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Defense and
Vol. 2, May 2
https://doi.org

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Analysis
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Tarik Šaban
1 Mechanical E
 
 
 

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Published by
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 DSS Vol. 2, May 2021, pp.79-85 

80 

coefficients based on the geometric characteristics of the projectile. 
The rapid advancement of computer technology has introduced a new discipline called computational fluid 
dynamics (CFD - computational fluid dynamics) which represents the third method in aerodynamics 
completing the previous two, pure theory and pure experiment. Computer fluid dynamics is the art of 
approximating major partial differential equations which describe fluid flow by simple algebraic expressions 
[2]. The end result of the application the CFD on the projectile flow is a picture of the flow field around the 
projectile which allows for a detailed analysis of the influence of individual parameters on the pressure 
(velocity) distribution and thus on aerodynamic drag coefficient. 

2. Influence of projectile shape on drag force coefficient  
In addition to the flow velocity (represented through Mach number), the aerodynamic drag coefficient in large 
extent depends on the geometric characteristics of the projectile. Slenderness of the projectile implies the ratio 
of the length of the projectile to the reference diameter of the projectile. Figure 1. shows the dependence of the 
drag coefficient on the Mach number for the three shapes of projectiles. The upper curve represents the drag 
coefficient for a spherical projectile, the middle curve corresponds to the drag coefficient for the projectile 
without the rear cone and with front part made with small slenderness, and the lower curve is the drag 
coefficient of a modern projectile. In terms of projectile slenderness, the sphere has the lowest slenderness and 
the modern projectile has the greatest slenderness. 

 
Figure 1. Coefficient of drag as a function of Mach number depending on the shape [3] 

 
All three curves shown in Figure 1. have the same trend. In the subsonic area the drag coefficient is constant 
(middle and lower curve) or changes very little, in the transonic area rises sharply to its maximum value and 
begins to slowly decline. The trend of a slight decrease in the drag coefficient with an increase in the Mach 
number continues in the supersonic range. It can also be noticed that projectiles with higher slenderness have 
smaller drag coefficient and that their maximum value of the drag coefficient occurs for the values Mach's 
number closer to one. 

3. Mathematical model 
Aerodynamics is a theoretical and experimental science, and represents a branch of fluid mechanics. 
Theoretically approach is based on the analytical solution of mathematical models of air flow. An analytical 
solution gives a complete insight into the physics of a problem, and once determined the analytical solution is 
suitable for the analysis of the influence of individual parameters in the mathematical model. Most airflow 
problems are described by nonlinear partial differential equations, which do not have a general analytical 
solution. This is especially true for turbulent flow, which due to the stochastic nature of that flow cannot be 
described analytically. With the development of computers, conditions were created for the numerical solution 
of mathematics model. Each simulation is based on a mathematical model, which denotes a mathematical 
notation physical model.  



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The mathematical model includes the following assumptions: 
 Air is a continuum. 
 Air is considered a homogeneous mixture of gases. 
 The physical properties of air are the same in all directions - the air is isotropic. 
 Air is a single-phase fluid. 
 Mass forces are neglected. 

The continuum assumption implies that the density of the fluid is large enough that even the infinitesimally 
small element of the fluid contains a satisfactory number of particles so it is possible to specify average 
velocity and average kinetic energy. That way they can determine flow characteristics (velocity, pressure, 
temperature, etc.) at each fluid point [4]. 
Continuity  behavior  can  be  described  by  transport  equations  based  on  the  basic  laws  of  mass  
conservation, momentum,  and  energy. The  equations  derived  from  the  given  laws  are  presented  in  
integral  form  for  an arbitrarily  selected  part  of  the  continuum,  the  volume Ω bounded  by  a  closed  area 
- the  limit  of  the  control volume dΩ (Figure 2). The surface element dS is defined by the unit vector of the 
normal 𝑛 [4]. 
 

 
Figure 1. Control volume [4] 

Law of mass conversation states: 
 𝛿𝛿𝑡 𝜌𝑑𝛺 + ρ(�⃗� ∗ 𝑛) 𝑑𝑆 = 0 (1)
 
Law of momentum conservation implies: 
 𝛿𝛿𝑡 𝜌�⃗� 𝑑𝛺 + ρ�⃗�(�⃗� ∗ 𝑛) 𝑑𝑆 = 𝜌𝑓 𝑑𝛺 − ρ(𝑛) 𝑑𝑆 + (𝜏̿ ∗ 𝑛) 𝑑𝑆 (2)
 
Law of energy conservation states: 
 𝜌𝐸 𝑑𝛺 + ∮ ρE(�⃗� ∗ 𝑛) 𝑑𝑆=  ∮ k(∇T ∗ 𝑛) 𝑑𝑆 + 𝜌𝑓 ∗ �⃗� + 𝑞 𝑑𝛺 − ∮ p(�⃗� ∗ 𝑛) 𝑑𝑆 + ∮ (𝜏̿ ∗ �⃗�) 𝑑𝑆 (3)
 
Here 𝑣 is velocity of airflow, p is pressure, ρ is density, T is temperature, E is total energy, and �̿� is stress 
tensor. The system of equations describing high-velocity flow cannot be solved analytically. To solve this 
system it is necessary to introduce a simplification or a problem or the equations. With the development of 
computers and computer fluid dynamics (CFD) numerical solution of equations describing the flow can be 
obtained. 

4. Solving equations using CFD 
Numerical solution of a mathematical model that describes the flow in the considered problem consists of 
three steps. In the first step, the area is discretized. Result of discretization of space is called a geometric grid. 
On the defined geometric grid it is necessary to discretize the partial differential equations of the mathematical 
model, respecting specific boundary conditions. The discretization of the equations is carried out by some of 



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the known methods (finite volume method, finite element method, finite difference method etc.). Result of 
discretization of partial differential equation on given geometric web is a system of algebraic equations. The 
nonlinear system of equations is solved iteratively by a procedure that involves solving a system of linear 
algebraic equations. A numerical solution is obtained, followed by its analysis, which includes a display of 
scalar, vector and tensor fields, integration of flow, force, moments, thermal flows, etc., and a diagram of the 
desired quantities. 
One of the main prerequisites for using a CFD model is validation model that takes place in several steps [4]: 

 Checking the program code. 
 Comparison of the obtained results with the available experimental data (predicting measurement 

errors). 
 Sensitivity analysis and parametric study. 
 Application of different models, geometry and initial/boundary conditions. 
 Reports on findings, model limitations, and parameter settings. 

The aerodynamic drag coefficient of the sphere was used here to verify the flow model for which 
experimental data are available [3]. The physical model of this case, the sphere in air flow, is shown in Figure 
3. 
 

 
Figure 2. Sphere in air flow 

The diameter of the sphere in the simulations was 2.54 mm and the flow was simulated at different velocities 
(2; 2.5; 2.64 and 3 Mach). Pressure and temperature of free air flow in all simulations were: p∞ = 101325 Pa 
and T∞ = 300 K. 
 

 

 
Figure 3. Drag coefficient as a function of Mach number for spheres with  different diameters 

Figure 4 shows a comparison of experimental and numerical results obtained. For free air flow velocity of Ma 
= 2.64 (point C) by numerical simulation the value of the coefficient of drag for the sphere, with diameter 2.54 
mm, was Cd =0.95, which is about 1% higher than the value obtained experimentally. Deviation (relative 
difference) observed were as follows: for M = 2 the deviation was 3%, for M = 2.5 the deviation was 4%, for 
M = 2.64 the deviation was 5% and for M = 3 the deviation was 4%.  
Comparing the values, very good agreement was observed between the experimental values and values 
determined using the Ansys Fluent software package.  



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5. Analysis of results  
The flow for four 5.56 mm projectiles with different (supersonic) velocities was considered in the research. 
The following assumptions were adopted for all simulations: 

 Working fluid is air, an ideal gas, which is modified in accordance with compressibility and changes 
in thermo-physical characteristics with temperature.  

 Density and viscosity depend on temperature, and cp and thermal conductivity are considered 
constant. 

 The parameters of free air flow were: p∞ = 101325 Pa and T∞ = 300K. 
 The flow around the projectile is considered compressible and turbulent. 
 Discretization of the spatial domain was performed by non-uniform unstructured mesh. 
 A "density-based solver" was used, developed for compressible high-speed flows. 
 The equations were linearized in implicit form, i.e. for given variable, unknown in each cell was 

calculated using relations that include existing and values from adjacent cells.  

The flow field around the projectile and the aerodynamic drag coefficient were determined by using FLUENT 
program for specified conditions. 
Four small-caliber projectile were used for num. simulations; for these projectiles experimentally determined 
values of aerodynamic drag coefficient [6] were known. Models used were: 
 5.56 mm SS109, 
 5.56 mm M855, 
 5.56 mm L110, 
 5.56 mm, M856. 

Simulations with free flow for each of the projectiles was performed for velocities in the range of 1.2 up to 3 
Mach for the following values of Mach numbers: 1.2; 1.5; 1.7; 2; 2.5; 2.64 and 3. 
Although all observed projectiles consist of a front part, a cylindrical part and a rear part, on projectiles SS109 
and M855 the rear part have the shape of a bevelled cone and on projectiles L110 and M856 the rear part have 
the edged shape. For this reason, two groups will be formed in the analysis of results. The first group consists 
of projectiles with a conical rear part, SS109 and M855, and the second group consists of projectiles with a 
edged rear end, L110 and M856. The results of the simulations, in the form of the drag coefficient values, for 
the first group of projectiles are shown in Figure 5. In Figure 5. are also presented the results of experimental 
tests for projectiles SS109 and M855. 
 

 
Figure 4. Drag coefficient vs Mach number for projectiles 5.56 mm, models SS109 and M855 [5] 

0.25

0.3

0.35

0.4

0.45

0.5

1 1.5 2 2.5 3

Cd

Ma

SS-109/M855

SS-109

M855

Experimental data SS109

Experimental data M855



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From the figure 5. it can be seen that the agreement of experimental and numerical data in supersonics is very 
good and the maximum rel. error is 2%. Agreement in transonics is somewhat lower, with rel. error of 10%. It 
can be noticed that the 5.56 mm SS109 projectile has a lower drag coefficient than the 5.56 mm projectile 
M855 in the supersonic range. Observing the pressure distribution [6] around the projectiles 5.56 mm, SS109 
and M855, shows that the SS109 projectile has a lower maximum pressure at the top of the front part in 
relation to the pressure at the top of the front part of the projectile 5.56 mm, M855; this also results in less 
drag on this part of the projectile. If we compare the pressure distribution at the rear for these two projectiles, 
it can be seen [6] that the M855 projectile has a lower bottom pressure than the SS109 projectile at the same 
velocity flow, which results in a higher coefficient of drag for this part and in accordance with the theoretical 
considerations 5.56 mm projectile SS109 has a slightly larger slenderness of the front part, significantly higher 
slenderness of the rear end and approximately the same overall slenderness in relation to the 5.56 mm M855 
projectile; although it has larger angle of inclination of the rear cone, which is unfavorable, it can be stated 
that it is better aerodynamically shaped from a 5.56 mm projectile, M855, for flight at supersonic speeds. 
Simulation results, drag coefficient vs Ma number, for the second group of projectiles, projectiles 5.56mm, 
L110 and M856, are shown in Figure 6. together with the experimental data.  
 

 
Figure 5. Drag coefficient vs. Mach number for 5.56 mm projectiles, models L110 and M856 [5] 

On Figure 6. it can be seen that the agreement of the experimental with the numerical results is very good and 
that many points actually match. Observing the pressure distributions [6] flowing around the front of the 5.56 
mm projectile L110, and around the front part of the projectile 5.56 mm M856, it can be noticed that the 
pressure at the top of the front of the projectile M856 is less than the pressure at the top of the projectile L110, 
which also results in less drag on this part. The front of the L110 projectile is also slimmer from the front part 
of the M856 projectile, resulting in a lower drag coefficient of the front part of the projectile. Considering the 
pressure distribution at the rear of the L110 projectile and at the rear projectile M856, a lower pressure is 
observed at the rear and at the bottom of projectile 5.56 mm L110. Although the M856 projectile has a greater 
slenderness of the rear and smaller coefficient of drag at the bottom, it can be concluded that the greatest 
influence on the total coefficient of drag in the supersonic area have the shape and slenderness of the front part 
of the projectile. Figure 6. shows that the 5.56 mm projectile L110 has a lower drag coefficient than projectile 
5.56 mm M856 in the considered velocity range. In the range of velocities up to Mach 1.5, the trend of the 
L110 projectile drag curve is the same as the trend of the coefficient curve for projectile M856. At the flow 
rate M = 1.7, the value of the drag coefficient projectile L110 approaches the value of the drag coefficient of 
the projectile M856, where the rel. difference is less than 5%. With a further increase in the flow rate, this 
difference decreases and for M = 3 it is less than 4%. 
More data from this research can be found in reference [6].    

0

0.1

0.2

0.3

0.4

0.5

1 1.5 2 2.5 3

Cd

Ma

L110/M856

L110

M856

Experimental data M856

Experimental data L110



 DSS Vol. 2, May 2021, pp.79-85 

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6. Conclusions 
In this paper, the influence of projectile shape for 5.56 mm ammo on aerodynamic drag coefficient is analyzed 
at supersonic velocities, using computational fluid dynamics. During these analyses, the following was found: 
 Shorter, blunt projectiles have a higher drag coefficient than slender projectiles. 
 In the supersonic range, the length of the fornt part has a large influence on the drag coefficient and 

with increasing slenderness of the front of the projectile comes a significant reduction of the 
coefficient of drag. 

 At higher (supersonic) velocities, the front of the projectile shaped like a secant has the lowest drag, 
less so than the conical and tangential front shapes. 

 The rounded tip of the projectile will result in less drag than the blunt tip. 
 Increasing the slenderness of the rear part of the projectile reduces the coefficient of drag and for 

supersonic velocities the optimal length of the rear part is between 0.5 and 1 caliber. 
 The optimal value of the rear cone angle, at supersonic speeds, is 7. 

In the Ansys FLUENT software package, based on a theoretical consideration of the supersonic projectile 
flow, a system of equations is chosen to describe the flow of air around projectiles. The domain in which the 
calculation is performed has been defined and boundary conditions have been set. From the offered software 
options for solving equations, the solving model was chosen and for each simulation the initial conditions 
were defined. Before num. simulations for projectiles, the model was verified based on the aerodynamic drag 
coefficients of the sphere for different flow velocities. 
Analysis of the influence of projectile geometrical characteristics on the drag coefficient at zero yaw angle 
was performed based on the results of numerical simulations. They are simulated for 5.56 mm projectile, 
SS109, M855, L110 and M856 models. These projectiles have similar outer shape: front part in the shape of a 
secant, cylindrical part and end part. They differ in the slenderness of individual parts of the projectile, in the 
slenderness of the whole projectile, by the radii of the ogive, by the shape of the rear part of the projectile.  
For each of the projectiles, 7 simulations were performed for Ma = 1.2; 1.5; 1.7; 2; 2.5; 2.64 and 3.  
The greatest influence on the drag of the projectile has the shape and slenderness of the front part of the 
projectile; 5.56 mm SS109 and M855 projectiles have lower slenderness compared to projectiles 5.56 mm 
L110 and M856. Regardless of the slenderness and how the SS109 and L110 projectiles are different they 
show similar aerodynamic characteristics in terms of the influence of their shape on drag; subsequent analysis 
of the results concluded that these two projectiles in some parts of the supersonic regime have the same drag. 
The drag analysis showed that the SS109 and L110 projectiles have less drag compared to projectiles M855 
and M856.  

References  
[1] S. Janković, Aerodynamics of projectiles, Faculty of Mechanical Engineering, Belgrade, 1979. 

[2] Anderson, J., Modern compressible flow, McGraw‐Hill Publishing, Singapore, 1990.   

[3] McCoy, R., Modern exterior ballistics: The launch and flight dynamics of symmetric projectiles, Schiffer 
Publishing Ltd, 2nd edition, Pennsylvania, 2012.  

[4]  Blazek, J., Computational fluid dynamics: Principles and appications, Elsevier Science Ltd, Oxford, UK, 
2001. 

[5]  Sabanovic T., “Analysis of influence of projectile shape on its drag coefficient for 5,56 mm ammunition 
using CFD”, Mechanical Engineering Faculty, University of Sarajevo, 2021. 


