An Approximate Model for the Theoretical Prediction of the Velocity Increase in the Intermediate Ballistics Period

Transcription

1 An Approxiate Model for the Theoretical Prediction of the Velocity Central European Journal of Energetic Materials, 205, 2(), ISSN An Approxiate Model for the Theoretical Prediction of the Velocity Increase in the Interediate Ballistics Period Radosław TRĘBIŃSKI, Marta CZYŻEWSKA Departent of Mechatronics and Aviation, Military University of Technology, Gen. S. Kaliskiego 2, Warsaw, Poland E-ails: Abstract: This paper is a continuation of an earlier work [] by the authors devoted to CFD odelling of the interediate ballistics period. A sei-analytical odel is proposed based on the analytical solution of the Prandtl-Meyer expansion for an approxiate assessent of the agnitude of the velocity increase in the interediate period. Two variants of the odel provide upper and lower estiates of the velocity increase. A coparison of these estiates with the results of CFD odelling gives a siilar agnitude for the velocity increase. A forula is proposed for a siple estiation of the velocity increase that can be used to calculate a correction to the uzzle velocity value obtained by internal ballistics odels. Keywords: interediate ballistics, internal ballistics, projectile velocity Introduction Our previous work [] was devoted to the presentation of the results of odelling of the interediate ballistics using CFD ethods. The ai of this odelling was to estiate the agnitude of the velocity increase Δu in the interediate ballistics period. It was shown that the relative increase in velocity is generally less than % and only for very low uzzle velocities does it exceed this value. Although the CFD odel predicts the velocity increase quite precisely, it cannot be used in the fire control systes, because it needs a long coputing tie. That is why a siple odel, requiring only several seconds of coputing tie,

2 78 R. Trębiński, M. Czyżewska is desirable. In this paper we present an approxiate, sei-analytical odel of the interediate ballistics. This odel is based on the sae physical odel as that presented in []. However, a copletely different approach to solving the atheatical odel is applied. The odel can be easily ipleented in fire control systes. Moreover, based on the results of [], a siple forula is proposed for estiating the value of Δu. The forula can be used for calculating a correction to the uzzle velocity calculated usinginternal ballistics odels. 2 Description of the Model The results of the odelling presented in [] proved that the period of tie in which the projectile undergoes acceleration after leaving the uzzle is very short. During this period the flow near the uzzle is doinated by the Prandtl-Meyer (P-M) expansion and its reflection at the axis (Figure ). Figure. Structure of the flow near the uzzle. An approxiate odel can be proposed for estiating the ean pressure acting on the tail of a projectile based on the analytical solution for plane P-M flow. The solution is given by the following relations [2]: u = c, u = 2 w+ u c, u = u + u ( ρ ) R R ϕ ρc p d c dp u ϕ = ϕ, w=, ϕ arccos arccos ρu = = c R ρ c ρ p ϕ ( M ) M M () where: u R, u φ ean noral and tangential coponents of the velocity of flow u, respectively; ρ density; p pressure; c sound velocity; Dw specific enthalpy increent; u, p, r, c, M, φ M uzzle velocity, pressure, density, velocity of sound, Mach nuber and Mach angle, respectively. For a perfect gas with the exponent k, the following relation between pressure p and the angle φ can be derived fro the relations ():

3 An Approxiate Model for the Theoretical Prediction of the Velocity { cos 2( ϕ ϕ ) 3 sin 2 ( ϕ ϕ ) } p = p w w w M M 2k k 2 + ( k ) M w =, w2 =, w3 = k k+ k + 2 w (2) Based on forula (2), we can find the value of the static pressure at the tail of the projectile at a point reached by a characteristic with inclination angle φ. In order to estiate the value of the dynaic pressure, we can ake use of the following forula for the axial coponent of the velocity of the flow: u = c 2 + M k+ s sinϕ+ s cos ϕ, s = p, w = w 2 2w4 w4 x 4 k k p (3) An approxiate value of the dynaic pressure can be calculated using the following forula: w u 2 5 x k p= ρc( ux u) = kps M, w5 = c k+ (4) In this forula the velocity of the projectile is assued to be equal to u, which is a good approxiation, taking into account the fact that the velocity increase is sall. The su of p and Δp gives an approxiate value of the pressure at a point at the tail of the projectile, which is reached by a characteristic having an inclination angle φ. However, this approxiation is valid only until the oent when the reflected wave reaches that point. In order to take into account the reflection, we used the acoustic approxiation. For this approxiation the reflection of a wave can be considered as a superposition of two waves. The waves are syetric against the reflecting plane (we used the solution for plane syetry). This eans that, at a given characteristic, both waves have the sae value of the pressure and the axial velocity coponent, while their radial coponents of velocity have equal values but opposite signs (see Figure 2). Therefore, we can copute p r and Δp r for the reflected wave using forulae (2) and (4).

4 80 R. Trębiński, M. Czyżewska Figure 2. Illustration of the idea of reflection of the wave at the plane of syetry. The procedure for the estiation of the ean pressure is as follows. For a given position of the projectile, pressure is calculated at n+ points, having radial coordinate r i = r (i-)/n. For each point, the inclination angles of the characteristics of incident and reflected waves are calculated (Figure 3 characteristics are assued to be rectilinear). Figure 3. Illustration of the procedure for the calculation of the inclination angles. x arctg r r x i ϕ =, i =, n, ϕr = arctg, i =, n+ π r + ri, i = n+ 2 (5) For a given value of φ, we calculate p and Δp. Then, for a given value of j r we calculate p r and Δp r. If φ <φ M, we put p = p, Δp = 0. Siilarly, if j r <φ M, we put p r = p, Δp r = 0. We then calculate the resultant pressure, assuing linear interference of the waves:

5 An Approxiate Model for the Theoretical Prediction of the Velocity... 8 pi = p+ pr p + p+ pr (6) The ean value of the pressure acting on the tail of the projectile is calculated as: n 2 2 ri p = ( r r )( p + p ), r = r av 2 i+ i i i+ i i= (7) Taking into account that x = u t (we can neglect the velocity increase in assessing the distance the projectile has oved), we can find the dependence of the ean pressure on tie and then integrate the equation of otion of the projectile. The results of [] suggest that the ean pressure value acting on the forward part of the projectile can be neglected. The odel described above does not take into account the cylindrical syetry. Due to this syetry, the pressure in the P-M flow decreases faster than for plane syetry. Therefore, we can expect that the approxiate odel will provide an upper estiate of the pressure values. We can take the cylindrical syetry into account in an approxiate anner. The continuity equation for cylindrical syetry has the for: ( ρ ) ( ρ ) ρ u u ρu = t x r r x r r (8) For an isentropic flow we can represent it in the for: ( ρv ) ( ρu ) p kpu = c t x r r 2 x r r (9) The first su and on the right hand side of forula (9) represents the pressure derivative for plane syetry. Thus, integrating forula (9) along the projectile trajectory we obtain: t kp( uτ, r, τ) ur( uτ, r, τ) p= pp dτ r 0 (0) We can treat the integral on the right hand side of forula (0) as a correction

6 82 R. Trębiński, M. Czyżewska ter to the solution for plane syetry. Therefore, we can calculate p P in the way described above and then correct its value. We estiate values of the radial velocity coponent u r fro the relation for plane syetry: 2 2 k 2w4 w4 u ( ϕ + r ) c M s cos s sin k k ϕ ϕ = + + () In the case of interference of the incident and reflected waves, we calculate the value of u r as: u = u ( ϕ) u ( ϕ ) r r r r (2) The integral in forula (0) is calculated nuerically, aking use of the values of p i and u ri in consecutive tie intervals. We can expect the above approxiation to give a lower estiate of the pressure because treating the characteristics as rectilinear causes a given characteristic to reach the projectile tail sooner than it should (Figure 4). This causes the pressure to drop faster than for the real characteristics. Figure 4. Illustration of the difference between rectilinear and real characteristics. The approxiate odel can be used with soe odifications in the case when the velocity of the propellant gases at the starting oent of the interediate ballistics period is subsonic. In this case, a rarefaction wave enters the uzzle. This accelerates the gases and as a result the flow at the uzzle becoes sonic. This eans that the first characteristic of the P-M flow is positioned at the uzzle. Therefore, we can use the approxiate odel described above to calculate the average pressure acting on the projectile. However, we need to know the new paraeters at the uzzle, which are different fro the results of the internal

7 An Approxiate Model for the Theoretical Prediction of the Velocity ballistics calculations. For this reason, we propose the following approxiation. We assue that at the initial oent, a centred rarefaction fan is generated inside the barrel (Figure 5). Based on this approxiation, we can calculate the velocity and pressure at the uzzle, based on the relation for the centred rarefaction wave: Figure 5. Wave diagra showing the rarefaction fan generated at the oent the projectile leaves the uzzle. 2 2 u0 + c0 = u + c k k (3) where u 0 and c 0 ean flow velocity and velocity of sound at the final oent of the internal ballistics period respectively. Because u = c at the uzzle, we can obtain the following forula fro (3): k 2 k 2 u = c = u0 + c0 = c0 M0 + k+ k k+ k (4) The values of the other paraeters at the uzzle can be calculated by the forulae: 2 2 k c ρc ρ0 p M c0 k ρ =, =, = (5) We can use the calculated values in the forulae derived earlier, except for forula (4), where we substitute u 0 and M 0 = u 0 /c for u and M, respectively.

8 84 R. Trębiński, M. Czyżewska 3 Results and Discussion The results obtained with the approxiate odel are copared with the results of the odel described in [] (subsequently referred to as the CFD odel) in Figure 6. This presents the tie dependence of the calculated ean pressure acting on the projectile tail for a 30 caliber launching syste. Zero tie corresponds to the oent when the projectile leaves the uzzle and the expansion of the propellant gases begins (this also applies to Figures 7 and 8). The results fro the approxiate odel differ considerably fro the CFD odel results. However, the variants of the approxiate odel (without and with the correction for cylindrical syetry) give upper and lower estiates of the pressure values. This eans that they also give upper and lower estiates of the velocity increase. The calculated Δu values for a 30 launching syste are given in Table. The first row refers to standard aunition, whilst the second and third rows refer to aunition with 20% larger and 20% lower powder ass respectively. In all cases, the values of the velocity increase calculated by the CFD odel are between the values calculated by the two variants of the approxiate odel and quite close to the average of the upper and lower estiates. This result strengthens our belief that the CFD odel properly predicts the agnitude of the velocity increase in the interediate ballistics period. Figure 6. p(t) plots for the CFD odel and two variants of the approxiate odel (30 launching syste).

9 An Approxiate Model for the Theoretical Prediction of the Velocity Table. Results of calculations of the velocity increase in the interediate ballistics period by the CFD odel and the two variants of the approxiate odel (plane and cylindrical) for a 30 launching syste Variant Dv CFD, [/s] Dv plane, [/s] Dv cylindr, [/s] Dv av, [/s] standard powder ass % larger powder ass % saller powder ass The results obtained for a subsonic case are illustrated by the p(t) and u(t) curves, calculated for a 9 launching syste, shown in Figure 7. In this case, the differences between the results of the approxiate odel and the CFD odel are uch larger than those for the supersonic case, especially for the variant for plane syetry which differs greatly fro the results for the CFD odel. The reason for this is obvious, because in the subsonic case the propellant gases are accelerated in the radial direction uch ore intensely than in the supersonic case. Figure 7. p(t) and u(t) plots for the CFD odel and the two variants of the approxiate odel (9 launching syste). The approxiate odel can be used for a rough estiation of the velocity increase in the interediate ballistics period. However, based on the results of the CFD odel, a uch sipler estiation can be proposed. In our previous paper [], we invoked results of easureents of the acceleration of projectiles presented in [3] and [4] in order to validate the

10 86 R. Trębiński, M. Czyżewska CFD odel. In these papers, the results of easureents of the acceleration of projectiles in the interediate period were used for deterining the pressure acting on the projectile. The dependence of the pressure on tie was approxiated by an exponential decay: p() t = p e βt 0 (6) We found that such an approxiation fits well to our results fro the CFD odelling. Making use of this approxiation, we obtained the following for of the projectile equation of otion: p du dt p = S pe p 0 βt (7) Integrating this equation, we obtained: S p u t = e (8) p 0 p () βp βt ( ) where S p eans the area of the tail of the projectile and p its ass. The coefficient β has the diension of the reciprocal of tie. Assuing that the uzzle velocity u 0 is a characteristic velocity, while the caliber d is the characteristic length, we can express the coefficient β by as a non diensional coefficient γ in the following way: u 0 β = γ d (9) In general, the coefficient γ ay be a function of several non diensional paraeters characterizing the process. However, we assue that it is only a function of the uzzle Mach nuber M. Fro forulae (8) and (9) we have: u ( M ) S pd/2 pax p u0 u = γ /2 (20) This forula suggests that the relative velocity increase is proportional to the ratio of the work done by the uzzle pressure on a path equal to half of the caliber to the kinetic energy of the projectile at the uzzle. In order to find the values of the coefficient γ (M ), the results of calculations presented in [] for

11 An Approxiate Model for the Theoretical Prediction of the Velocity various launching systes are used. These are presented in Table 2. Based on the Δu CFD values, the coefficients γ have been estiated and the results are shown in Figure 8. Table 2. Results of calculations of the velocity increase in the interediate ballistics period by the CFD odel for various launching systes and by forulae (20) and (2) Caliber, [] Δu CFD, [/s] Δu pax, [/s] Error, [%] 5.56 () * (2) ** * Assault rifle Beryl; ** New assault rifle under developent. Figure 8. Dependence of the coefficient γ on the Mach nuber M. The dependence of γ on the Mach nuber M is approxiated by the linear relation: ( M ) γ = + M (20)

12 88 R. Trębiński, M. Czyżewska The assuption that the coefficient γ is only a function of the Mach nuber M is a rough approxiation. This is deonstrated by the differences between the regression line in Figure 8 and the points corresponding to the calculated values of γ. Nevertheless the calculated values of the velocity increase differ fro the results fro the CFD odel by no ore than 20% (see Table 2). This precision is acceptable because the relative increase in the velocity, as a rule, does not exceed % of the uzzle velocity. Therefore, the proposed forulae can be used to calculate a correction to the uzzle velocity deterined by the internal ballistics odels, which provide values of u 0, p 0 and M 0. 4 Conclusions The ain results of the work can be suarized as follows:. The estiates obtained for the projectile velocity increase in the interediate ballistics period give an upper and a lower liit of the results obtained by the use of the CFD odel described in []. 2. The average value of the two estiates gives a good approxiation of the results obtained by the CFD odelling for the case when the flow of propellant gases at the uzzle is supersonic. In the case of subsonic flow, only the lower estiate can be used as an approxiate value of the velocity increase. 3. The proposed approxiate forula approxiates, with acceptable accuracy, the results of the CFD odelling for various launching systes. 5 References [] Trębiński R., Czyżewska M., Estiation of the Increase in the Projectile Velocity in the Interediate Ballistics Period, Cent. Eur. J. Energ. Mater., 205, 2(), [2] Landau L.D., Lifshitz E.M., Fluid Mechanics, Pergaon Press, Oxford, 963. [3] Carlucci D., Vega J., Epirical Relationship for Muzzle Exit Pressure in a 55 Gun Tube, WIT Trans. Modell. Siul., 2007, 45, [4] Carlucci D., Frydan A.M., Cordes J.A., Matheatical Description of Projectile Shot Exit Dynaics (Set-Forward), J. Appl. Mech., 203, 80,