Originally Posted by
The Schwartz
In a nutshell, fluid flow-fields around tumbling bullets are extremely complex to model and, as velocity increases turbulence (chaotic, stochastic flow) does so too. Trying to do it by hand (say, on a Texas Instruments calculator) would be impractical and time-consuming at best. For a task like that, CFD (computational fluid dynamics) software like ANSYS CFX would be needed to process the differential equations needed to model flows like that.
Besides penetration depth, there are other physical parameters that merit examination and quantification. I'd respectfully offer that permanent wound cavity volume, pulverized tissue mass, and instantaneous/exit velocity are among them.
I agree. With physical gelatin models that are the dynamic equivalent of soft tissue, measurement of maximum penetration depth is easy enough and one of the attributes that makes those mediums (namely, 10% ordnance gelatin) so attractive. It is widely accepted that tumbling increases effectiveness, but putting a "hard" number in that is impossible. At the risk of opening the proverbial "can of worms", the Bio-Physics Division of the US Army Ballistic Research Laboratory at Aberdeen Proving Grounds (BRL) produced a mathematically-based predictive personnel incapacitation equation based upon 7,898 wound data accumulated by the WDMET (Wound Data Munitions Effectiveness Team) during the Vietnam War. The US Army personnel incapacitation equation relies upon an incremental kinetic energy expenditure parameter (ΔE15) of a random munition strike to the center of mass (COM) of a combatant’s or assailant’s body over a penetration depth of 1 – 15 centimeters to predict a projectile’s probability of incapacitation, represented symbolically as P[I/H]. Greater values of ΔE15 equate to greater strain energy storage within surrounding tissues produced by the bullet’s passage through them. Increased strain energy storage increases the likelihood of proximate tissue damage and with that damage, an increased probability of incapacitation. Of course, there are a few individuals in the terminal ballistics research community who dispute the model's validity, but the US Army relies on these P[I/H] models even to this day in their ORCA (Operational Requirement-based Casualty Assessment) computer code which still contains the BRL P[I/H] model as its ballistic insult subroutine MUVES-S2 (Multiple UNIX-based Vulnerability Estimation Suite). MUVES-S2 is embedded in ORCA and is used to evaluate the vulnerability/lethality of munitions in terms of injury to personnel within target geometry. The ORCA and MUVES-S2 vulnerabity code has been reviewed by peers in the medical and biological fields and by the U.S. Army and Navy and found to be adequate in producing injury, impairment, and operational casualty assessments. For that reason, I prefer to use the BRL P[I/H] model as a measure of lethality within its documented limitations. I am not insisting that anyone here accept the BRL P[I/H] model without doing research of their own; that responsibility is up to each one of us.
Your use of "effective diameter" to address the phenomena of tumbling, while admitted "simplistic", is probably the best "guess" that we can make without bringing expensive multi-physics software suites to the table. Given the magnitude of uncertainty and variation that is seen with yawing/tumbling bullets, it is probably as accurate of a representation as we'll ever see of the phenomena without getting lost in the minutæ that would inevitably be required.
In the end, "how useful" tumbling bullets might be for close-quarters civilian self-defense is going to be a "one-for-one" proposition. Given the nearly infinite variability of the human body, every single event will be a law unto itself.