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The drag coefficientThe drag coefficient c_{D} is the
most important aerodynamic coefficient and generally depends on
- 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 neglectionIt can be shown, that with the exception of very low velocities, the Re dependency of c_{D} can be neglected.2. d dependencyDepending 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) * d^{2}/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 coefficientThe zero-yaw drag coefficient as a function
of the Mach number Ma is generally determined experimentally either
by wind tunnel tests or from Doppler Radar measurements.
Fig.: Zero-yaw drag coefficient for two military bullets
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 functionsGenerally 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
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