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Application of flight mechanics for bullets. Timo Sailaranta Aalto University School of Science and Technology. Timo Sailaranta. Fluid Dynamics Licenciate Seminar . Kul-34.4551. Contents. Objective of Study Background Simulation scheme Bullet Geometry Aerodynamic model

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application of flight mechanics for bullets

Application of flightmechanicsforbullets

Timo Sailaranta

Aalto University

School of Science and Technology

Timo Sailaranta

Fluid Dynamics Licenciate Seminar




Objective of Study


Simulation scheme

Bullet Geometry

Aerodynamic model

Trajectory model

Bullet turning



objective of study
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

background of study 1 2
Background of Study [1][2]


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

background of study 1 21
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

simulation scheme
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

bullet data
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)

aerodynamic model
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

aerodynamic model case simulated
Aerodynamic model – case simulated


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

aerodynamic model case simulated1
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 cfd results
Aerodynamic model – CFD Results


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

aerodynamic model cfd results1
Aerodynamic model – CFD Results


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

aerodynamic model cfd results2
Aerodynamic model – CFD Results


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

aerodynamic model cfd results3
Aerodynamic model – CFD Results


Example: Axial force coefficient

aerodynamic model cfd results4
Aerodynamic model – CFD Results


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

aerodynamic model cfd results5
Aerodynamic model – CFD Results


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

trajectory model
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 model2
Trajectory model


Rotationally symmetric bullet geometry

Example: Normal force components

frequency domain analysis
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)

bullet turning at apex
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 apex1
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

bullet turning magnus moment resonance
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
Results - Terminal velocities

NSCM24 Helsinki

Resonance = matching of fluid and bullet body frequencies

Timo Sailaranta Jaro Hokkanen & Ari Siltavuori

angular velocity 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
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
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



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



[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.

[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.