The aerodynamics of arrow flight is really complicated. At launch the arrow experiences a 500 G acceleration causing all sort of complex superimposed motions - vibrational, rotational and sideways. These all impact arrow aerodynamics but ultimately distill down to just two important characteristics - arrow drag and directional stability.
Arrow aerodynamics is briefly covered here to convey a feeling for the issues without going into excessive detail. The detail may still look excessive, but be assured, it is greatly simplified!
An arrow is an assembly of parts, each designed for a specific function.
Most bow energy is transferred as kinetic energy in the forward motion of the arrow. However, a small proportion of this energy is miss-directed causing the arrow to have a combination of:
a bending resonance in one or more modes,
moving slightly sideways,
rotating about the center of gravity,
During launch the nock is assumed to move in a straight line, but this is rarely the case. Imbalance of limbs and cams, off-center shot bow, miss-alignments of components, vibration and balance, archer-form, bow twist and release issues. All these combine to make it difficult to predict the precise condition of the arrow as it leaves the string. This means the initial conditions for the aerodynamic calculations are also vague.
When describing an arrow's flight, we need to share a common terminology and positional reference system. The axis system we use has its origin at the arrow's center of gravity (CoG), the x-axis aligned with the air-stream, the y-axis horizontal and at right-angles to the air-stream, and the z-axis at right-angles to the other two axis:
Arrow axis system showing the yaw and pitching planes. Note that the axis system aligns air-stream, not the arrow. The diagram shows the arrow aligned with the air-stream.
The arrow's angle of attack or offset from air-stream is the angle between the arrow's longitudinal (spin) axis and the x-axis.
Note that in arrow aerodynamics there is no concept of roll. Arrow spin is assumed but largely ignored except for the induced drag during spin-up.
Archery has a long tradition of specialist terms for various arrow flight characteristics but they can be vague or ambiguous. To help resolve this, aeronautical terms are used where appropriate.
Drag is the aerodynamic force that acts to slow an arrow in the direction of travel. It is proportional to the frontal area, the square of the speed (assuming turbulent flow) and the aerodynamic form expressed as the drag coefficient. The standard drag equation is:
FD = 1/2 ρ v2 CD A
The details of this are covered below.
Despite its sleek appearance, the typical arrow's aerodynamic form is not particularly good. The drag can be estimated from our knowledge aerodynamics, but presently no one can claim high precision from theory. Direct or indirect measurement is required for reasonable accuracy.
Lift is the force that keeps an aeroplane in the air. But it not only does that, it provides the force needed to turn an aeroplane when banking. It is closely related to drag but at right-angles to the direction of travel. (In reality lift and drag are components of a single force acting on the arrow, that we arbitrarily choose to describe this force).
The standard lift equation is similar to the drag equation:
FL = 1/2 ρ v2 CL A
where the lift coefficient CL is proportional to the angle of attack for small angles. The details of this are covered below.
A well launched and stable arrow will not experience lift, however should the arrow fish-tail (yaw) or porpoise (pitch) lift is generated in the direction of the angle of attack. The magnitude of this lift is not large, but sufficient to result in path deviations and the drag it induces slows the arrow. It is lift on the fletching that provides most of the arrow's orientation correction.
It is interesting to note that a bare shaft with an angle of attack will experience lift and hence path deviation in flight.
The center of pressure is the point on the arrows axis where the aerodynamic forces are deemed to be balanced and applied. This point may move a little in flight depending on flight speed and angle of attack, potentially allowing a marginally stable arrow to become unstable. Larger fletching will move the CoP backward and broadhead blades will move it forward.
For stability the CoP should always be behind the center of gravity otherwise the arrow will oscillate or even reverse orientation during flight with significant deviation of flight path. See Measurements, Tips and Tricks to find out how to determine CoP.
Front of Center indicates the location of the Center of Gravity (CoG) of an arrow expressed as a percentage of arrow length forward of the arrow center. Typical FoC values range from 7% to 18%. Usually the standard length (nock valley to front of shaft) is used, although in reality the aerodynamic length would be more appropriate.
See Measurements, Tips and Tricks to find out how to determine CoP
Fletching significantly impacts an arrow's aerodynamics. Its principal purpose is to move the center of pressure rearward for improved stability. Other functions include inducing spin and aesthetics.
Generally, fletches provide drag and lift, both of which provide a correcting moment to rotated the arrow into alignment with the air stream. The lift provides the most efficient stabilizing action as the correcting force and associated induced drag is only generated when required. (Note that some flu-flu fletches provide little or no lift, only drag, so are deliberately less efficient in this regard).
The sum of the lift and drag components gives the correcting moment Mc :
Mc = L d + D d tan( α )
where d is the distance between the CoM and the CoP of the fletches. Also, both the lift L and drag D forces are approximately proportional to the angle of attack α.
The optimal proportion of drag to lift is an interesting question with no definitive answer. Most modern fletching ensures lift is the major corrective force as it is generated only when required, reducing drag on the arrow when aligned to the air stream.
Over sized low drag fletches can cause overshoot and even continuous oscillation due to excessive correction and low dampening.
Straight offset fletches are a simple compromise to the slightly more aerodynamically efficient helical fletch. With these fletches the offset angle is the same at all radii, with the result that when spinning, the actual angle of attack will increase with distance from the arrow surface. This is because the velocity vector due to the spin increase with distance from the arrow spin axis. The result is higher drag at the steady state speed because there will be a slight negative pitch at the inner part of the fletch working against the slight positive pitch at the fletch tips.
Helical fletches are notionally more aerodynamically correct in the sense that the angle of attack is the same at all distances from the arrow surface (at the steady state spin rate). The result is the steady state spinning drag is more or less the same as if there were not offset with straight fletches.
Helical fletches are suitable for any type of archery, but in general for most archers the small reduction in drag is probably not worth the additional complexity in attaching then to the shaft.
There seem to be persistent reports of helically fletched arrows dropping a little more than straight offset fletches of a similar size and shape. The only explanation would appear to be incorrect design or mounting that generates higher drag.
Flu-flu fletching is the aerodynamic opposite to the helical fletching. Flu-flu fletches provide high drag and are designed to reduce an arrows range and are particularly useful for aerial shots. These fletches may be adaptive in that the initial launch acceleration and drag may deform them to a relatively low drag shape and then their natural resilience restores their higher drag form after a short time. Good flu-flu fletches provide high initial speed then abruptly appear to drop as the high drag kicks in.
Flu-flu fletching tends to lack the fineness of standard fletches, however their effectiveness - that is accuracy at short ranges and the suddenness of the drop - depends on the material and arrangement. Achieving the adaptiveness involves selecting a material of the right mass and stiffness so the form recovery time is in the order of 200 ms. A heavier point may be required to counter the extra mass and maintain the FoC in the acceptable range.
Spin is the rotation of an arrow about its longitudinal axis. Typical spin speeds are 500 to 4,000 rpm and is not particularly critical. The spin is generated by a slight angular offset, typically 0.5° to 4°, of the fletching which creates a wind turbine effect. Spin is used to:
gyroscopically stabilise,
ensure the arrow's aerodynamic profile is averaged out over a short travel distance,
provide increased drag in early flight,
possibly improve grouping in a cross wind.
Spin speed is directly proportional fletch angle and flight speed and independent of the number of fletches. However, spin is more complex than it may first appear.
Interestingly the gyroscopic effect on the arrow is minimal and can safely be ignored for most purposes. When and if it does come into play, its precession effect is more negative than helpful. It is only mentioned here as it is often the first thing that comes to mind. It is far more important with gun ballistics where the spin may be 100x faster.
An arrow's longitudinal profile is variable due to fletching, nock and boardheads if used. Spinning the arrow eliminates a directional bias on a yawing or pitching arrow by averaging out these variations. This is particularly desirable for two blade broadheads.
During spin-up, a little forward kinetic energy is transferred into rotary kinetic energy with some short term induced drag. As the arrow approaches its steady spin speed, the drag drops back to a value similar to that at zero offset for helical fletches but a slightly greater value for straight offset fletches. This is a most convenient characteristic as the extra drag occurs when maximum yawing and pitching is likely to occur so has a useful dampening effect. Three parameters determine this effect:
Fletch area determines the spin torque for a given offset. A larger total area will produce more torque, achieve spin speed quicker, so will present higher initial drag that will drop off more quickly.
Fletch offset determines steady state spin speed and the spin torque available. Higher offsets will have higher spin speeds and higher initial drag.
Arrow spin moment of inertia will determine how quickly an arrow can accelerate its spin with the applied torque.
Thus, there may be some value in tuning this effect for best dampening and minimum drag (arrow drop).
It is possible that as an arrow slows, some angular momentum could be converted back to forward momentum by thrust from a propeller effect!
Due to the Magnus effect a spinning shaft that is at an angle to the air stream will experience a small lateral force called the Kutta-Joukowski lift. It is referred to as a lift because it is equivalent to an aerofoil. This is the curved path effect seen with spinning balls in most ball based sports.
The Kutta-Joukowski force is approximated:
Fm = w
r2 ρ v L
where w is the spin rate in revolutions per second (rps), r
the arrow radius, ρ the air density, v the air velocity
and L the effective exposed length. In practice this force is
small and only comes into play with significant fishtailing or porpoising.
If the spin rate (or the fletch sweep rate) is similar to the arrow's lateral resonance frequency (40 to 150 Hz) then possible cross-coupling effects may sustain or even enhance the vibration resulting in an abrupt change in drag. This would be a good reason to limit the spin speed.
Broadhead users should be aware that higher spin speeds will increase drag unless the broadhead has similar angular offset to the fletches. A one sided sharpen bevel on the broadhead will help a little at lower spin speeds.
It has been reported that cross wind grouping is improved with spin. The rational for this observation may be that the cross-wind drag (and hence wind sensitivity) is dependent on orientation about the longitudinal axis. Slow spin would then widen the group.
Typically, the nock can be shown to contribute 10% of the arrow's drag. This is higher than one might expect and is due to the suction effect. It would seem to be a good candidate for drag reduction, however, its is not easy. The simplest approach is the taper the shaft to the smallest practical nock.
In some ways the point is the easiest part of an arrow to model as it operates in clean air and is usually a simple shape. Normal target points contribute less than 5% of an arrow's drag, but can determine boundary layer formation (see below).
The shaft contributes overwhelmingly to an arrow's drag. In general, the thinner the better, especially with barrelling.
A significant complication in arrow aerodynamics is the laminar-to-turbulent air flow transition. Various studies have shown this transition is not well defined and certainly not well understood. Wind tunnel tests have shown flow can be laminar with an angle of attack less than 2 degrees. However, it is widely accepted that for normally launched arrows, the flow is always turbulent. The reality is probably in between, with a transition to turbulent flow occurring at some distance down the shaft. Further, the transition from lamina to turbulent and back to lamina can show some hysteresis in the transition speed.
It seems turbulent flow is induced at launch by arrow vibration and / or initial off axis movement. If a perfect release on a center shot bow is achieved (which is everyone's goal) then it is possible that laminar flow is established. Such a perfect launch would then be susceptible to air turbulence inducing turbulent flow at any point in its flight with the result of a change of drag and hence grouping. A solution would be to ensure turbulent flow by introducing boundary layer energisers or surface roughness (like golf ball dimples?) near the point.
Like all flying objects, arrows develop a boundary layer of slower laminar or turbulent air, that usually grows in thickness along the shaft. A typical thickness may be 10 mm at the nock for a perfect arrow flight (i.e. no resonance, pitching or yawing). This means the fletches will be significantly immersed in this boundary layer. This in turn means the fletching and shaft should be analyzed as one - but that is very complex.
Non-ideal motion of the arrow will also significantly modify the boundary layer geometry with time, and it is this effect that makes precision arrow aerodynamics such a difficult subject.
Launching an arrow induces significant vibration into arrows shaft. While
many modes and harmonics are present, the fundamental lateral mode remains
dominant with a decay rate measured in seconds. The amplitude may be as
high as 30 mm peak to peak at the nock of the arrow. The amplitude at the
point is generally less as it is constrained by the concentrated mass of
the point.
The impact of this vibration on drag is difficult to predict, but it is
likely to be small as the oscillation involves alternate drag and thrust
quarter cycles. A greater effect may well be to ensure turbulent flow over
the full length of the arrow.
The arrow's fundamental resonant frequency
is closely related to the arrow's dynamic spine. It is typically in the
40 to 150 Hz range. You can get a feel for the resonance by holding the
arrow vertically between the thumb and forefinger at about a 20% down
from the nock and flick the shaft at its midpoint. You will feel a
vibration last for several seconds. By moving you hold point up or down
the resonance may be more pronounced and last longer. Your hold point is
then close to a null, where your fingers can have very little dampening
effect. Invert the arrow and repeat - and you will find the
second null closer to the point due to the
anchoring effect of
the point’s mass. In
both cases you will feel the same resonant frequency - it is the arrow's
fundamental.
An arrow's stability is the result of net effect of the above factors. A fundamental necessity is the CoP must always be behind the center of gravity to create a positive stability. The relative distance of the CoP behind the CoG determines the magnitude of the corrective moment.
A fletchless arrow has the CoP very close to the CoG with the result of neutral stability. A neutrally stable arrow is assumed to have no corrective forces acting on the arrow, so on leaving the bow any angular offset or angular rotation will be maintained to become obvious.
The bare arrow will still experience drag and lift. If the is no angle of attack, the drag will be reduced and there will be no lift. If there is some angle of attack, there will additional induced rag and some lift at right angle to the air stream in the direction of the angle of attack.
If an arrow leaves with angular rotation speed, the rotation will continue, so the angle of attack will increase down range. The resultant lift will then cause a deviation from the expected ballistic path. Sufficient angular rotation speed can cause the arrow to completely reverse its orientation.
It should be noted that a broadhead arrow with undersized fletching can result in neutral of even negative stability. This can be dangerous as the larger plan area can result in significantly increased lift and a rogue flight path.
Fletches move the CoP rearward. This invites the question "what is the optimal fletch arrangement?" In general, the aim would be for rapid correction of an arrow launched with angular offset and /or angular rotation speed and minimum path deviation in the process.
The fletches are part of a feedback system where a proportional force is applied to correct an error.
Mc = L d + D d tan( α )
which for small a (±10°) can be approximated to:
Mc = Kc d α
where Kc is a catch
all constant of proportionality.
The corrective moment acts against the arrow's moment of inertia, so the
relative magnitudes determines the speed of correction. Without dampening
the arrow will oscillate - the corrective moment will cause overshoot,
which will be corrected only to overshoot again and again. This is very
similar to a swinging pendulum.
So what is going to dampen this oscillation? Well, not much! The oscillation is likely to continue through to target with minimal amplitude reduction. Typically, the dampening time constant is 0.5 to 2 seconds for the amplitude to drop to one third. The best solution is to minimize the oscillation at launch by tuning out the angular offset and the rotation from the start. Feather fletching is said to provide dampening; however, the author suspects higher drag is the reason.
The aim of the arrow design then is to minimize the path swing as the arrow fishtails or porpoises. This swing is proportional to the arrow's lift coefficient, oscillation amplitude and inversely proportional to its frequency.
With the addition wing area of a broadhead, the CoP will move forward. This must be compensated for with a larger fletch area.
We have an arrow with a complex set of motions on a ballistic trajectory. The aerodynamics are non-trivial and most analysis make significant simplifications. Despite the simplifications useful results are still possible.
Interestingly, there is no completely satisfactory drag equation for arrows. One can argue that there are at least three components to drag:
pressure drag proportional to sectional area,
frictional drag proportional to skin area and
induced drag proportional to plan area and angle of attack
These are all relatively easily evaluated. However, there are other sources of drag that are usually either ignored or somehow bundle with the above. These include drag due to arrow vibration, lamina to turbulent transitions, spin and fletch deformation.
The commonly used drag equation is:
FD = 1/2 ρ v2 CD A
where
FD is the drag
force
ρ is the air density
v air velocity
CD the drag
coefficient and
A is a reference area.
This assumes turbulent flow over most of the arrow.
The reference area A can be a bit fuzzy. Just how it is defined depends on the conventions of the application field. For aircraft it is typically wing area. For bullets and cars, it is the net frontal sectional area. In archery there is no well-established convention, however when dealing with arrow components, each component is treated using the most appropriate method. For a fully assembled arrow we use the shaft's maximum frontal sectional area.
For a particular arrow component and atmosphere
FD = K v2
where K an all encompassing constant.
All this seems simple, but wait, the detail is hidden in CD which is a function of the mysterious Reynolds Number** which in turn is also a function of velocity. By definition the Reynolds number is:
Re = ρ v L / μ = v L / κ
where
μ
is the dynamic or shear viscosity
κ is the kinematic viscosity = μ / ρ
L is the characteristic length
The L term is interesting in that it is proportional to size, and provides an understanding of the effect of changing the size (but not the shape) of an arrow.
In practice, the Reynolds number, drag coefficient and the resultant drag force of each component (point, shaft, fletches and nock) of the arrow are calculated at the launch speed. These drag components are added for the total drag and the arrow's composite drag coefficient calculated. It is this composite drag coefficient that is used by the ballistics engine to calculate the actual drag at different air speeds.
From a practical point of view, this approach is found to be the best of the options, but it is not perfect. The main limitations are that it fails to consider the aerodynamic interaction between the components and it makes no attempt to model lamina to turbulent transitions.
However, a significant advantage is that the impact of changes to components can be determined with useful precision.
To link theory to the reality, it is best to calibrate the system with real flight data. This allows us to compensate for the known and unknown limitations of the theory.
The arrow's composite drag equation is slightly modified by a proportionality factor determined by calibration:
FD = 1/2 ρ v2
CD A k
where
k is the
proportionality factor and
A is the arrow is the shaft's cross sectional area.
To calculate k, the arrow's speed needs to be measured at several different times after launch. This is most simply done indirectly by measuring arrow drop for several different ranges. In FlyingSticks this can be obtained from sight data or by direct drop measurement in a process called Reverse Ballistics. FlyingSticks hides the k factor from the user as it is seen as an internal calibration factor. For individual ballistic trajectory evaluation, the drag equation is reduced to:
FD = K v2
where K combines all the non-varying values into a single constant for a particular flight:
K = 1/2
ρ CD A k
For area A, FlyingSticks uses the maximum shaft sectional area, so for a high performance arrow this might yield a CD ~= 2.1. However, if the net sectional area (i.e. shaft + fletches + broadhead) were to have been used a lower CD ~= 0.9 would be expected.
The advantage in using the shaft sectional area is that aerodynamic inefficiencies are more intuitively expressed as a higher CD. For example, a high performance arrow fitted with a LARP point may have a CD of ~45, but if the net sectional area were used the CD would change little from ~0.9 even though the drag would have increased greatly.
In reviewing the published literature, it appears no one has seriously studied the aerodynamics of a bent arrow, little alone a vibrating arrow. We have assumed a vibrating arrow has increased drag due to an increased in effective sectional area.
Spin drag is modelled as induced drag from the fletch rotational torque "lift" that decays exponentially with time as the spin reaches steady state. It uses the standard lift equation applied to each fletch, as if were a wing immersed in the shaft's boundary layer. However, arrow vibration will periodically move some fletches into clean air, changing their lift and drag characteristics. This latter effect in not modelled.
The exponential decay rate is calculated from the fletch lift and the arrow's spin moment of inertia and presented to the user as the spin time constant.
Conventional understanding from wind tunnel testing indicates a typical arrow is likely to experience lamina flow for Re < 10,000 and turbulent flow for Re > 22,000 and in between, some arrangement where the transition point progressively moves long the shaft towards the point.
In practice it seems the flow is always turbulent. There are likely to be several factors at work. Firstly, it appears that the lamina to turbulent flow transition along the arrow shaft can be very critical on point shape.
Secondly, a finger release or non-center-shot bow will induce significant lateral vibration that almost certainly induces turbulent flow.
A flight with lamina flow will experience significantly lower drag, but the probability of attaining this state is low so best avoided entirely by NOT having a streamlined point and even adding some surface roughness.
There are some conditions when lamina flow might be reliably achieved - such as a well tuned, low speed center-fire bow with thin stiff arrows using a mechanical release in still air. This approach could be useful for indoor rounds.
From a knowledge of the components in drag equation it is possible to conduct a sensitivity analysis to indicate the likely drag impact of making a change to the arrow's geometry and the environment. This impact is expressed as a change at the target (in offset, strike velocity and penetration) and the sight's range-tape.
Lift is an important force in arrow flight when yawing or pitching occurs for several reasons:
lift generated by the fletching provides a correction moment to "stabilize" the arrow,
lift experienced by the arrow causes the arrow to deviate from its ballistic path in a dampened oscillatory fashion,
additional drag (induced drag) while generating lift.
The lift equation is similar to the drag equation:
FL = 1/2 ρ v2 CL A
where the lift coefficient is typically a linear function of angle of attack. Generating lift also generates induced drag that must be added to normal drag.
** Note on characteristic length L and Reynolds number:
The drag equation's area term L and the Reynolds number are intimately related. The Reynolds number is primarily used for comparing similar shapes but of different sizes or different speeds or in different fluids. In other words, the flow over similar bodies with the same Reynolds number will be very similar. The important issue is to be consistent in the way L is defined for the purpose it is used, hence a degree of fuzziness is acceptable!