Drag Variability and the use of the “Axial Form Factor” in the Hornady 4DOF™
Trajectory Calculator
For some time the “standard practice” within the shooting sports industry has been to use a fixed Ballistic Coefficient (BC), or drag simulation, for any given projectile. It has always been assumed that the BC/Drag Coefficient (Cd) of the projectile was the BC of the projectile no matter what. Recent work with Doppler radar, which is able to provide high resolution plots of the projectile drag performance, have shown that this is not the case. This paper will discuss and show projectile drag data and it’s variability with changes in the weapon system and ammunition. Variables that were tested were, rifling form, muzzle brakes, different propellant types with the same projectile/firearm and barrel twist rate. As will be shown significant changes in the projectile drag were seen across these variables. The use of the “Axial Form Factor” will be discussed as a way to account for these variables in order to achieve the highest fidelity firing solution using 4DOF™ with an individual firearm and load.
Rifling form:
Two calibers were used to perform testing on rifling form. A 300 Winchester Magnum (WM) was used with several different 26” Bartlein barrels , all chambered with the same physical reamer. All barrels were 1-10” twist. Rifling forms used were 5R, Polygonal and a truncated 5R with .303” bore and .308” grooves. A prototype 225 gr bullet was used for all testing. 6.5 Creedmoor was tested with a 24” 5R Thompson Center (TC) barrel and a Wiseman 6 groove square rifled barrel both in 1-8” twist with the 6.5 mm 140 ELD-Match™.
As can be seen from the plots noticeable differences in the drag are seen in both calibers, particularly in the 6.5. The 30 caliber shows small but noticeable differences particularly in the low supersonic mach numbers. In the 6.5 mm barrels the TC 5R barrel showed significantly greater and earlier onset of significant limit cycle yaw than did the 6 groove square rifled Wiseman barrel. Conjecture as to why could be done but for the purposes here we are simply trying to show differences. More testing will have to be done to quantify possible differences between 5R and square rifling.
Muzzle Brakes:
Five different muzzle brakes were tested to see what effects different designs would have on the projectile. The picture below shows the types used which will be referred to as A through E. A 26”, 1-10” twist Bartlein, 300 WM barrel was used for all testing. The same .30-225 gr prototype projectile was used in all testing. Top and side views off the five muzzle brakes are shown below.
The Cd plots show significant impacts to the drag of the projectile versus no brake, especially at high supersonic Mach numbers, right out of the muzzle. Most of the brakes show significant “tipoff” of the projectile at the muzzle which causes yaw and takes substantially differing amounts of time to damp. Most of the brakes tested will have an adverse effect on the drag and associated performance of the projectile. Brake D may have some advantage over the no brake because of only a small impact to the high supersonic drag and significant reduction in drag at lower supersonic speeds. If a brake is a necessity Brake D would certainly be the one to use with this setup.
Propellant effects:
Five different propellants were used with the 300 WM 1-10” 5R Bartlein barrel test rifle and the 225 grain prototype projectile, RL-19, RL-22, H4350, H4831 and H1000. No muzzle brake was used.
Differences here are substantial. RL-22 and H1000 show definite muzzle tipoff, likely a result of higher muzzle exit pressures and longer envelopment by the muzzle gas cloud. However, it is very interesting that both these propellants show significantly lower low supersonic drag. Perhaps less bullet upset/distortion, in-bore because of slower rise times and lower maximum acceleration. For ranges of out to about 1000 yards the data shows it would be very hard to beat RL-19. A longer barrel, in order to reduce muzzle exit pressures, with the RL-22 or the H4831 might produce the optimum results, especially for long range firing.
Barrel twist rate:
The 300 WM test rifle was used with 5 different twist rate barrels and the prototype 225 grain projectile. All barrels were 26”, 5R Bartlein barrels chambered with the same physical reamer. Twist rates of 1-7”, 1-8”, 1-9”, 1-10” and 1-11” were used. Twist rate data is displayed in combination with calculated muzzle Gyroscopic Stability Factor (Sg). We will be presenting another paper with more data on a wide range of bullet weights and a detailed discussion of twist rates, Sg and bullet performance.
The plot shows substantial differences in the drag performance of the projectile as a function of the twist rate and associated Sg. Again, all Sg numbers are for muzzle exit conditions. An Sg of 1.4 is a good number for a lower limit, ensuring adequate projectile stabilization under anything other than extraordinary atmospheric conditions, and has been a standard for some time. Higher Sg numbers, at least 2.0, will certainly result in improved projectile performance, particularly at high supersonic Mach numbers. This trend has been suggested by Applied Ballistics and is certainly corroborated by our data. This data suggests that the high supersonic Mach number performance of projectiles becomes substantially the same for an Sg of around 2.0. Small improvements in performance continue to be gained to an Sg=2.5 throughout the Mach number range. The data shows substantial improvements in low supersonic projectile performance with an Sg above 3.0.
Examination of the drag data shows the higher Sg values likely reduce initial yaw and coning from the muzzle and progressively delay and reduce limit cycle yaw and magnus effects. The shape of the drag curve significantly flattens and changes above an Sg=3.0. Because of this behavior we will begin adding a second projectile file, to appropriate projectiles in the 4DOF™ data base, for those who plan on or are using twist rates giving an Sg above 3.0 to account for the change in shape of the drag curve. It must be pointed out however, there are no free lunches. The higher Sg will increase aerodynamic jump and spin drift, but this will be predicted by 4DOF.
There are practical limits to faster and faster twist rates. All jacketed bullets have a point at which the jacket will fail from the in-bore torque forces and the external centrifugal forces. As stated above we will be shortly present another paper with much more data and discussion of different bullet performance. As a teaser the effects continue to show with lighter, shorter bullets but are not as pronounced.
The use of “Axial Form Factor”:
As can be seen in the above data, significant changes in the drag of a projectile can occur depending on the type or condition of the rifling in the barrel, the type of muzzle brake, the powder being used and the twist rate. Current Cd data in the 4DOF™ projectile data files are an average of different guns, cartridges, loads, in many cases plain muzzles and muzzle brakes and twist rates that give an Sg between 1.5 - 2.0. As we discussed in the twist rate section we will also begin offering two data files, for appropriate extreme range projectiles, that include drag data based on the projectile performance with an Sg=3.0 or higher.
The most accurate way to ensure the best result with any weapon ammunition system is of course to test it with Doppler radar and use the drag data for that exact combination. Unfortunately most of us can’t justify spending over $100K for our own Doppler radar. In order to account for the variability across the spectrum of firearm setups/condition and loads out there we decided to use the “Axial Form Factor” to allow adjustment of the supplied average Cd curves in 4DOF. As can be seen from the above data, the shape of most of the Cd curves remain substantially the same just small differences in value, until we get to the previously mentioned high Sg situations. This allows for a simple multiplier to deal with the variability from system to system for tuning the Cd curve to the weapon for its actual drag performance. The figure below displays what the Axial Form Factor is actually doing
Trajectory Calculator
For some time the “standard practice” within the shooting sports industry has been to use a fixed Ballistic Coefficient (BC), or drag simulation, for any given projectile. It has always been assumed that the BC/Drag Coefficient (Cd) of the projectile was the BC of the projectile no matter what. Recent work with Doppler radar, which is able to provide high resolution plots of the projectile drag performance, have shown that this is not the case. This paper will discuss and show projectile drag data and it’s variability with changes in the weapon system and ammunition. Variables that were tested were, rifling form, muzzle brakes, different propellant types with the same projectile/firearm and barrel twist rate. As will be shown significant changes in the projectile drag were seen across these variables. The use of the “Axial Form Factor” will be discussed as a way to account for these variables in order to achieve the highest fidelity firing solution using 4DOF™ with an individual firearm and load.
Rifling form:
Two calibers were used to perform testing on rifling form. A 300 Winchester Magnum (WM) was used with several different 26” Bartlein barrels , all chambered with the same physical reamer. All barrels were 1-10” twist. Rifling forms used were 5R, Polygonal and a truncated 5R with .303” bore and .308” grooves. A prototype 225 gr bullet was used for all testing. 6.5 Creedmoor was tested with a 24” 5R Thompson Center (TC) barrel and a Wiseman 6 groove square rifled barrel both in 1-8” twist with the 6.5 mm 140 ELD-Match™.
As can be seen from the plots noticeable differences in the drag are seen in both calibers, particularly in the 6.5. The 30 caliber shows small but noticeable differences particularly in the low supersonic mach numbers. In the 6.5 mm barrels the TC 5R barrel showed significantly greater and earlier onset of significant limit cycle yaw than did the 6 groove square rifled Wiseman barrel. Conjecture as to why could be done but for the purposes here we are simply trying to show differences. More testing will have to be done to quantify possible differences between 5R and square rifling.
Muzzle Brakes:
Five different muzzle brakes were tested to see what effects different designs would have on the projectile. The picture below shows the types used which will be referred to as A through E. A 26”, 1-10” twist Bartlein, 300 WM barrel was used for all testing. The same .30-225 gr prototype projectile was used in all testing. Top and side views off the five muzzle brakes are shown below.
The Cd plots show significant impacts to the drag of the projectile versus no brake, especially at high supersonic Mach numbers, right out of the muzzle. Most of the brakes show significant “tipoff” of the projectile at the muzzle which causes yaw and takes substantially differing amounts of time to damp. Most of the brakes tested will have an adverse effect on the drag and associated performance of the projectile. Brake D may have some advantage over the no brake because of only a small impact to the high supersonic drag and significant reduction in drag at lower supersonic speeds. If a brake is a necessity Brake D would certainly be the one to use with this setup.
Propellant effects:
Five different propellants were used with the 300 WM 1-10” 5R Bartlein barrel test rifle and the 225 grain prototype projectile, RL-19, RL-22, H4350, H4831 and H1000. No muzzle brake was used.
Differences here are substantial. RL-22 and H1000 show definite muzzle tipoff, likely a result of higher muzzle exit pressures and longer envelopment by the muzzle gas cloud. However, it is very interesting that both these propellants show significantly lower low supersonic drag. Perhaps less bullet upset/distortion, in-bore because of slower rise times and lower maximum acceleration. For ranges of out to about 1000 yards the data shows it would be very hard to beat RL-19. A longer barrel, in order to reduce muzzle exit pressures, with the RL-22 or the H4831 might produce the optimum results, especially for long range firing.
Barrel twist rate:
The 300 WM test rifle was used with 5 different twist rate barrels and the prototype 225 grain projectile. All barrels were 26”, 5R Bartlein barrels chambered with the same physical reamer. Twist rates of 1-7”, 1-8”, 1-9”, 1-10” and 1-11” were used. Twist rate data is displayed in combination with calculated muzzle Gyroscopic Stability Factor (Sg). We will be presenting another paper with more data on a wide range of bullet weights and a detailed discussion of twist rates, Sg and bullet performance.
The plot shows substantial differences in the drag performance of the projectile as a function of the twist rate and associated Sg. Again, all Sg numbers are for muzzle exit conditions. An Sg of 1.4 is a good number for a lower limit, ensuring adequate projectile stabilization under anything other than extraordinary atmospheric conditions, and has been a standard for some time. Higher Sg numbers, at least 2.0, will certainly result in improved projectile performance, particularly at high supersonic Mach numbers. This trend has been suggested by Applied Ballistics and is certainly corroborated by our data. This data suggests that the high supersonic Mach number performance of projectiles becomes substantially the same for an Sg of around 2.0. Small improvements in performance continue to be gained to an Sg=2.5 throughout the Mach number range. The data shows substantial improvements in low supersonic projectile performance with an Sg above 3.0.
Examination of the drag data shows the higher Sg values likely reduce initial yaw and coning from the muzzle and progressively delay and reduce limit cycle yaw and magnus effects. The shape of the drag curve significantly flattens and changes above an Sg=3.0. Because of this behavior we will begin adding a second projectile file, to appropriate projectiles in the 4DOF™ data base, for those who plan on or are using twist rates giving an Sg above 3.0 to account for the change in shape of the drag curve. It must be pointed out however, there are no free lunches. The higher Sg will increase aerodynamic jump and spin drift, but this will be predicted by 4DOF.
There are practical limits to faster and faster twist rates. All jacketed bullets have a point at which the jacket will fail from the in-bore torque forces and the external centrifugal forces. As stated above we will be shortly present another paper with much more data and discussion of different bullet performance. As a teaser the effects continue to show with lighter, shorter bullets but are not as pronounced.
The use of “Axial Form Factor”:
As can be seen in the above data, significant changes in the drag of a projectile can occur depending on the type or condition of the rifling in the barrel, the type of muzzle brake, the powder being used and the twist rate. Current Cd data in the 4DOF™ projectile data files are an average of different guns, cartridges, loads, in many cases plain muzzles and muzzle brakes and twist rates that give an Sg between 1.5 - 2.0. As we discussed in the twist rate section we will also begin offering two data files, for appropriate extreme range projectiles, that include drag data based on the projectile performance with an Sg=3.0 or higher.
The most accurate way to ensure the best result with any weapon ammunition system is of course to test it with Doppler radar and use the drag data for that exact combination. Unfortunately most of us can’t justify spending over $100K for our own Doppler radar. In order to account for the variability across the spectrum of firearm setups/condition and loads out there we decided to use the “Axial Form Factor” to allow adjustment of the supplied average Cd curves in 4DOF. As can be seen from the above data, the shape of most of the Cd curves remain substantially the same just small differences in value, until we get to the previously mentioned high Sg situations. This allows for a simple multiplier to deal with the variability from system to system for tuning the Cd curve to the weapon for its actual drag performance. The figure below displays what the Axial Form Factor is actually doing
