application of flight mechanics for bullets

APPLICATION OF FLIGHT MECHANICS FOR BULLETS

Timo SailarantaAalto UniversitySchool of Science and Technology

Timo SailarantaFluid Dynamics Licenciate Seminar

Kul-34.4551 12.3.2012

Contents

• Objective of Study• Background• Simulation scheme• Bullet Geometry• Aerodynamic model• Trajectory model• Bullet turning• Results• Conclusions

Helsinki12.3.20122

Objective of Study

• 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

Helsinkixx.xx.20113

Background of Study [1][2]

• Incidences of celebratory firing 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 traveling

slowly, they do travel fast enough to cause significant injury and

death

• Estimated lethal energy 40-80 J, skull penetrating velocity 60

m/s

Helsinkixx.xx.20114

Background of Study [1][2]

• 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

Helsinkixx.xx.20115

Simulation scheme

• 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

Helsinkixx.xx.20116

Bullet Geometry Studied

Helsinkixx.xx.20117

Bullet data

• 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)

Helsinkixx.xx.20118

Aerodynamic model

• 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

Helsinkixx.xx.20119

Aerodynamic model – case simulated

• Table 1 Freestream 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

Helsinkixx.xx.201110

Aerodynamic model – case simulated

• 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


Aerodynamic model - Grid


Aerodynamic model – flow field 45 AoA


Aerodynamic model – CFD Results


• Magnus-moment coefficient time histories AoA 135 deg (pd/2V=0.479)



• Magnus-moment coefficient time histories AoA 90 deg (pd/2V=0.479)



• Magnus-moment model for trajectory simulations (pd/2V=1)



• Example: Axial force coefficient



• Example : Normal force coefficient fit CN=2sin(α)+0.8sin2(α)



• Example : Pitching moment coefficient fitTakashi 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

Trajectory model

• 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)


Trajectory model


Trajectory model

• Rotationally symmetric bullet geometry• Example: Normal force components


Trajectory model [3]

Helsinkixx.xx125

Frequency domain analysis [5]


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Frequency domain analysis

• 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)


1'1' in 2'2

' in

Bullet turning at apex

• 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)


Bullet turning at apex

• 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

NSCM24 Helsinkixx.xx.201129

Bullet turning - Magnus moment resonance• 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


Results - Terminal velocities

Resonance = matching of fluid and bullet body frequencies

NSCM24 Helsinki Timo Sailaranta Jaro Hokkanen & Ari Siltavuorixx.xx.201131

Velocity histories (launch angle 86 deg)


AoA histories (launch angle 86 deg)


Theta histories (launch angle 86 deg)


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

flight)


Magnus moment direction

• 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)


Shooter hit probability

• 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


Conclusions

• 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


References

[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. h

ttp://ats.ctsnetjournals.org/cgi/content/full/83/1/283

[2] Jaro Hokkanen, Putoavan luodin lentomekaniikka ja iskuvaikutukset, 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.


http://ats.ctsnetjournals.org/cgi/content/full/83/1/283

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