Application of flight mechanics for bullets

1. Application of flightmechanicsforbullets Timo Sailaranta Aalto University School of Science and Technology Timo Sailaranta Fluid Dynamics Licenciate Seminar Kul-34.4551

2. Contents Helsinki Objective of Study Background Simulation scheme Bullet Geometry Aerodynamic model Trajectory model Bullet turning Results Conclusions

3. Objective of Study Helsinki The objective of this paper is a) to study flight of an upwards fired bullet – focus on turning at the apex and the terminal velocity b) to estimate danger caused by the falling bullet The analysis is computational The bullet effect on human is estimated based on literature

4. Background of Study [1][2] Helsinki Incidences of celebratoryfiring a major public health concern internationally In Los Angeles (1985-1992) 118 victims, 38 of them died Although the bullets falling at terminal velocity are travelingslowly, they do travelfast enough to cause significant injury and death Estimated lethal energy 40-80 J, skull penetrating velocity 60 m/s

5. Background of Study [1][2] Helsinki A new bullet geometry is searched for in order to slow down the bullet falling velocity A redesigned base area might provide a way to do the task – potential geometry could be an hexagonal/octagonal base The modification causes a large Magnus-moment at subsonic speeds  nose down falling bullet tumbling and velocity retardation The phenomena studied at first with an ordinary geometry

6. Simulation scheme Helsinki Separate flow and trajectory simulations Bullet aerodynamic model created at first CFD, engineering method and experimental results utilized Table look-up approach during the trajectory simulation – based on simple closed-form fits Bullet/flow time-dependent interaction realisation adequate ? – combined simulation might be needed

7. Bullet Geometry Studied Helsinki

8. Bullet data Helsinki Bullet mass 9.5 g Diameter 7.62 mm Length about 28 mm Estimated inertias Ix=0.6e-007 kgm2 Iy=0.4e-006 kgm2 Launch velocity 850 m/s Rifle twist 1:12” (initial spin about 3150 rounds/s)

9. Aerodynamic model Helsinki Two separate CFD codes were used to carry out the computations (OpenFOAM and Fluent) Used to find out the high angle of attack aerodynamic interaction called Magnus –phenomena Magnus-moment particularly important for a bullet stability/turning at apex Results compared with experimental ones if available Small angle aerodynamics obtained using an engineering code

10. Aerodynamic model – case simulated Helsinki Table 1Freestream flow parameters and reference dimensions.   Only one case at altitude 1000 m simulated Velocity V = 50 m/s Pressure p = 89875 Pa Density ρ = 1.1116 kg/m3 Dynamic viscosity μ = 17.58ˑ10-6 kg/ms Temperature T = 281.65 K Reference length d = 7.62ˑ10-3 m Reference area S = 4.56ˑ10-5 m2 Reynolds number Red = 24 000 Spin rate 6283 rad/s (1 000 rps) Angles of Attack 45, 90, 110 and 135 degrees

11. Aerodynamic model – case simulated Helsinki Reynold’s number Red< x00000  subcritical case (2D theoretical 330000) Body boundary layer laminar Flow separates at about 90 – 100 degrees circumferential location Large wake region and about constant cross flow drag coefficient f(Re) Cdc =1.2

12. Aerodynamic model - Grid Helsinki

13. Aerodynamic model – flow field 45 AoA Helsinki

14. Aerodynamic model – flow field 90 AoA Helsinki

15. Aerodynamic model – flow field 135 AoA Helsinki

16. Aerodynamic model – CFD Results Helsinki Magnus-moment coefficient time histories AoA 135 deg (pd/2V=0.479)

17. Aerodynamic model – CFD Results Helsinki Magnus-moment coefficient time histories AoA 90 deg (pd/2V=0.479)

18. Aerodynamic model – CFD Results Helsinki Magnus-moment model for trajectory simulations (pd/2V=1)

19. Aerodynamic model – CFD Results Helsinki Example: Axial force coefficient

20. Aerodynamic model – CFD Results Helsinki Example : Normal force coefficient fit CN=2sin(α)+0.8sin2(α)

21. Aerodynamic model – CFD Results Helsinki Example : Pitching moment coefficient fit Takashi Yoshinaga, Kenji Inoue and Atsushi Tate, Determination of the Pitching Characteristics of Tumbling Bodies by the Free Rotation Method, Journal of Spacecraft, Vol. 21, No. 1, Jan.-Feb., 1984, pages 21-28

22. Trajectory model Helsinki Two separate 6-dof trajectory codes were used to carry out the computations Spinning and non-spinning body-fixed coordinate system ICAO Standard atmosphere Spherical Earth (Coriolis acceleration and centrifugal acceleration included)

23. Trajectory model Helsinki

24. Trajectory model Helsinki Rotationally symmetric bullet geometry Example: Normal force components

25. Trajectory model [3] Helsinki

26. Frequency domain analysis [5] Helsinki

27. Frequency domain analysis Helsinki Complex roots are obtained The period time and the time-to-half/double are computed A stability parameter was defined as inverse of the time-to-half (stable case, negative) or time-to-double (unstable case, positive)

28. Bullet turning at apex Helsinki The bullet turning at around the apex is mostly determined by Magnus-moment [4] The bullet effective shape non-symmetric due to spin and viscous phenomena  aerodynamic moment vector is no more oblique to the level defined by the bullet symmetry axis and velocity vector Magnus-moment behavior varied in this study (no other coefficients despite some time-depencies)

29. Bullet turning at apex NSCM24 Helsinki Magnus-moment behavior in trajectory simulations depicted Average value negative (or zero) at high AoA the bullet lands in stable manner base first if no resonance present

30. Bullet turning - Magnus moment resonance Helsinki Magnus-moment oscillation frequency 1000 Hz (CFD) Bullet fast mode oscillation frequency 180 HZ (freq domain analysis) Resonance will take place if these adjusted to match for a short time (coupling frequency region very narrow) Assumed to be possible in reality also since the CFD-analysis carried out extremely limited Resonance evokes the bullet fast mode oscillation causing increasing coning motion with drag penalty and low impact velocity

31. Results - Terminal velocities NSCM24 Helsinki Resonance = matching of fluid and bullet body frequencies Timo Sailaranta Jaro Hokkanen & Ari Siltavuori

32. Velocity histories (launch angle 86 deg) Helsinki

33. AoA histories (launch angle 86 deg) Helsinki

34. Theta histories (launch angle 86 deg) Helsinki

35. Angular velocity histories (launch angle 86 deg) Helsinki A short time resonance is seen at right (about after 20 s flight)

36. Magnus moment direction Helsinki Bullet turning would always take place even without resonance if the corresponding average moment was taken positive at high AoA Positive moment affects to the direction of coning motion (clockwise seen from behind)  always nose first landing and high velocity > 120 m/s Only experimental data found for terminal velocity of 7.62 cal bullet is about 90 m/s, which is close to the base first landing results obtained (about 85 m/s)

37. Shooter hit probability Helsinki The bullet landing area diameter ≈ 1000 m when the elevation angle 90±5 deg (≈ upwards fired) The bullet Landing area at least 1000000 times larger than the shooter projected area small hit probability Also the bullet landing velocity typically small when fired upwards

38. Conclusions Helsinki The bullet turning at the apex depends on Magnus-moment (aerodynamic interaction) direction and/or oscillation frequency Skull penetrating velocity 60 m/s (216 km/h) mostly exceeded - redesigned bullet base might limit the terminal velocity below that value – subsonic Magnus caused small AoA instability is searched for More sophisticated aero-model and/or simulation scheme is possibly needed in the future

39. References Helsinki [1] Angelo N. Incorvaia, Despina M. Poulos, Robert N. Jones and James M. Tschirhart, Can a Falling Bullet Be Lethal at Terminal Velocity? Cardiac Injury Caused by a Celebratory Bullet. http://ats.ctsnetjournals.org/cgi/content/full/83/1/283 [2] Jaro Hokkanen, Putoavanluodinlentomekaniikkajaiskuvaikutukset, kandidaatintyö, 2011, Aalto-yliopisto. [3] Peter H. Zipfel, Modeling and Simulation of Aerospace Vehicle Dynamics, AIAA Education Series, AIAA, 2000. [4] Timo Sailaranta, Antti Pankkonen and Ari Siltavuori, Upwards Fired Bullet Turning at the Trajectory Apex. Applied Mathematical Sciences, pp 1245-1262, Vol. 5, 2011, no. 25-28, Hikari Ltd. [5] Timo Sailaranta, Ari Siltavuori, Seppo Laine and Bo Fagerström, On projectile Stability and Firing Accuracy. 20th International Symposium on Ballistics, Orlando FL, 23-27 September 2002, NDIA.