The drag coefficient

- bullet geometry (symbolic variable B),
- Mach number Ma,
- Reynolds number Re,
- the angle of yaw d

The following asssumptions and simplifications are usually made in ballistics:

1. Re neglection

It can be shown, that with the exception of very low velocities, the Re dependency of c_D can be neglected.

2. d dependency

Depending on the physical ballistic model applied, an angle of yaw is either completely neglected (d=0) or only small angles of yaw are considered. Large angles of yaw are an indication of instability.
For small angles of yaw the following approximation is usually made:

a) c_D(B,Ma,d) = c_Do(B,Ma) + c_Dd(B,Ma) * d²/2

Another theory which accounts for arbitrary angles of yaw is called the "crossflow analogy prediction method". A discussion of this method is far beyond the scope of this article, however the general type of equation for the drag coefficient is as follows:

b) c_D(B,Ma,d) = c_Do(B,Ma) + F(B,Ma,Re,d)

3. Determination of the zero-yaw drag coefficient

Fig.: Zero-yaw drag coefficient for two military bullets
M80 (cal. 7.62 x 51 Nato)
SS109 (cal. 5.56 x 45)

There is also software available which estimates the zero-yaw drag coefficient as a function of the Mach number from bullet geometry. The latter method is mainly applied in the development phase of a new projectile.

4. Standard drag functions

Generally each bullet geometry has its own zero-yaw drag coefficient as a function of the Mach number. This means, that specific - time-consuming and expensive - measurements would be required for each bullet geometry. A widely used simplification makes use of a "standard drag function" c_Do ^standard which depends on the Mach number alone and a form factor i_D which depends on the bullet geometry alone according to:

c_Do(B,Ma) = i_D(B) * c_Do ^standard(Ma)

If this simplification is applicable, the determination of the drag coefficient of a bullet as a function of the Mach number is reduced to the determination of a suitable form factor alone. It will be shown that the concept of the ballistic coefficient, widely used in the US for small arms projectiles follows this idea.

Abbreviations

cD	Drag coefficient; cD(B,Ma,Re,d)
cDostandard	Zero-yaw standard drag function
iD	Form factor

More Abbreviations