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The effic h, diameter an stly used tod ng Sabot), w ammunition N 2744-1741 inal Research s s y armored warehouse, f anti-tank ercing Fin et and with nd disables tic energy, luenced by o the laid fficient), it al ballistic maximum wo models ent) of an STANAG all types of xial force is the CFD D, e energy of a iency of thi nd density o day in armie which is made are shown 1 a s f s e n 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 = 53 − 23 𝑀√ (𝑀 ≥ 1) (4) 𝑘 = 0,9 − 0,9𝑅𝑅 + 𝑅𝑅 (5) 𝑟, = 𝑙,𝑑 + 𝑑4 (6) The axial force coefficient of the base 𝑪𝒙𝒃 is computed from formula (7) [6]: 𝐶 = −𝐶 𝑑𝑑 (7) The pressure coefficient 𝐶 is computed as [6]: 𝐶 = 𝐶 𝑑𝑑 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 = 𝜇𝜌 (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 = 𝑒7ℎ 𝐶 𝑒ℎ < 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 𝑙𝑙 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] A 2.1.2. Axia The axial for Reynolds nu the projectile The axial for (𝐶 ), ax as a sum of t The wave d according to where: K is area (see Fig Figu The formula ˄LEfin , wher expression [ The drag coe Figure 7. E Axial force reference are al force com rce coefficie umber is intr e cylindrical rce coefficie xial force co the following𝐶 drag at supe o the formula shape correc g. 6), 𝑛 is ure 5. Airfoi a (21) is app re ˄LEfi is sw 6]: efficient valu Exposed win coefficient o ea (S) and fin mputation m ent of fins (𝐶 roduced via s l part. ent of fins co efficient of t g form [6]: = 𝐶 ersonic spee a (21) [6]: 𝐶 ction factor ( number of fi l shape corre plied when M weep angle of ue is taken to ng geometry of leading ed n dimensions ethods for p ) is comp skin friction nsists of the trailing edge + 𝐶 eds for fins = 𝑛 (see Fig. 5), fins. ection factor MLE ≥ 1 (Mac f fin leading 𝜇 = 𝑎𝑟𝑐𝑠𝑖 o be constant without bod dge is obtaine s (Fig. 8) into 6 projectile fin puted in the . The referen wave drag ( e (𝐶 ) a + 𝐶 with sharp 𝐾𝑀 𝑡𝑐 𝑆𝑆 is average K [6] ch number n g edge (see F 𝑖𝑛 1𝑀 t down to fre dy [6] Fig [6] ed by multip o account [6 DSS ns function of M nce area is a (𝐶 ), a and viscous + 𝐶 leading/taili fin thicknes F normal to lea Fig. 7). The M ee-stream Ma . 8. Schemat plying 𝐶 by ]: S Vol. 1, No. Mach numbe a circle area b axial force co force coeffic ing edges (s s ratio, S is r Figure 6. Pro ading edge). Mach angle ach number 1 ic of fin blun the number 1, December er, but also so based on the oefficient of cient of fins see Fig. 5) reference are ojected area o MLE is supe μ is then co 1 if μ  ˄Lefin nt leading ed of fins (𝑛 2020, pp.1-15 ome effect o e diameter o leading edge (𝐶 ) (20) is calculated (21) ea, 𝑆 is fin of fins [6] ersonic if μ ˃ mputed from (22) n [6]. dge geometry ), taking the 5 f f e ) d n ˃ m y e A The blunt lea Parameter 𝐶 Axial force reference are The average formula [6]: Viscous forc case 𝑆 consider to b The turbulen where: 𝑅𝑒 ̅ i 2.2. CFD-m Numerical s because they experimenta computation geometricall The main d physical / m In the genera  Prob mod ading edge a 𝐶 is estimate coefficient o ea (S) and fin e pressure co ce coefficien is the wette be turbulent nt skin frictio is Reynolds n model (Comp simulation m y compleme al approach, nal approach ly complex) disadvantage mathematical al case, a num blem identif deling option 𝐶 average press ed by utilizin of trailing ed n dimensions𝐶 Figure 9. S oefficient on nt of fins is c ed surface are [6]. 𝐶 on coefficien𝐶 = (lo number - ,̅ puted Fluid methods, usin ent experime most of the h, most of th and later ana of the com model [7]. merical comp fication invo ns, which ph = 𝐶 𝑡 sure coefficie𝐶 = cos ˄ ng formula [6𝐶 = dge is obtain s (Fig. 9) into= 𝐶 𝑡 Schematic of a fin blunt 𝐶 = −0,65 calculated by ea of fins, in 𝐶 = nt 𝐶 is comp0,455og 𝑅𝑒 ̅) , (1 , 𝑐̅ is Mean A Dynamics) ng computed ental and an e time is spe he time is s alyzing the re mputer appro putational si olves defini hysical mod 7 𝑡 𝑏𝑆 𝑛 ent 𝐶 is esti˄ 𝐶 6]: 𝑘1 ned by multi o account [6𝑡 𝑏𝑆 𝑛 f fin blunt tra trailing edge 5 𝑀 , y same formu nstead of the 𝐶 𝑆 𝑆 puted by form1 + 0,21𝑀 ) Aerodynamic d fluid dynam nalytical mod ent designing spent on gen esults. oach is limit imulation con ing the obje dels will be DSS imated by uti iplying it by ]: ailing edge ge e is compute ulae (10) as wetted surfa mula [6]: ) , c Chord (see mics, are an dels, reducin g the experi nerating a g tation to pro nsists of seve ectives of n included in S Vol. 1, No. ilizing formu the number eometry [6] ed at superso in projectile ace area of pr Fig. 7), 𝜈 is important as ng total time iment and m geometric m oblems for w eral main ste numerical si n the analysi 1, December ula (24) [6]: of fins (𝑛 onic regions e skin friction rojectile bod kinematic v spect of mod e and labor making the m mesh (if the which there eps (Fig. 10): imulation, w is (viscosity 2020, pp.1-15 (23) (24) (25) ), taking the (26) according to (27) n drag. In fin dy and flow i (28) (29) iscosity. dern research costs. In the model. In the flow area i is a reliable : what are the y, turbulence 5 e o n s h e e s e e e, com wha  Prep solv  Solv conv adju  In th extra field 2.2.1. Mat Each simula model. The m  Air  Air  The  Air  Mas Continuity b momentum, arbitrarily se volume dΩ ( mpressibility) at accuracy is processing i ver used [8]. ver settings vergence mo ustment on a he post-proc act useful da d of pressure thematical m ation is base mathematica is a continuu is considered physical pro is a single-ph ss forces are behavior can and energy elected part (Figure 11). , what simp s required an involves def include s onitoring (sta specific part cessing proce ata. Visualiza s or velocitie Figu model ed on a math al model incl um. d a homogen operties of ai hase fluid. neglected. n be describe y. The equati of the contin The surface plifications c nd how long i fining the g olver type ability analys t of the doma ess, the resul ation tools in es, to visualiz ure 20. Proces hematical mo ludes the foll neous mixtur ir are the sam ed by transp ions derived nuum, the vo element dS i Figure 1 8 can be used, it takes to ge geometry, me selection, sis), and accu ain) [8]. lts are exami n numerical p ze flow vect ss of numeri odel, which lowing assum re of gases. me in all dire port equation d from the g olume Ω bo is defined by 1. Control vo DSS whether use et results [8]. esh (space discretizatio uracy check ined and ana programs ma tors, to predic cal simulatio denotes the mptions [7]: ctions - the a ns based on t given laws a unded by a y the unit vec olume [7] S Vol. 1, No. er-defined fu discretization on scheme, (mesh indep alyzed to und ake it possib ct the positio on [8] mathematic air is isotropi the basic law are presented closed area ctor of the no 1, December unctions sho n), physical solution i pendence che derstand the ble to gain in on of shock w cal notation o ic. ws of mass c d in integral - the limit o ormal 𝑛 [7]. 2020, pp.1-15 ould be used model, and initialization eck and mesh solution and sight into the waves. [8]. of a physica conservation form for an of the contro 5 d, d n, h d e al n, n ol The characte each point in  Law  Law  Law 𝜕𝜕𝑡 𝜌𝐸 where �⃗� is v 2.2.2. Sim The followin  The  The chan temp  The  Spat  The cont spee  The will  A un In order to r (Figure 12). the outer bou and 6 length eristics of th n space and a w of mass con w of momentu𝜕𝜕𝑡 𝜌 w of energy c 𝐸𝑑Ω + 𝜌 velocity of air mulation of a ng will be ad problem wil working flu nges in therm perature, wh flow around tial domain d numerical tinuity, amou ed flows. equations w be compute niform air flo reduce the n The calculat undary set a hs from the to Fi he air flow (p at any time, b nservation: 𝜕𝜕𝑡 um conserva �⃗�𝑑Ω + 𝜌 onservation: 𝜌𝐸(�⃗� ∙ 𝑛)𝑑𝑆 rflow, p is pr air flow arou dopted for all ll be observe uid is air, an mo-physical here cp and th d the projecti discretization density-base unt of mom will be lineari d using relat ow encounte number of fin tion domain at distances o op (cylinder- igure 12. 3D pressure, tan by a system o 𝜌 𝑑Ω + ation: 𝜌�⃗�(�⃗� ∙ 𝑛)𝑑𝑆 = 𝑘(∇𝑇 ressure, ρ is und projectil l simulations ed as 3D geom n ideal gas, characterist hermal condu ile is compre n will be with ed solver wi mentum and e ized in impli tions that inc ers a projectil nite elements is limited by of 3 projectil -shaped mesh model of AP 9 ngential stres of differentia 𝜌(�⃗� ∙ 𝑛)𝑑𝑆 = − 𝑝𝑛𝑑 ∙ 𝑛)𝑑Ω − density, T is le s [9]: metry proble which is m tics with the uctivity are c essible and tu h a hybrid m ill be used, energy. This icit form, i.e. clude both ex le at a zero y s, only the p y the externa le lengths fro h) to avoid d PFSDS proje DSS ss, velocity, al equations 𝑆 = 0 𝑑𝑆 + (𝜏̿ 𝑝(�⃗� ∙ 𝑛)𝑑𝑆 temperature em (because modified in a e temperature onsidered co urbulent (k-ε mesh. which simu s method wa . for given va xisting and un yaw angle. projectile seg l boundary o om the shell, disturbances i ectile 120 mm S Vol. 1, No. temperature [9]: ∙ 𝑛)𝑑𝑆 𝑆 + (𝜏̿ ∙ �⃗� e, E is total en of fins). accordance w e. Density a onstant. model of tur ultaneously as developed ariables; unk nknown valu gment was ta of the project 7 projectile in free stream m M829A2 [ 1, December e, etc.) are de �⃗�)𝑛𝑑𝑆 nergy, 𝜏̿ is st with compre and viscosity rbulence was solves the d for compr known value ues from adja aken - at an tile, symmetr e lengths from m [9]. [9] 2020, pp.1-15 etermined, a (30) (31) (32) tress tensor. essibility and y depends on s used). equations o essible high s in each cel acent cells. angle of 60 ry planes and m the bottom 5 at d n f h- ll 0 d m 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 [10]. 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 [10]. 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 Figure project 3.2. CFD vs The results o results of PR 3, where the %. The perc coefficients Based on the the projectile it arises as a through the phenomena: Figure 18. C (P The pressure the free stre oblique shoc behind the p During supe causes the s supersonic f circular mo characterized change disc large. In fron 0 0.2 0.4 0.6 0.8 1 1.2 3 C x 0 0.2 0.4 0.6 0.8 1 1.2 1.4 3 C X e 16. Coeffic tile (Projecti s. PRODAS of the numer RODAS (fun e difference centage diff obtained by e obtained re e 120 mm, M a result of th e atmosphere fluid viscos Coefficients o Projectile 120 e and velocit eam pressure ck waves ap projectile. ersonic flow streams to b flow, the ben vement of d by pronou ontinuously nt of the sho 3.5 M 3.5 M ients of axia le 120 mm, M rical simulat nction Cx = f is about 8.4 ference of th applying the esults of num M829A2 was he action of e. The sour ity, shock w of axial force 0 mm, M829 ty fields, sho e is not the s ppear at the , underpress bend toward nt extension air is creat unced shock (extremely) ock wave, the 4MACH 4MACH l force of the M829A2) [9 ion of air flo f (M)) are sho %. As the M he coefficien e PRODAS m merical simu s done. The r f a normal fo rce of fluid aves (at velo e of the proje 9A2) [9] wn in Figure same for dif top of the p ure is create ds the projec of the boun ed. General k waves, ext in a very s ere is a zone 4.5 PRODAS STANAG 465 4.5 PROD CFD 12 e 9] ow around pr own in figur Mach numbe nts obtained model is give ulations, the a resistance for orce and a ta d resistance ocities M ≥ 1 ectile Figu es 20 and 21 fferent Mach projectile, w ed behind th ctile axis. N ndary layer d lly speaking tremely narr short time in e of undisturb 5 55 5 DAS Fig PR DSS rojectile (12 re 18. The lar er increases, by the num en in figure 1 analysis of th rce of a fluid angential for to body m ), and turbul ure 19. Relat Projec , are comple h numbers. A while a chara he rear part Near the axi draws air fro g, the supers row areas of nterval, with bed flow, wh gure 17. Rela RODAS) - Pro S Vol. 1, No. 0 mm, M829 rgest deviatio the differenc merical simu 19 [9]. he pressure a d that oppose rce on the su motion are p lence flow be tive differenc ctile 120 mm x. From Figu As expected acteristic und of the projec s, the stream m the rear o sonic flow f fluid in wh h pressure gr hile behind it tive differen ojectile 120 m 1, December 9A2), in para ons are reco ce decreases ulation in re and velocity es the motion urface of a b practically t ehind the bo ces (CFD vs m, M829A2 [ ure 20, it can for the supe derpressure z ctile. This u ms must be of the projec regime of which the flo radients bein t there is a z nces (STANA mm, M829A 2020, pp.1-15 allel with the rded for M = s by almost 1 elation to the y field around n of a body in body moving three natura dy [9]. PRODAS) - [9] n be seen tha ersonic flow zone appear underpressure nd again. In ctile so that a any body i ow propertie ng extremely one in which AG vs A2 [9] 5 e = 1 e d n g al - at w, s e n a s s y h DSS Vol. 1, No. 1, December 2020, pp.1-15 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] DSS Vol. 1, No. 1, December 2020, pp.1-15 14 Figure 21. Field of velocities around the projectile 120 mm, M829A2 for different Mach numbers [9] DSS Vol. 1, No. 1, December 2020, pp.1-15 15 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). References [1] A. Ćatović: Anti-tank projectiles, Manual for students, University of Sarajevo, Mechanical engineering faculty, Defense Technologies Department, Sarajevo, 2019. [2] W. Odermatt: http://longrods.ch/compo.php, October 2020 [3] Z. Huang, Z. Chen: Numerical investigation of the tree-dimensional dynamic process of sabot discard, Journal of Mechanical Science and Technology, Vol 28, No 7, 2637-2649, 2014. [4] B. Zečević: Anti-tank Ammunition, Lectures for students, University of Sarajevo, Mechanical engineering faculty, Defense Technologies Department, Sarajevo, 2018. [5] S. S. Kadić: Prediction of drag force at zero yaw angle for conventional artillery projectiles, Master thesis, University of Sarajevo, Mechanical engineering faculty, Defense Technologies Department, July 2007. [6] STANAG 4655 Ed.1: An engineering model to estimate aerodynamic coefficients, NATO Standardization Agency, 18 January, 2010. [7] S. S. Kadić: Aaerodynamic, interior and exterior ballistic request, optimization in base bleed projectile design, PhD thesis, University of Sarajevo, Mechanical engineering faculty, Defense Technologies Department, Jul 2014. [8] A. Ćatović.: Prediction of terminal-ballistic parameters for natural fragmenting high-explosive projectiles using experimental data and numerical methods, PhD thesis, University of Sarajevo, Mechanical engineering faculty, Defense Technologies Department, Sarajevo, 2019. [9] Blazek, J.: Computational fluid dynamics: principles and applications, Elsevier Science Ltd, Oxford, United Kingdom, 2001. [10] A. Trakić: Axial force coefficients of APFSDS projectiles, Master thesis, University of Sarajevo, Mechanical engineering faculty, Defense Technologies Department, July 2020. [11] PRODAS V3, http://www.prodas.com/XQ/ASP/P.400/QX/webPageXML4.htm, Arrow Tech, 2020.