrow
shaft. We need to look at the behaviour of a bareshaft arrow at an offset
angle to its flight direction. In the following discussion the drag forces are described in relation to the arrow and to the direction of flight of the arrow as appropriate. From the archer's viewpoint you may be looking at an offset angle in the vertical or horizontal planes. To keep it simple the effect of gravity changing the speed and direction of flight of the arrow is ignored.
The total drag force on the shaft, acting at a right angle to it, acts to decelerate
the arrow in the direction of flight and to accelerate the arrow sideways
(upwards in the diagram). The total drag force on the pile, acting along the axis of the arrow, acts to decelerate the arrow in the direction of flight and accelerate the arrow sideways (downwards in the diagram). Ignoring for the moment any arrow rotation then
the net effect is a deceleration of the arrow in its direction of flight
and a sideways (upwards) acceleration. If the arrow stays with a fixed
orientation then it would end up flying in a curve.
In practice the arrow will rotate under the net effect of the drag and vortex shedding torques. As these rotate the arrow in opposite directions, depending on which is the stronger effect the pile of the arrow will rotate towards the direction of flight
(drag the winner) or away from the direction of flight (vortex shedding
the winner). The resulting flight of the arrow is very different with different
directions of rotation.
If the arrow pile rotates towards
the direction of flight then as the arrow flies along in a curved path
the offset angle steadily reduces to zero and hence the 'sideways' component of the total drag reduces to zero. Because the arrow has acquired rotation (angular momentum)
then it keeps rotating until the overall drag torque brings it to a halt. The arrow
ends up with an offfset angle opposite to the one it started with and so
the process repeats with the 'sideways' drag force now in the opposite
direction. The arrow therefore flies in an "S" shaped pattern about a mean
flight path.
The mean flight path of the arrow is not a straight line but a very gradual curve. This curvature results from the arrow gradually losing velocity to the total drag and hence the 'sideways' drag forces on the arrow also decrease. Each time the arrow zigs and then
zags the sideways movement on the zig is greater than that on the zag because
the arrow is travelling faster when zigging than zagging. With each zig-zag
combination the arrow ends up fractionally moved sideways. This effect
cumulates over the arrow flight to produce an overall displacement. This
displacement even at 90 metres is only very small and most archers’ essentially
regard the flight as straight.
A second point is that the fishtailing/porpoising of the arrow represents
an energy oscillation between rotational energy and pressure potential energy
(think of a swinging pendulum). This 'fishtailing energy' is lost via the
drag on the fletching surface and the appropriate section of arrow shaft
so the fishtailing/porpoising is steadily reduced by this drag damping mechanism.
If the arrow pile rotates away from the direction of flight then as the arrow flies along the offset angle steadily increases and hence the overall drag force keeps increasing. The
arrow is decelerated more and more in the original direction of flight
and accelerated more and more sideways. In effect the direction of flight
itself rotates. The arrow continues to rotate away from the current flight
direction. The overall effect, assuming the arrow does not hit the ground,
would be for the arrow to end up flying in an ever increasing spiral
as its speed and hence drag forces keep decreasing.
What determines which way an arrow
rotates is the position of the centre of gravity. As previously discussed
as the centre of gravity moves towards the pile end of the arrow the shaft
drag contribution to arrow rotation rapidly increases. The torque from vortex
shedding, which acts at one point, slowly increases as the centre of gravity
moves forward as its 'lever arm' gets longer. The relative strengths of
the two rotational effects as it relates to the centre of gravity is shown
in the following diagram.
The diagram illustrates how the turning effect from drag (red lines)
and vortex shedding (blue lines) vary as the centre of gravity is moved
forwards from the arrow centre. If the centre of gravity is behind point
"A" then the vortex shedding effect is the stronger and the arrow will
fly in a curved path. If the centre of gravity is in front of point "B"
then the arrow will fly straight in the "S" pattern as the drag torque
effect is stronger. If the centre of gravity lies between "A" and "B" then
the arrow can fly either way at any specific instant depending on the arrow's
current rotational characteristics. Each torque is shown as two lines because
the arrows' rotation has an effect on the air velocity and hence
drag and vortex shedding torques.
The cross over point for actual arrows between flying straight or in a curve
corresponds roughly to around an 8% FOC. It depends a fair bit on the geometry of the
nock as this will effect the magnitude of the Munk moment. So with aluminium arrows
if a 7% FOC pile is fitted the odds are that the arrow will fly in a curve, with
a 9% FOC pile the odds are that it will fly straight. One would expect most if not
all carbon arrows to fly straight as their centre of gravity is further
forward. A side effect of this behaviour is that it's much easier to bare shaft
tune a low FOC arrow as it curves away a lot more than a high FOC arrow.
In the mists of time when hunters used pointed sticks as arrows it was
difficult to hit distant targets because their arrows flew in a curve.
The along came some early Einstein who discovered that if you stuck a bunch
of feathers on the back of the arrow it would fly in straight line over
any distance. Adding fletchings significantly increases the drag torque
rotating the pile towards the direction of flight. The arrow will always
fly straight irrespective of where the centre of gravity is.
The total drag force on the shaft, acting at a right angle to it, acts to decelerate
the arrow in the direction of flight and to accelerate the arrow sideways
(upwards in the diagram). The total drag force on the pile, acting along the axis of the arrow, acts to decelerate the arrow in the direction of flight and accelerate the arrow sideways (downwards in the diagram). Ignoring for the moment any arrow rotation then
the net effect is a deceleration of the arrow in its direction of flight
and a sideways (upwards) acceleration. If the arrow stays with a fixed
orientation then it would end up flying in a curve.
If the arrow pile rotates towards
the direction of flight then as the arrow flies along in a curved path
the offset angle steadily reduces to zero and hence the 'sideways' component of the total drag reduces to zero. Because the arrow has acquired rotation (angular momentum)
then it keeps rotating until the overall drag torque brings it to a halt. The arrow
ends up with an offfset angle opposite to the one it started with and so
the process repeats with the 'sideways' drag force now in the opposite
direction. The arrow therefore flies in an "S" shaped pattern about a mean
flight path.
If the arrow pile rotates away from the direction of flight then as the arrow flies along the offset angle steadily increases and hence the overall drag force keeps increasing. The
arrow is decelerated more and more in the original direction of flight
and accelerated more and more sideways. In effect the direction of flight
itself rotates. The arrow continues to rotate away from the current flight
direction. The overall effect, assuming the arrow does not hit the ground,
would be for the arrow to end up flying in an ever increasing spiral
as its speed and hence drag forces keep decreasing.
What determines which way an arrow
rotates is the position of the centre of gravity. As previously discussed
as the centre of gravity moves towards the pile end of the arrow the shaft
drag contribution to arrow rotation rapidly increases. The torque from vortex
shedding, which acts at one point, slowly increases as the centre of gravity
moves forward as its 'lever arm' gets longer. The relative strengths of
the two rotational effects as it relates to the centre of gravity is shown
in the following diagram.