GAS-SOLID FLOW MODEL FOR MEDIUM CALIBER NAVAL GUN

Transcription

1 GAS-SOLID FLOW MODEL FOR MEDIUM CALIBER NAVAL GUN HAZEM EL SADEK School of Enery and Power Enineerin Nanjin University of Science and Technoloy, Nanjin CHINA XIAOBING ZHANG School of Enery and Power Enineerin Nanjin University of Science and Technoloy, Nanjin CHINA MAHMOUD RASHAD School of Enery and Power Enineerin Nanjin University of Science and Technoloy, Nanjin CHINA Abstract:- In this work the comutational fluid dynamics for two-hase flow of the interior ballistic rocess is modeled. This model is solved numerically in order to redict the interior ballistic erformance. The numerical simulation is carried out by MacCormack s technique deendin on the overnin equations of the two-hase flow. A self-adatin method is used to exand the comutational domain of the rojectile motion. The movin control volume conservation method is used to adat the movin boundary. This aroach is alied to 76mm medium caliber naval un with uided rojectile. The simulation of the two-hase flow model with the rojectile motion ives a ood areement between simulation results and exerimental results. Key-Words: - Guided rojectile, interior ballistic, two-hase flow modelin, comutational fluid dynamics, movin boundary. 1. Introduction Solid roellant is the most traditional roellant in un systems used to accelerate the rojectile inside the bore, after inition of the solid roellant a lare amount of enery is enerated inside the combustion chamber of the un, hence, ressure and temerature of the roducts increase. Once the ressure reaches the startin ressure value, the rojectile starts to move inside the barrel due to the exansion of the combustion roducts. The simulation of the interior ballistic rocess for 76mm medium caliber naval un was carried out before by usin the Lumed Parameter Model [1], but the two-hase Flow simulation model will be better as it has more details for the interior ballistic erformance with more accuracy, in addition; it ives the chance to study the interaction rocess between the as hase and the solid roellant hase. In order to describe the interior ballistic erformance, the two-hase flow simulation is required, the rimary urose of interior ballistic simulation is the rediction of the rojectile muzzle velocity and the eak ressure in the un system, the two-hase flow of interior ballistic rocess is very difficult to model, but it is a very challene in the interior ballistic field, due to many inter hase interaction, in addition, the comlex chemical and hysical henomena which occurred in a very short time under hih temerature and hih ressure conditions. At the initial time, the mixture of asroellant rains is laced in the combustion chamber and limited by breech at the left end, and E-ISSN: Volume 9, 2014

2 by the rojectile base at the riht end. The scheme of that initial eometry is shown in Fi.1 The momentum conservation equation of solid hase: [(1 ua] [(1 u u A] t [(1 R A] f A m u A (1 A s c (4 The enery conservation equation of the as hase: Fi.1. Schematic of initial eometry in the un system In this work, the theoretical and numerical simulation of the two-hase flow is carried in order to study the interior ballistic henomena, the simulation will be helful to describe the hysical and the chemical rocesses of the interior ballistic henomena; in addition it will reduce the number of the exeriments which is required for obtainin the otimal arameters for interior ballistic, the interior ballistic rocess were well demonstrated by some aers [2-6]. 2. Mathematical Model 2.1 Governin Equations The two-hase flow mathematical model is established to describe the hysical and chemical rocess in the interior ballistic henomena [7], in this model; it is considered that as hase and solid hase are continuum flows. The overnin equations are made for the mass and momentum conservation laws for both hases (as hase and solid hase, and the enery conservation for the as hase. These equations are shown as follows: The mass conservation equation of the as hase: ( A ( u A mc A min A t The mass conservation equation of the solid hase: [(1 A] [(1 u A] mc A t The momentum conservation equation of as hase: ( ua ( uua t f A m u A m u A ( A s c in in (1 (2 (3 [ A( e u u 2] [ u A( e t u u 2] Q A f u A m A( e s c ( A u u 2 minhin A t Where; is the volume fraction of the as hase, u, u are the as and solid velocity,, are the as and the solid density, (5 e, are the ressure and internal enery of the as hase, mc is the rate of as mass eneration due to roellant combustion, m is the mass flow rate of as from in vent holes of the initer, Hin is the stanation enthaly of the as flow from vent holes, fs, R, Q are interhase dra, interranular stress, and interhase heat transfer resectively. The above overnin equations, Eqs.(1-5, for one-dimensional two-hase flow of a nonlinear hyerbolic artial differential equations can be written in a vector form of conservation laws as: U E S t x (6 Where, U, E, S are the conserved variables, the flux vector and the source vector resectively. The comonents of U are the conserved variables: U1 A U (1 2 A U U 3 ua (7 U (1 4 ua 2 U ( / 2 5 e u A The comonents of E are the flux functions: E E1 E u A (1 u A 2 2 E 3 ua 2 E (1 ( 4 u R A 2 E ( / 2 / 5 u e u A (8 E-ISSN: Volume 9, 2014

3 And the comonents of S are the source term functions: S1 S2 S S 3 S4 S 5 mc A min A mk A ma c fs A mcu A minuin A mkuk A ( A fs A mcu A (1 A Q A fsu A mc A( e u u 2 ( A min Hin A mk H k A t ( (12 Where, 0 and are the initial and critical values of the as orosity resectively, and c can be exressed as a function in the as orosity as follows: 1 c1 0 k ( 0 c( c1e 0 0 ( Constitutive Relations The two-hase flow simulation needs some constitutive relations to close the above overnin equations. These relations describe the interaction between the as hase and the solid hase such as Nobel-Abel equation of state, interranular stresses, interhase dra force, and the interhase heat transfer Nobel-Abel equation of state The erfect as equation of state is inaroriate in interior ballistic rocess, hih temerature and hih ressures requires the real as equation of state, such as Nobel-Abel Equation: P( v RT (10 Where; P is the ressure, v is secific volume, is co-volume, R is secific as constant, and T is the temerature Interranular stresses The interranular stresses between the solid roellant articles can be calculated as follows: 2 0 Pc Pc 2 k (1 RP 1 e 0 2 k(1 0 ( Interhase dra force The interhase dra force can be calculated as follows: 1 fs u u u u * d (14 Where, u and u are as velocity and roellant articles velocity resectively, is as density, and d is the equivalent diameter of the roellant Interhase heat transfer The interhase heat transfer can be calculated as follows: Q (1 S q / M (15 Where, is the density of the solid roellant, S is the total surface are of the roellant, q is the heat transfer, and roellant. M is the remainin mass of The theoretical and numerical modelin of the interior ballistic cycle for the used initer is similar to the modelin of the two-hase flow in the un chamber, the assumtion and full details of the initer modelin are resented in [2]. E-ISSN: Volume 9, 2014

4 Density Velocity Pressure WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS 3. Numerical Solution 3.1 Numerical Technique The overnin equations are discretised in a finite volume manner on a reular mesh with movin boundary to solve the Riemann roblem, this solution is carried out by MacCormack technique which is an exlicit finite difference method with second order accurate in both sace and time. This technique consists of two stes, redictor ste and corrector ste. The redictor ste is based on the riht hand side with forward difference: t U U E E ts x tt t t t t i i i1 i i (16 The Corrector ste is based on the riht hand side with rearward difference and substitutin the t t redicted values of the time derivative U i at time (t t : t t 1 *[ t t t Ui Ui Ui 2 t E E ts x t t tt tt i i1 i ] ( Numerical Verifications The overnin equation in Eq. (6 contains two Euler equations for as hase and solid hase couled with the source term, the numerical solution for the overnin equations should be tested to verify the accuracy of the CFD code, in this work, Sod Shock Tube test is used [8], this test consists of one dimensional Riemann roblem with a comuted domain 1 x 1, this solution is comuted with 500 mesh cells, final time t=0.2, and CFL is 0.9, the closed-form exact solution of the Reimann roblem does not exist therefore an iterative scheme was carried out by Toro to obtain the exact solution of the Reimann roblem [9]. The initial states of the Reimann roblem are shown as: Where, is the density of the fluid, is the ressure, and u is the fluid velocity. The comarison between the numerical solution by usin MacCormack technique and the exact solution of Sod Shock Tube test is illustrated in Fi.2. As shown in Fi.2, the solution of the Reimann roblem by usin MacCormack technique is reasonable with the exact solution. (a Exact Test Distance (b (c 1.0 Exact Test Distance Exact Test Distance Fi.2. Verification of MacCormack technique by exact solution of the Riemann Problem 1.0 for x 0 ( x, for x for x 0 x (,0 0.1 for x 0 0 for x 0 ux (,0 0 for x 0 ( Grid Adatation In order to comute the osition of the rojectile base x which describe the extension of the last mesh cell behind the rojectile base, we aly the fundamental rincile of dynamics: dv dt A P m (19 1 E-ISSN: Volume 9, 2014

5 Where; v is the rojectile velocity, A is the barrel cross-section, P is the ressure at the rojectile base, 1 is coefficient of secondary enery losses, and m is the rojectile mass. The comutational domain has a fixed mesh cells with lenth x before movin of the rojectile, once the rojectile start to move, the comutational domain beins to chane as the lenth of the last mesh cell beins to exand with lenth x1 at time t, the rid adation alorithm is carried out to solve this dynamic mesh roblem. On the next iteration the time is denoted byt and the lenth of the last mesh cell will be shown in Fi.3. ( a ( b J-1 J x1 Fi.3. Schematic of Grid Adation t x2 The dislacement of the rojectile at the new time ste t t will be calculated as follows: x2 x1 v t (20 The lenth of the last mesh cell at the rojectile base will increases at every time ste, the Self- Adation alorithm is carried out to adat this exansion and enerate a new mesh cell accordin to the followin conditions: If x2 1.5 x1, then: The number of mesh cells will be the same, exand the lenth of the last cell x1to be x2. If x2>1.5 x1, then: Add a new cell with lenth x, and of x1 will take the value x2- x. 3.4 Boundary Conditions In this work, the reflective boundary conditions are used for both left and riht ends of the comutational domain before the rojectile start to move. Once the rojectile beins to move, the left v t J-1 J x2 At time t At time t t as end will be fixed and the reflective boundary condition is carried out, but the riht end need to move behind the rojectile, and another suitable boundary condition method is required, the movin control volume conservation method is used to handle the motion of the riht end at the rojectile base[10]. 4. Results and Analysis In this work, the two-hase flow code is carried out to 76 mm medium caliber naval un, the detailed data of the used un is illustrated as follows in Table Pressure Distribution As shown in Fi.4, once the ressure at Vent- Holes inside the initer reaches 20 MPa, the Vent- Holes rutures and the flame jet flows from the initer to the chamber enetratin the roellant at time 1.08 ms, the roellant starts the inition at the broken Vent-Holes and the ressure will increase radually inside the chamber, once the ressure at the rojectile base reaches 30 MPa, the rojectile starts to move inside the bore, and the ressure continue in increasin till it reaches the maximum ressure inside the un at time 5.4 ms, then the ressure decreases radually until the rojectile exit from the muzzle and the interior ballistic rocess ends. Table 1 Data of 76 mm Naval Gun Parameter Value Unit Gun Geometry Gun caliber m Tube lenth m Chamber lenth 0.38 m Projectile mass 5.9 K Chamber volume m 3 Proellant (Double Base 7-Perforated Mass 2.66 K Imetus force J/K Inition tem. 615 K Co-volume m 3 /K Density 1550 K/ m 3 E-ISSN: Volume 9, 2014

6 Fi.4. Pressure history on x-t diaram.. Fi.6. Solid Phase Velocity Profile on x-t diaram. 4.2 Velocity Profile Fiures 5 and 6 Show the velocity rofile for as and solid hases in the interior ballistic rocess. Due to the jet flow from initer to chamber throuh the Vent-Holes, the as roducts will be formed. Durin the time from 1ms to 2ms some neative values for velocity are observed, as the as roducts move towards the breech of the un and the rojectile base to allow the roellant inition as shown in Fi.5. After the rojectile stars to move the as velocity behind the rojectile will take the value of rojectile velocity, leavin the solid roellant behind it, thus it is observed that the values of solid velocity are less than the values of as velocity as shown in Fi Gas Volume Fraction Profile The as volume fraction starts in the initial stae by the initial value accordin to the roellant weiht and the combustion chamber volume. Once the Vent-Holes ruture, the as volume fraction increases in the Vent-holes reion as the releasin of the as from initer to chamber. The roellant bed in the chamber starts to inite due to the left and riht travellin waves of the flame leadin to decreasin of the roellant orosity and increasin of the as volume fraction until the comlete burnin stae. At this stae, the as orosity attains its maximum value as illustrated in Fi.7. Fi.7. Gas Volume Fraction Profile on x-t diaram Fi.5. Gas Phase Velocity Profile on x-t diaram. 4.4 Temerature Profile At the initial hase of combustion, the as temerature increases raidly until it reaches the maximum value, 2450 K, at time 6.5ms. Then it starts to decrease radually as the volume behind the rojectile increase due to the rojectile movement down the un bore as shown in Fi. 8. E-ISSN: Volume 9, 2014

7 Tab.2, this table shows an accetance and areement between the exerimental and simulation results. Table 2. Comarison between exerimental and numerical results: Fi.8. Gas Temerature distribution on x-t diaram IB Parameter Maximum chamber ressure [MPa] Muzzle velocity [m/s] Exerimental Results Simulation Results While, at the beinnin of the inition rocess of the main roellant chare around the initer function the solid roellant articles near to the Vent-holes will firstly inite, then the flame roaates in both sides of the un chamber causin the inition of the whole roellant. The comlete inition takes lace at about time 1.5 ms as shown in Fi. 9. At that time the whole roellant is burned u and delivers all its enery to the rojectile as a kinetic enery excet some enery lost due to heat transfer to the un barrel. 6. Conclusions The numerical simulation for the two-hase flow of the interior ballistic cycle is carried out for 76mm naval medium caliber un with uided rojectile, the overnin equations are discretised in a finite volume manner on a reular mesh with movin boundary, the movin control volume conservation method is used to handle the motion of the rojectile inside the barrel, the Self-Adation alorithm is used to handle the exansion of the comutational domain behind the rojectile base. The solution is executed by MacCromack technique, this technique has been tested and validated to ensure that it is accurate and can solve the twohase flow code with movin boundary. The twohase flow simulation ives a ood areement results comared with the exerimental results. The simulation will be helful to achieve the required interior ballistic erformance for the uided rojectile by chanin the interior ballistic arameters to increase the muzzle velocity and decrease the eak ressure which are required for the uided rojectiles. Fi.9. Solid Temerature distribution on x-t diaram 5. Validation of the simulation results The results from the interior ballistics exeriment are limited, the velocity can be measured only at the muzzle, and the ressure can be measured at the breech, the comarison between the exerimental and simulation results are shown in Acknowledements This work is suorted by the Natural Science Foundation of Jiansu Province (No. BK References [1] H. Elsadek, X.B. Zhan and M. M. Rashad, Parametric study on mixed roellant for the medium caliber naval un uided rojectile, Journal of enineerin and alied science, Vol.60, No.1, 2013, E-ISSN: Volume 9, 2014

8 [2] Y.X. Yuan and X.B. Zhan, Multihase hydrokinetic foundation of hih temerature and hih ressure, Publishin Comany of Harbin Institute of Technoloy, Harbin, [3] N.C. Markator, Modelin of two-hase transient flow and combustion of ranular roellants, Int. J. Multihase Flow, Vol 12, No.6, 1986, [4] T. Sheu, S.M. Lee, Analysis of combustion rocesses in a un interior ballistics, International Journal of Comutational Fluid Dynamics, Vol.4, No. 1, 1995, [5] J. Nussbaum, P. Helluy, Numerical simulations of as-article flows with combustion, Flow Turbulence And Combustion, Vol. 76. No. 4, 2006, , doi: /s [6] H. Miura, A. Matsuo, Numerical rediction of interior ballistics erformance of rojectile accelerator by solid/as two hase reactin flow simulation, 48th AIAA Aerosace Sciences Meetin Includin the New Horizons Forum and AerosaceExosition, Vol. 48, No. 1, 2010, , doi: / [7] C. Lowe, CFD Modelin Of Solid Proellant Inition, 'PhD thesis', Cranfield University; Cranfield, [8] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyerbolic conservation laws, Journal of Comutational Physics, Vol. 27, No. 1, 1978,. 1-31, doi: / ( [9] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, sriner, [10] S.G. Ahmed, A new alorithm for movin boundary roblems subject to eriodic boundary conditions, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 16,No. 1, 2006, , doi: / E-ISSN: Volume 9, 2014