When an arrow is shot from a bow a major proportion of the energy stored in the bow limbs is transferred to the arrow as energy of motion (Kinetic Energy). The arrow velocity depends
on how much kinetic energy it has, the higher the velocity the higher the
kinetic energy. As the arrow travels through the air there is a momentum change between the air and the arrow. The air gains momentum the arrow loses it. The momentum change results in a change of arrow velocity, either the arrow speed and/or the arrow travel direction. A momentum change is the result of the action of a force. The momentum change on the arrow resulting from aerodynamic effects is called the drag force.
There are a number of drag force mechanisms that act on an arrow, popularly known as viscous (shear) drag, pressure (inertial) drag, the Magnus effect and
the Leaf Fall (Munk or rudder) effect. The Magnus effect and the Munk effect both relate to pressure forces created by asymmetric separation of the fluid flow from the arrow
shaft surface. They are considered on other pages (Vortex Shedding and the Magnus effect).
Shear Drag is an energy loss through air friction. Because the air is viscous, as the arrow shaft and fletchings fly along they drag the air around them along as well. This air drags the adjacent air along as well and so on. Around the arrow you have layers
of air travelling at different velocities and this generates friction.
The following diagram illustrates how this layer of air affected
by the arrow's passage grows along the shaft and fletchings of an arrow
flying perfectly straight. The angle of the 'cone' of affected air to the
shaft is about one degree, i.e. at the back of a typical arrow the
shaft affected air zone runs to about 1 to 1.5 centimetres from the shaft.
Shear drag acts in the direction of the air flow.
Form drag results from the creation of a turbulent wake behind the arrow. At it's core it is the reaction to a change of momentum to the air flow. As the air flows over the arrow pressure and shear forces create eddies or vortices behind
it which drop off the arrow forming a wake. The faster the arrow is travelling the more turbulence is created and therefore the higher the drag force. The bigger the
wake is then the more turbulence it has and therefore the higher the drag force. Form drag is regarded as acting at a right-angle to the arrow surface.
Form drag is in many applications broken down into components referred to as Drag, Lift and Pitching Moment each with their own "Coefficients".
An alternative view of form drag, hence the pressure drag description, is that it is the consequence of the dynamic flow pressure acting on the body. With a
stationary arrow the air pressure acting over its surface all cancels out and so there is no net force. A moving body experiences a dynamic flow pressure
P = 0.5DV2 force per unit area where D is the fluid density and V the fluid velocity. Total pressure on a moving body is just the dynamic flow pressure
multiplied by the area 'A' and by a momentum transfer efficiency coefficent Cd. This gives you usual engineering equation for pressure drag i.e.
Pressure drag force = 0.5DV2ACd.
Another factor relating to form drag is the flow regime within the boundary layer. You basically have two possibilities;
Laminar boundary and turbulent wake
Turbulent boundary and turbulent wake
Which regime you have basically depends on the flow Reynolds number. When the boundary layer switches from laminar to turbulent you get a significant drop in the
drag on the object (result of delayed flow separation). There has been almost zero investigation of recurve arrow aerodyamics but as far as I know no-one has
observed the drop in an arrow drag coefficient associated with a boundary layer laminar to turbulent transition. (Which doesn't establish it never happens :) ).
For an arrow, form drag is many thousands of times stronger than shear drag. If the arrow is flying at any offset angle to its direction of flight then the Form drag and Shear drag essentially combine into a total drag as the air flow is across the shaft
(ref).
Shear drag can therefore usually be ignored as a separate effect. It
would only need to be looked at separately if the arrow was travelling
very slowly or the archer was shooting in a tank full of e.g. treacle.
Further consideration of the effect of drag on an arrow is done in terms
of Form drag acting at a right-angle to the surface.
Sometimes you see the drag force quoted as being proportional to the velocity and sometimes as being proportional to the square of the velocity. Which one is correct? Well they both are, it depends on the situation. There are two main drag forces, viscous (shear) drag proportional to the velocity and inertial (form) drag proportional to the square of the velocity. At very low velocities (strictly speaking low Reynolds number) the only
significant drag force comes from viscosity and so the drag force is proportional
to the velocity (this situation is called the 'Stokes Regime' and is described
by Stokes equation for viscous drag ). For an arrow typically travelling
at 150-350fps the inertial (form) drag is thousands of times greater than the
viscous drag so the overall total drag is proportional to the square of the velocity. This is why the usual basic drag force equation includes the fluid density
but there is no term for the fluid viscosity.
Drag acts on the arrow pile, shaft and fletchings. The most complicated effect is on the arrow shaft and this will be discussed first. Drag effect on the pile and fletchings
is then easily described.
If you push an arrow at any point along its shaft then the effect is to rotate the arrow about a specific point. The push generates a torque, defined as the
force multiplied by the distance to the point of rotation. When you push an arrow (Fa) at
a distance "a" from the arrow centre of gravity then the arrow rotates
about a point at a distance "b" the other side of the centre of gravity.
The relationship between "a" and "b" is that if you multiply them together
then for a specific arrow you always get the same answer. In other words
the nearer to the centre of gravity you push the arrow then the further
away is the point about which the arrow rotates.
The torque along an arrow shaft from drag at distance "a"(Ta)is given by:
Ta = Fa (a + k/a )
Where "a" is the distance of the drag force point from the centre of gravity
Ta is the torque on the arrow at "a"
Fa is the drag force on the arrow at distance "a"
k is a constant value ("a" multiplied by "b")
(a + k/a ) is the distance from the applied drag force to the point of arrow rotation
The following graph illustrates how the torque from drag varies along the arrow shaft. Basically the torque drops sharply from infinity at the centre of gravity down to a minimum
value at some point and starts to steadily increase again. Effectively
the only thing that makes the drag force vary along the shaft is arrow
rotation i.e. with rotation the air velocity will vary along the shaft.
As the rotational velocity of an arrow is much smaller than the linear
velocity the drag force can be regarded as essentially constant along the
shaft.
There are two key points to notice. Firstly the direction of rotation from drag from the front of the shaft to the centre of gravity (CG) is in the opposite direction
to the drag from the CG to the nock end of the shaft. Secondly the variation
of the size of the torque around the centre of gravity is close to being
symmetrical. All the torque on the arrow from the front of the shaft to
around twice the distance from the front of the shaft to the CG cancels
out. The effect of the drag force on this section of the shaft is to move
the arrow as a whole. The drag on the remaining length of the shaft at
the rear, as well as moving the arrow as a whole, generates a torque to rotate the arrow. The further forward the
centre of gravity of the arrow is then the higher the proportion of the total drag on the shaft that acts to create a turning moment on the arrow.
(This is why the FOC of an arrow is such an important property).
The behaviour of drag force on the arrow pile is exactly the same as the effect on the shaft.
In general because of the shape of the pile, its symmetry about its centre of gravity and being
at right-angles to the shaft there is no resulting rotational
torque and the direction of the drag force is along the axis of the
shaft and hence through the arrow's centre of gravity. The drag force on
the pile therefore has the effect of moving the arrow as a whole
in the direction along the axis of the arrow . Drag on the pile can induce
a rotational torque on the arrow if the pile is "bent" or has a shape such
that the direction of the pile drag force is not directed through the overall
centre of gravity of the arrow when having a high enough offset angle.
(A broadhead is the obvious example).
Drag on the fletching surface acts exactly as the drag on the rear part of the shaft, it all acts to rotate the shaft. Frictional (viscous) drag effects are negligible.
Fletchings have some thickness and will have an overall 'edge' area depending
on the thickness, number of fletchings and fletching profile.
Drag on the edge area acts like drag on the pile, it has the effect
of moving the arrow as a whole.
How you define drag on an object is somewhat arbitrary. For example with the currently best investigations into arrow aerodynamics (references on the site
home page) Drag, lift and pitching moment drag coefficents are used as applying to the arrow as a complete unit. This is really the only sensible approach on a
practical experimental basis as the various arrow components are interactive in an aerodynamic sense. Talking about the individual drag properties, as was commonly
done in the early 20 century and as I have done above is not really a valid approach.
Shear Drag is an energy loss through air friction. Because the air is viscous, as the arrow shaft and fletchings fly along they drag the air around them along as well. This air drags the adjacent air along as well and so on. Around the arrow you have layers
of air travelling at different velocities and this generates friction.
The following diagram illustrates how this layer of air affected
by the arrow's passage grows along the shaft and fletchings of an arrow
flying perfectly straight. The angle of the 'cone' of affected air to the
shaft is about one degree, i.e. at the back of a typical arrow the
shaft affected air zone runs to about 1 to 1.5 centimetres from the shaft.
Shear drag acts in the direction of the air flow.
Form drag results from the creation of a turbulent wake behind the arrow. At it's core it is the reaction to a change of momentum to the air flow. As the air flows over the arrow pressure and shear forces create eddies or vortices behind
it which drop off the arrow forming a wake. The faster the arrow is travelling the more turbulence is created and therefore the higher the drag force. The bigger the
wake is then the more turbulence it has and therefore the higher the drag force. Form drag is regarded as acting at a right-angle to the arrow surface.
Form drag is in many applications broken down into components referred to as Drag, Lift and Pitching Moment each with their own "Coefficients".
