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Defense and Security Studies  ISSN 2744-1741 
Vol. 1, No. 1, December 2020, pp.1-15            Original Research 
https://doi.org/10.37868/dss.v1i1.63 

This work is licensed under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/ ) that allows 
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authorship and initial publication in this journal. 

 1 

 
 

Axial force coefficient of APFSDS projectile 
 
Ammar Trakić1 

1 Defense Technologies Department, Mechanical Engineering faculty, University of Sarajevo; Bosnia and Herzegovina  
 
 

*ammar.trakic@hotmail.com 

© The Author  
2020. 
Published by  
ARDA. 

Abstract 
Armor-piercing ammunition is primarily used to combat against heavy armored 
targets (tanks), but targets can be light armored vehicles, aircraft, warehouse, 
structures, etc. It has been shown that the most effective type of anti-tank 
ammunition in the world is the APFSDS ammunition (Armor Piercing Fin 
Stabilized Discarding Sabot).  The APFSDS projectile flies to the target and with 
his kinetic energy acts on the target, that is, penetrates through armor and disables 
the tank and his crew. Since the projectile destroys target with his kinetic energy, 
then it is necessary for the projectile to have the high impact velocity.  

The decrease in the velocity of a projectile, during flight, is mainly influenced by 
aerodynamic forces. The most dominant is the axial force due to the laid 
trajectory of the projectile. By knowing the axial force (axial force coefficient), it 
is possible to predict the impact velocity of the projectile, by external ballistic 
calculation, in function of the distance of the target, and to define the maximum 
effective range from the aspect of terminal ballistics.  In this paper two models 
will be presented for predicting axial force (the axial force coefficient) of an 
APFSDS projectile after discarding sabot. The first model is defined in STANAG 
4655 Ed.1. This model is used to predict the axial force coefficient for all types of 
conventional projectiles. The second model for predicting the axial force 
coefficient of an APFSDS projectile, which is presented in the paper, is the CFD 
model (eng. Computed Fluid Dynamics). 

 

Keywords: APFSDS, axial force coefficient, STANAG 4655 Ed.1, CFD, 
PRODAS 

1. Introduction 

Ammunition based on the use of kinetic energy (KE) penetrates through targets primarily with the energy of a 
projectile body or sub-projectile (penetrator), which is made of high-density metal. The efficiency of this 
ammunition depends on the impact velocity of the projectile at the target, the length, diameter and density of 
the penetrator. A representative of modern kinetic energy ammunition, which is mostly used today in armies 
around the world, is APFSDS ammunition (Armor Piercing Fin Stabilized Discarding Sabot), which is made 
of an alloy of tungsten or depleted uranium [1]. 
The components of modern armor piercing fin stabilized discarding sabot ammunition are shown 
schematicallyy in Figure 1: 

1. penetration with fin stabilizer, 
2. sabot, 
3. combustible case, 
4. propellant, 
5. primer assembly, and 
6. case base. 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

2 

 

Figure 1. Main components of APFSDS ammunition [2] 

During the movement of the APFSDS projectile through the barrel of the weapon, the energy of the propellant 
charge acts on the bottom of the projectile (the penetrator is still connected to the sabot) and drives it. After 
the projectile comes out of the barrel, due to the difference in resistance and mass of the sabot and penetrator, 
the sabot separates and the penetrator continues to fly towards the target (Figure 2). When moving through the 
barrel of the cannon, the projectile with the sabot reaches supersonic velocities (up to 5 Mach). 
APFSDS projectiles are statically stabilized projectiles. Stabilization is provided using aerodynamic surfaces-
wings. When determining the axial force (axial force coefficient), the projectile body and the wings are 
observed separately. The sum of these two components of the axial forces on the APFSDS projectile gives the 
total axial force (axial force coefficient) [4]. 
 

        

Figure 2. Process of discarding sabot from APFSDS projectile [3] 

2. Models for prediction of coefficient of axial force 

There are many methods for predicting the axial force coefficient of APFSDS projectiles, but all of them are 
based on the application of three general methods: 

1. experimental (wind tunnel or polygon), 
2. theoretical, and 
3. numerical (CFD - Computational Fluid Dynamics). 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

3 

Experimental methods determine aerodynamic coefficients in an air (aero) tunnel or on the basis of measuring 
the movement of a projectile in flight (as a material point or a rigid body). This method gives the most 
realistic values for the axial force coefficient; however, the disadvantages of the tunnel experiment are [5]: 

 high price, 
 scaling problems if the model is not life-size, 
 interference from tunnel walls, 
 measurement difficulties. 

Assumptions and simplifications are necessary in theoretical methods for the solving problems. This includes 
simplifying geometry and simplifying equations. Equations known as Navier-Stokes equations, along with the 
energy equation and the continuity equation, describe the flow of fluid around a body. They are analytically 
unsolvable in closed form, but can be simplified for specific geometry or flight conditions [5]. 
Numerical methods are new; they have been used since the advent of computers during World War II. 
Advanced CFD codes numerically solve Navier-Stokes equations and can show the complete flow field 
around an object for specific flight conditions. With these methods, problems arise in determining the 
boundary conditions, because the initial conditions must be defined with great accuracy [5]. 
In the continuation of this chapter, two models for predicting the axial force coefficient APFSDS projectiles 
will be presented. The first model is presented in the STANAG 4655 Ed.1 standard. The second model for 
predicting the axial force coefficient of an APFSDS projectile is the numerical model (CFD). The program to 
be used for the numerical simulation of projectile flow is ANSYS Fluent. The presentation of the models as 
well as the results of the calculations will be shown below. 

2.1. Model defined in standard STANAG 4655 (Ed.1) 

The standard, STANAG 4655, shows an engineering model for prediction of the aerodynamic coefficients of 
conventional projectiles. The details of the standard are given and are divided into three parts [6]: 

1. Body Aerodynamics 
2. Fin Aerodynamics 
3. Generalized Yaw Aerodynamics 

The axial force of a projectile can be divided into two parts: pressure axial force and viscous (friction) axial 
force. The complete axial force coefficient Cx is finally obtained by summing up the relevant, separately 
calculated pressure axial force components and the viscous axial force obtained for entire wetted area. The 
total axial force coefficient of APFSDS projectile (without sabot) is [6]: 

 𝐶 = 𝐶 + 𝐶   (1) 

where 𝐶  is axial force coefficient of projectile body and 𝐶  is axial force coefficient of fins. 

2.1.1. Axial force computation methods for projectile body 

The axial force of a projectile consists of the pressure axial force of the nose, base (including possible tail 
boom), protruding (driving band, grooves and steps), and of the viscous axial force as a sum of the following 
form [6]: 

 𝐶 = 𝐶 + 𝐶 + 𝐶 + 𝐶   (2) 

where 𝐶  is axial force coefficient of the nose, 𝐶  is axial force coefficient of the base, 𝐶 is axial force 

coefficient of the protruding and 𝐶  is viscous axial force coefficient. 
The axial force coefficient of the nose at supersonic region for a cone is calculated according to the formula 
(3) giving the pressure coefficient on the nose surface [6]. The second term takes into account the nose shape 
on drag force (see Fig. 3). 

 
𝐶 = 𝐶 = 𝑘 sin 𝜀 + 𝑘

sin 𝜀 (cos 𝜀) ( ,  , )

𝑀 + [𝑅𝑅(1 − 𝑅𝑅)] , cos 𝜀
     (𝑀 ≥ 1)   (3) 

The coefficient k1 is an average pressure coefficient on a blunt projectile face behind a normal shock wave and 
the coefficient k2 takes into account the shape of the nose as a function of radius ratio parameter RR [6].  



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

4 

 
Figure 3. Variable RR (Radius Ratio) [6] 

 
The coefficient k1 is computed from (4), and the coefficient k2 is computed according to (5). The radius ratio 
RR in the formulae is an inverse of the ratio of the true radius of curvature and the tangent-ogive radius r’ 
(formula 6). The nose contour line is to be extended to the projectile center line in case of blunted nose (see 
Figure 3). The extended nose length is used in formula (6). The ratio RR is zero for cones [6]. 

 
𝑘 =

5

3
−

2

3 𝑀√
        (𝑀 ≥ 1) (4) 

 𝑘 = 0,9 − 0,9𝑅𝑅 + 𝑅𝑅  (5) 

 
𝑟 , =

𝑙,

𝑑
+

𝑑

4
  (6) 

The axial force coefficient of the base 𝑪𝒙𝒃
 is computed from formula (7) [6]: 

 

𝐶 = −𝐶
𝑑

𝑑
  (7) 

The pressure coefficient 𝐶  is computed as [6]: 
 

                                               𝐶 = 𝐶
𝑑

𝑑
       𝑥 = 2 when 𝑀𝑎 < 0.9, otherwise 1  (8) 

The pressure coefficient on the base of a long cylinder 𝐶  is computed at supersonic speeds (1.1 < Ma) [6]: 
 

𝐶 = −0,31𝑒 ,   (9) 

Viscous axial force coefficient 𝑪𝒙𝒇
 is calculated by formula [6]: 

 
𝐶 = 𝐶

𝑆

𝑆
  (10) 

where 𝐶  is average friction coefficient (11 or 12) for a smooth flat, 𝑆  is computed wetted surface area 

and S is reference area .   

The turbulent boundary-layer friction coefficient 𝐶  is computed by equation [6]: 
 

𝐶 =
0,455

(log 𝑅𝑒) ,
(1 + 0,21𝑀 ) ,   (11) 

where 𝑅𝑒 is Reynolds number , 𝑙 is projectile/nose length and 𝑣 is kinematic viscosity. 

The laminar boundary-layer friction coefficient 𝐶  is computed by equation [6]: 

 
𝐶 =

1,328

√𝑅𝑒
(1 + 0,21𝑀 ) ,   (12) 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

5 

The kinematic viscosity 𝑣 is computed from [6]: 
 

𝑣 =
𝜇

𝜌
  (13) 

The air density ρ is computed according to ICAO standard atmosphere. The dynamic viscosity μ is obtained 
from the Sutherland formula [6]: 
 

𝜇 =
𝐶 𝑇 ,

𝑇 + 𝐶
  (14) 

where  𝐶 = 1,458𝑒  
/

, 𝐶 = 110,4  [𝐾] and T is air temperature, obtained from ICAO atmosphere 

model. 

The axial force coefficient of protruding 𝑪𝒙𝒑𝒓
 is computed by estimating the forward and backward facing 

surface pressure drag separately [6]: 
 

𝐶 = 𝐶 + ∆𝐶   (15) 

The pressure coefficient sum (of the backward and forward facing parts) 𝐶  will change linearly 
between the sum and 0 when the ratio e/h (width/depth of groove) goes from 7 to 0 [6]. 
 

𝐶 =
𝑒

7ℎ
𝐶             

𝑒

ℎ
< 7  (16) 

The formulae for the pressure coefficients at velocities above speed of sound are [6]: 
 

𝐶 = (−0,067(𝑀 − 1) + 0,4) sin 𝜗 (17) 

 
𝐶 = −0,65𝑀 ,  (18) 

where 𝜗 is angle of grove profile. 
Certain types of finned projectiles have a relatively large groove pattern on the surface of the cylindrical part 
of body. These grooves are needed at the internal ballistic phase and after launch, the grooves cause an 
unfavorable flow retarding effect.  
The axial force coefficient of excessive amount of grooves (see Figure 4) is computed in from formula [6]: 
 

∆𝐶 = 1,6
𝑙

𝑙
𝐶 (𝐶 − 1) (19) 

The coefficient 𝐶  is the viscous drag coefficient (equation 10) of body cylinder part and the coefficient 

𝐶  is used to take into account the groove depth on drag. The coefficient is the surface area ratio of 
grooved cylinder length to that of same length cylinder without grooves; the incremental drag will be zero in 
case the surface coefficient 𝐶  is 1 [6]. 

 

Figure 4. Groove pattern area on the surface of the cylindrical part of body [6] 



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6 

2.1.2. Axial force computation methods for projectile fins 

The axial force coefficient of fins (𝐶 ) is computed in the function of Mach number, but also some effect of 
Reynolds number is introduced via skin friction. The reference area is a circle area based on the diameter of 
the projectile cylindrical part.  
The axial force coefficient of fins consists of the wave drag (𝐶 ), axial force coefficient of leading edge 

(𝐶 ), axial force coefficient of trailing edge (𝐶  ) and viscous force coefficient of fins (𝐶 ) 

as a sum of the following form [6]: 
 

𝐶 = 𝐶 + 𝐶 + 𝐶 + 𝐶   (20) 

The wave drag at supersonic speeds for fins with sharp leading/tailing edges (see Fig. 5) is calculated 
according to the formula (21) [6]: 
 

𝐶 = 𝑛
𝐾

𝑀

𝑡

𝑐

𝑆

𝑆
 (21) 

where: K is shape correction factor (see Fig. 5),  is average fin thickness ratio, S is reference area, 𝑆  is fin 

area (see Fig. 6), 𝑛  is number of fins. 

               

        Figure 5. Airfoil shape correction factor K [6]                             Figure 6. Projected area of fins [6] 

The formula (21) is applied when MLE ≥ 1 (Mach number normal to leading edge). MLE is supersonic if μ ˃ 
˄LEfin , where ˄LEfi is sweep angle of fin leading edge (see Fig. 7). The Mach angle μ is then computed from 
expression [6]: 
 

𝜇 = 𝑎𝑟𝑐𝑠𝑖𝑛
1

𝑀
 (22) 

The drag coefficient value is taken to be constant down to free-stream Mach number 1 if μ  ˄Lefin [6]. 

                          

   Figure 7. Exposed wing geometry without body [6]      Fig. 8. Schematic of fin blunt leading edge geom. [6] 

Axial force coefficient of leading edge is obtained by multiplying 𝐶  by the number of fins (𝑛  ), taking the 
reference area (S) and fin dimensions (Fig. 8) into account [6]: 
 

𝐶 =
𝐶 𝑡 𝑏

𝑆
𝑛  (23) 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

7 

The blunt leading edge average pressure coefficient 𝐶  is estimated by utilizing formula (24) [6]: 
 

𝐶 = cos ˄ 𝐶  (24) 

Parameter 𝐶  is estimated by utilizing formula [6]: 
 

𝐶 = 𝑘1 (25) 

Axial force coefficient of trailing edge is obtained by multiplying it by the number of fins (𝑛 ), taking the 
reference area (S) and fin dimensions (Fig. 9) into account [6]: 
 

𝐶 =
𝐶 𝑡 𝑏

𝑆
𝑛  (26) 

 
Figure 9. Schematic of fin blunt trailing edge geometry [6] 

The average pressure coefficient on a fin blunt trailing edge is computed at supersonic regions according to 
formula [6]: 
 

𝐶 = −0,65 𝑀 ,  (27) 

Viscous force coefficient of fins is calculated by same formulae (10) as in projectile skin friction drag. In fin 
case 𝑆  is the wetted surface area of fins, instead of the wetted surface area of projectile body and flow is 
consider to be turbulent [6]. 
 

𝐶 = 𝐶
𝑆

𝑆
 (28) 

The turbulent skin friction coefficient 𝐶  is computed by formula [6]: 
 

𝐶 =
0,455

(log 𝑅𝑒 ̅)
,

(1 + 0,21𝑀 ) ,  (29) 

where: 𝑅𝑒 ̅ is Reynolds number - 
̅
, 𝑐 ̅is Mean Aerodynamic Chord (see Fig. 7), 𝜈 is kinematic viscosity. 

2.2. CFD-model (Computed Fluid Dynamics) 

Numerical simulation methods, using computed fluid dynamics, are an important aspect of modern research 
because they complement experimental and analytical models, reducing total time and labor costs. In the 
experimental approach, most of the time is spent designing the experiment and making the model. In the 
computational approach, most of the time is spent on generating a geometric mesh (if the flow area is 
geometrically complex) and later analyzing the results.  
The main disadvantage of the computer approach is limitation to problems for which there is a reliable 
physical / mathematical model [7].  
In the general case, a numerical computational simulation consists of several main steps (Fig. 10):  

 Problem identification involves defining the objectives of numerical simulation, what are the 
modeling options, which physical models will be included in the analysis (viscosity, turbulence, 
compressibility), what simplifications can be used, whether user-defined functions should be used, 
what accuracy is required and how long it takes to get results [8]. 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

8 

 Preprocessing involves defining the geometry, mesh (space discretization), physical model, and 
solver used [8]. 

 Solver settings include solver type selection, discretization scheme, solution initialization, 
convergence monitoring (stability analysis), and accuracy check (mesh independence check and mesh 
adjustment on a specific part of the domain) [8]. 

 In the post-processing process, the results are examined and analyzed to understand the solution and 
extract useful data. Visualization tools in numerical programs make it possible to gain insight into the 
field of pressures or velocities, to visualize flow vectors, to predict the position of shock waves. [8]. 

 

Figure 20. Process of numerical simulation [8] 

2.2.1. Mathematical model 

Each simulation is based on a mathematical model, which denotes the mathematical notation of a physical 
model. The mathematical model includes the following assumptions [7]: 

 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. 

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 11). The surface element dS is defined by the unit vector of the normal  𝑛 [7]. 

 
Figure 11. Control volume [7] 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

9 

The characteristics of the air flow (pressure, tangential stress, velocity, temperature, etc.) are determined, at 
each point in space and at any time, by a system of differential equations [9]: 

 Law of mass conservation: 
 

𝜕

𝜕𝑡
𝜌 𝑑Ω + 𝜌(�⃗� ∙ 𝑛)𝑑𝑆 = 0 (30) 

 Law of momentum conservation: 
 

𝜕

𝜕𝑡
𝜌 �⃗�𝑑Ω + 𝜌�⃗�(�⃗� ∙ 𝑛)𝑑𝑆 = − 𝑝𝑛𝑑𝑆 + (�̿�  ∙ 𝑛)𝑑𝑆 (31) 

 Law of energy conservation: 

 
𝜕

𝜕𝑡
𝜌𝐸𝑑Ω + 𝜌𝐸(�⃗� ∙ 𝑛)𝑑𝑆 = 𝑘(∇𝑇 ∙ 𝑛)𝑑Ω − 𝑝(�⃗� ∙ 𝑛)𝑑𝑆 + (�̿�  ∙ �⃗�)𝑛𝑑𝑆 (32) 

where �⃗� is velocity of airflow, p is pressure, ρ is density, T is temperature, E is total energy, 𝜏̿ is stress tensor. 

2.2.2. Simulation of air flow around projectile 

The following will be adopted for all simulations [9]: 

 The problem will be observed as 3D geometry problem (because of fins). 
 The working fluid is air, an ideal gas, which is modified in accordance with compressibility and 

changes in thermo-physical characteristics with the temperature. Density and viscosity depends on 
temperature, where cp and thermal conductivity are considered constant. 

 The flow around the projectile is compressible and turbulent (k-ε model of turbulence was used). 
 Spatial domain discretization will be with a hybrid mesh. 
 The numerical density-based solver will be used, which simultaneously solves the equations of 

continuity, amount of momentum and energy. This method was developed for compressible high-
speed flows. 

 The equations will be linearized in implicit form, i.e. for given variables; unknown values in each cell 
will be computed using relations that include both existing and unknown values from adjacent cells. 

 A uniform air flow encounters a projectile at a zero yaw angle. 

In order to reduce the number of finite elements, only the projectile segment was taken - at an angle of 600 
(Figure 12). The calculation domain is limited by the external boundary of the projectile, symmetry planes and 
the outer boundary set at distances of 3 projectile lengths from the shell, 7 projectile lengths from the bottom 
and 6 lengths from the top (cylinder-shaped mesh) to avoid disturbances in free stream [9]. 
 

 
Figure 12. 3D model of APFSDS projectile 120 mm M829A2 [9] 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

10 

The following types of boundary conditions were selected (Figure13) [9]: 
 The "wall" boundary condition, which is used to separate the regions of fluid and solid matter, is 

placed on the outer boundary of the projectile. At the "wall" boundary condition, the "stationary wall" 
and "no-slip" options were chosen, because in the case under consideration, viscous effects cannot be 
ignored.  
The mass flux through the "wall" boundary is zero, and the pressure values at this boundary are 
obtained by extrapolation from inside the solution domain. 

 The "symmetry" boundary condition was used as a plane of axisymmetric geometry. 
 The "pressure far field" boundary condition, which is used to model the parameters of the 

compressible free stream at infinity, is set at the outer boundary of the calculation domain for given 
problem. 

 
Figure 13. Generated mesh around projectile 120 mm, M829A2, and boundary conditions applied [9] 



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

11 

3. Results and discussion 

For prediction of the accuracy of the engineering model (from the NATO related STANAG 4655 standard) 
and CFD numerical model, a comparison was made in the research with the PRODAS model. For calculation 
of axial force coefficients, APFSDS projectile, 120 mm, M829A2, was chosen. The reason for choosing this 
projectile model is because the aerodynamic coefficients are available from PRODAS program for this 
projectile model. 
The PRODAS software was developed to satisfy a need for rapid performance evaluation of ammunition 
characteristics. The development of an effective design/analysis tool for use by the design engineer in the 
development and evaluation of projectiles has been a multi-year project which began at General Electric in 
1972 and has continued at Arrow Tech Associates, Inc. since 1991. The developed tool is now called 
PRODAS which is an acronym for the Projectile Design/Analysis System Error! Reference source not 
found.. 
From the smallest match bullets, to GPS guided artillery shells, PRODAS brings together: 

 Modeling - Building a model from a drawing or even a picture. 
 Aerodynamics - Comparing aerodynamic coefficients from multiple aero estimators. 
 Launch Dynamics - Interior ballistics, balloting and jump. 
 Trajectories - Fly 4DOF, 6DOF and Body Fixed and Guided Trajectories. 
 Terminal Effects - Estimate penetration of KE projectiles and lethality of fragmenting or shaped 

charge warheads. 
 System Effectiveness - Using focused analysis or general purpose macros, compare projectiles or even 

GN&C algorithms Error! Reference source not found.. 

3.1. STANAG 4655 vs. PRODAS 

The axial force coefficients of the projectile body, predicted using the model from STANAG 4655 and 
PRODAS models are shown in Figure 14. The axial force coefficients of the projectile body predicted by the 
model from STANAG (Figure 14) shows a significant difference in the range of Mach 3 to 5. It can be seen 
that this difference decreases with increasing Mach number [9].  
The axial force coefficients of the projectile fins, predicted using the model from STANAG 4655 and 
PRODAS models are shown in Figure 15. The downward trend in the value of the axial force coefficient of 
the fins in the STANAG model is higher than in the PRODAS model. The differences between the values 
decrease slightly with increasing Mach number (Figure 15) [9]. 

Figure 14. Coefficients of axial force of the 
projectile body (120 mm, M829A2) [9] 

Figure 15. Coefficients of axial force of the projectile 
fins (120 mm, M829A2)  

The axial force coefficients of the projectile obtained by the model from the STANAG 4655 standard, are 
smaller than the coefficients obtained by the PRODAS model (for projectile 120 mm, M829A2). The 
difference between the values of the coefficients obtained with model from STANAG and PRODAS, 
decreases with increasing Mach number (Figure 16).  
The percentage difference of the coefficients obtained by the model from the STANAG in relation to the 
coefficients obtained by applying the PRODAS model is given in figure 17.   
From figure 17 it can be noticed that the largest percentage difference between the predicted values of the 
model from the STANAG and PRODAS is 16.3 % in the range of 3 to 5 Mach [9]. 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3 3.5 4 4.5 5

C X
_b
od

y

MACH

PRODAS
STANAG 4655

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

3 3.5 4 4.5 5

CX
_f
in

MACH

PRODAS
STANAG 4655



 DSS Vol. 1, No. 1, December 2020, pp.1-15 

12 

 
Figure 16. Coefficients of axial force of the 
projectile (Projectile 120 mm, M829A2) [9] 

 

3.2. CFD vs. PRODAS 

The results of the numerical simulation of air flow around projectile (120 mm, M829A2), in parallel with the 
results of PRODAS (function Cx = f (M)) are shown in figure 18. The largest deviations are recorded for M = 
3, where the difference is about 8.4 %. As the Mach number increases, the difference decreases by almost 1 
%. The percentage difference of the coefficients obtained by the numerical simulation in relation to the 
coefficients obtained by applying the PRODAS model is given in figure 19 [9]. 
Based on the obtained results of numerical simulations, the analysis of the pressure and velocity field around 
the projectile 120 mm, M829A2 was done. The resistance force of a fluid that opposes the motion of a body in 
it arises as a result of the action of a normal force and a tangential force on the surface of a body moving 
through the atmosphere. The source of fluid resistance to body motion are practically three natural 
phenomena: fluid viscosity, shock waves (at velocities M ≥ 1), and turbulence flow behind the body [9]. 
 

 
 

Figure 18. Coefficients of axial force of the projectile 
(Projectile 120 mm, M829A2) [9] 

Figure 19. Relative differences (CFD vs PRODAS) - 
Projectile 120 mm, M829A2 [9] 

The pressure and velocity fields, shown in Figures 20 and 21, are complex. From Figure 20, it can be seen that 
the free stream pressure is not the same for different Mach numbers. As expected for the supersonic flow, 
oblique shock waves appear at the top of the projectile, while a characteristic underpressure zone appears 
behind the projectile.  
During supersonic flow, underpressure is created behind the rear part of the projectile. This underpressure 
causes the streams to bend towards the projectile axis. Near the axis, the streams must bend again. In 
supersonic flow, the bent extension of the boundary layer draws air from the rear of the projectile so that a 
circular movement of air is created. Generally speaking, the supersonic flow regime of any body is 
characterized by pronounced shock waves, extremely narrow areas of fluid in which the flow properties 
change discontinuously (extremely) in a very short time interval, with pressure gradients being extremely 
large. In front of the shock wave, there is a zone of undisturbed flow, while behind it there is a zone in which 

0

0.2

0.4

0.6

0.8

1

1.2

3 3.5 4 4.5 5

C x

MACH

PRODAS
STANAG 4655

0

0.2

0.4

0.6

0.8

1

1.2

1.4

3 3.5 4 4.5 5

C X

MACH

PRODAS
CFD

Figure 17. Relative differences (STANAG vs 
PRODAS) - Projectile 120 mm, M829A2 [9] 



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13 

there are differences in the values of pressure, speed, temperature and density. Figure 21 shows the boundary 
layer that forms around the projectile in flight. It can also be seen that the angle of the oblique shock wave 
decreases with increasing velocity at which the flow is simulated [9]. 
 

 

Figure 30. Pressure field around the projectile for different Mach numbers [9] 

 



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Figure 21. Field of velocities around the projectile 120 mm, M829A2 for different Mach numbers [9] 



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4. Conclusion 

Based on theoretical considerations and analysis of available models (STANAG 4655 and CFD) for predicting 
the aerodynamic coefficient of axial force for wing-stabilized projectiles, the prediction of the axial force 
coefficient for APFSDS projectile 120 mm, M829A2 was performed. The data obtained using the engineering 
model (from the STANAG 4655 standard), and the data obtained by numerical simulation of projectile flow 
with the available data from the PRODAS database were compared. The following was stated: 

 The total axial force coefficients of the APFSDS projectile (provided with the model from the 
STANAG 4655 standard) have a satisfactory agreement with the total coefficient from the PRODAS 
model. The largest difference between the values is about 16.3 %. As the Mach number increases, the 
difference decreases. 

 The advantage of the STANAG 4655 model is that it allows the calculation of coefficients based on 
the geometric characteristics of the projectile without the use of computers. 

 The CFD model gives very good results, the values of the axial force or the axial force coefficient. 
Good agreement between the results of the CFD model and PRODAS indicates that the initial and 
boundary conditions are well set. 

 The accuracy of the CFD model depends on the mesh, initial and boundary conditions. The accuracy 
of the CFD model can be increased by modifying the mesh (i.e. by increasing the number of finite 
elements). 

Declaration of competing interest 

The authors declare that they have no any known financial or non-financial competing interests in any 
material discussed in this paper. 

Funding information  

No funding was received from any financial organization to conduct this research. 

References 

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faculty, Defense Technologies Department, Sarajevo, 2019. 

[2] W. Odermatt: http://longrods.ch/compo.php, October 2020 

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[9] Blazek, J.: Computational fluid dynamics: principles and applications, Elsevier Science Ltd, Oxford, 
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[10] A. Trakić: Axial force coefficients of APFSDS projectiles, Master thesis, University of Sarajevo, 
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