Peng-fei Li,Xiao-bing Zhang

School of Energy and Power Engineering,Nanjing University of Science and Technology,Nanjing,210094,PR China

Keywords: Muzzle flow Muzzle brake Secondary combustion Chemical reaction

ABSTRACT The simulation of the artillery interior and intermediate ballistics problem is performed to investigate the in fluence of a gas dynamics device,muzzle brake,on the muzzle hazard phenomena,such as flash and blast waves.The correlation of the chemical reactions with the characteristics of the muzzle flow field is analyzed by the simulation for a further understanding of the secondary combustion phenomenon of the muzzle flow.The novel structure of muzzle flow caused by the muzzle brake is presented by the simultaneous solution of the interior ballistics model and multi-species Navier-Stokes equations in order to analyze the in fluence of the muzzle brake structure on the chemical reactions.The secondary combustion of the muzzle flow due to the oxygen-supplement chemical reactions is obtained by the chemical reaction kinetic model.The interaction of the blast waves released from the muzzle brake is illustrated in detail and the mechanism of the formation of muzzle flash is analyzed.This research provides a reference for the studies on the suppression of the muzzle flash.

1.Introduction

Muzzle brake is an indispensable device for the lightweight design of artillery.However,the application of some speci fically structured muzzle brake results in violent muzzle flash and severe impingement on the equipment and people behind the muzzle.Therefore,a detailed investigation of the in fluence of the muzzle brake is essential to control the shock wave and the secondary combustion of the muzzle flow.This study laid the focus on the effect of the speci fic structure of a muzzle brake and the reaction mechanism of the propellant gas.

Muzzle flow has been widely investigated both experimentally and numerically for decades as the explosive releasing of the propellant gas causes a variety of fatal hazards[1],and many reliable numerical methods were proposed[2-5].Besides,the interaction between the precursor flow and the propellant gas flow was analyzed[6-10].Schmidt et al.studied the muzzle flow considering various realistic factors,these studies provided sufficient reference and theoretical basis for the research on the muzzle flow[11-13].With further research,scholars paid more attention to the negative effects of the muzzle flow[14-16].The problem becomes more complicated when a muzzle brake is used for better operational performance.Muzzle brake is a device in the muzzle part of the cannon barrel which employs the forward momentum of the propellant gas to balance the recoil impulse on the barrel.But the side effect of using a muzzle brake is the increase of the overpressure of muzzle flow around the muzzle because the propellant gas is de flected and guided by the lateral slots[17].As a consequence,the shock waves caused by the de flected propellant gases pose a threat to the personnel and equipment on the side and rear of the muzzle.Recently,a lot of studies have been carried out to investigate the details about muzzle brake,such as the stress distribution of muzzle brake,recoiling ef ficiency,and the optimization of the structure to suppress the severe muzzle flow impingement[18-22].In this study,the secondary combustion phenomenon of the muzzle flow is investigated by the numerical method considering the in fluence of the lateral slots of the muzzle brake.The numerical method adopted in this study is the multi-model coupling method,which takes into account the entire shooting process of the artillery and the special characteristics of the muzzle flow field with a muzzle brake.Besides,the relationship between the chemical reactions and the particular structure of the muzzle flow field is revealed by a detailed analysis of multiple computational data.

In terms of the combustion of the propellant gas,Klingenberg et al.studied the distribution of the muzzle flows and identi fied the developing process of the muzzle flash as three stages,which are the primary flash,intermediate flash and secondary flash[23,24].It is concluded that the ignition of the secondary combustion is determined by the shock heating effect of the gas pressure and the static gas temperature.The oxygen is mixed with the gas rest with the turbulence of muzzle flow[25].Choi and Shin investigated the combusting process of the base bleed projectile,the detailed numerical method and discussion were presented[26].In their research,the equations of the propellant components were employed to calculate the parameters of the propellant gas based on the theory provided by Gordon and McBride[27].Further investigation was conducted by Zhuo and Feng,the dynamic overlapping grids method was applied to simulate the muzzle flow with a base bleed projectile,and the chemical kinetics model was employed to calculate the chemical reactions in the combustion region of the muzzle flow[28].Qin and Zhang described the secondary combustion phenomenon in detail and analyzed the in fluence of the precursor flow with an inert gas labeling method[29].

The methods and simpli fied models proposed in previous investigations provide abundant references for the study on the complicated muzzle flow.In this study,the entire shooting process of the cannon equipped with a speci fic muzzle brake is simulated.For an accurate result,the simulation is conducted by Fluent(Ansys Inc.)finite volume code coupled with the interior ballistics model.The chemical reaction kinetics model is employed to describe the secondary combustion phenomenon, the chemical nonequilibrium Navier-Stokes equations are solved by the AUSM+scheme and the finite-rate chemistry interaction model[30].The turbulence in the flow field is simulated byk-εmodel.Based on the result,the scale and the value of the muzzle flow are measured,the conclusion is that the flows from the lateral slots of the muzzle brake interact with each other,and more oxygen is provided for the secondary combustion.The development and the mechanism of muzzle flash are presented by tracing the heat which is generated by the chemical reactions.Besides,the in fluence of the muzzle brake structure is analyzed,the conclusion is that spacing and the orientation angle between the lateral slots are essential factors for the suppression of the muzzle flash.

2.Mathematical models

2.1.Interior ballistics model

In order to obtain the muzzle flow accurately,the interior ballistics model is coupled with the simulation of the flow field to de fine the velocity of the moving projectile.The interior ballistics model is based on several assumptions and simpli fications of reality conditions.It has been proved to be an accurate and ef ficient model for calculating the interior ballistics parameters[31].The interior ballistics model consists of five equations:

(1).Propellant form function

In this investigation,the propellant particle with seven perforations is adopted.Based on the geometric law of burning,the burning process of the propellant particle is divided into three stages,which are identi fied by the value of the parameterZ:

whereZis the relative thickness of the propellant particle that has burnt.ψis the mass fraction of the propellant particle that has burnt.χ,λandμare the form characteristic parameters of the propellant particle.χsandλsare the form characteristic parameters when the surface of the propellant particle starts to decrease during the burning process.

(2).Burning function

whereu1is the burning coef ficient.eis half of the thickness between two perforations.nis the burning exponent.pis the mean pressure in the barrel.

(3).Motion equation

where v is the velocity of the projectile.lis the distance the projectile has moved.

(4).Momentum equation

whereSis the inner section area of the barrel.pdis the gaseous pressure on the base of the projectile.Fqis the resistance in front of the projectile.Ffis the frictional resistance.φis the minor work coef ficient.mis the mass of the projectile.

(5).Energy equation

wherefis the energy capacity of the propellant particle.ωis the total weight of the propellant.θ=γ-1,γis the adiabatic exponent.lψis the diameter shrunk length of the free volume of the chamber.lψis defined as:

wherel0=.αis the covolume of the propellant.ρpis the density of the propellant particle.Δis the mean density in the chamber volume.V0is the total volume of the chamber.

In the computing program of the interior ballistics model,all equations should be converted to the dimensionless equations:

These dimensionless equations are solved by the fourth-order Runge-Kutta method.The interior ballistics model is coupled with the Fluent finite volume code as a User-De fined Function.Then,the velocity of the moving boundary is calculated at every time step.

2.2.Governing equations of the flow field

In this study,the computational model is assumed to be twodimensional axisymmetric.As the parameters during the interior ballistic process are determined by the interior ballistics model,the multi-species and chemical reactions are neglected.Therefore,the Navier-Stocks equations are employed to simulate the development of the precursor flow ahead of the projectile.After the projectile leaves the muzzle,the propellant gas mixes with the air.The chemical reactions depend on the temperature and the pressure of the propellant gas.The Navier-Stokes equations with chemical reactions are employed to simulate the development of the muzzle flow:

whereUis the vector of the conservation variables.f=（E-Ev）·i+（F-Fv）·j,EandFare the vectors of the convection fluxes,EvandFvare the vectors of fluxes caused by viscidity.nis the normal vector of the area element of the control volume.HandHvare the vectors of the source termcaused by axial symmetry.Qis the vector of the source term caused by chemical reactions.The equations are as follows:

whereρis the density of the propellant gas which is generated from the combustion of the propellant particle.wi（i=1，2，…，n）is the mass fraction of gas speciesi.uis the velocity component in the axial direction,v is the velocity component in the radial direction.eis the total energy per unit of the gas mass.pis the static pressure of the gas.τxx,τxy,τyyandτθθare the viscous stresses,qxandqyare the transmission of heat.（i=1，2，…，n）is the generation rate of gas speciesifrom the chemical reactions.These variables are expressed as follows:

whereμis the viscosity coef ficient,λis the thermal conductivity of the propellant gas,Diis the diffusion coef ficient of speciesi.

2.3.Chemical reaction kinetic model

In this simulation,considering the major components and the conditions under which the chemical reaction occurs,the chemical reaction kinetic model is expressed by 12 steps H2-CO-O2reactions,which are proposed by Gibeling and Buggeln[32].The detail of this model is shown in Table 1.The mass fraction of the species is given in Table 2,which is consistent with the composition of the propellant gas.The processes of the reactions are solved by the finiterate chemical reaction package in Fluent finite volume code.In this chemical reaction kinetic model,the general form of reaction r is as follows:

Table 1 H2-CO-O2 reactions.

Table 2 The species of the propellant gas and the mass fraction.

wherenis the number of the chemical species in the system,v’i，ris the stoichiometric coef ficient for reactantiin reactionr,Miis the symbol denoting the speciesi,v’’i，ris the stoichiometric coef ficient for productiin reactionr.kf，ris the forward rate constant of reactionr,kb，ris the backward rate constant of reactionr.

The chemical reactions in the muzzle flow are regarded as a reversible reaction,the molar rate of creation of speciesiin reactionris as follows:

whereΓis the effect of the third body on the reaction rate,which is neglected in this study.Cj，ris the molar concentration of speciesjin reactionr,η’j，ris the rate exponent for reactant speciesjin reactionr.

The forward rate constant of reactionris given by the Arrhenius expression

whereAris the pre-exponential factor of reactionr,βris the temperature exponent of reactionr,Eris the active energy for reactionr,Ris the universal gas constant.

The backward rate constant of reactionris expressed as follow:

whereKris the equilibrium constant for reactionr,patmrepresents atmospheric pressure(101325 Pa),andrepresent the changes in Gibbs free energy,and they are computed as follows:

whereSiandhiare the entropy and enthalpy of the speciesievaluated at temperatureTand atmospheric pressure.

In Table 1,Ais the pre-exponential factor,βis the temperature exponent,Eis the activation energy of the reactions.Mrepresents the third body collision.

Table 3 The measured results and simulated results of the interior ballistic process.

3.Computational model

Fig.1.The simpli fied computational model.

In this study,the lateral slots of the muzzle brake are approximately circular and the in fluence of ground is neglected.Therefore,the two-dimension axial-symmetric model is used in this simulation.The scale and boundary conditions of the computational model are shown in Fig.1.In order to provide a sufficient scale for the muzzle flow and the travel of the projectile,the total length of the computational domain isL2=2.7 m and the radius isR=0.715 m.The length of the barrel isL1=1.7 m,the length of the projectile travel is 1.502 m,the caliber of the barrel isd=0.03 m.The boundary conditions are defined as annotations.All surfaces of the solid structures are defined as “wall” .The boundaries of the computational domain outside the barrel are defined as “pressure outlet” [19,29].Before the start of the projectile,the pressure of the area ahead of the projectile and the area outside the muzzle is 101325 Pa,the temperature is 300 K.The initial conditions and the boundary conditions are derived from the assumptions of the actual surrounding conditions[4,29,33].In this simulation,the interior ballistics model works as a user-defined function program to de fine the velocity of the moving projectiles.When the base of the projectile leaves the muzzle,the velocity of the projectile is 895.4 m/s.The pressure and the velocity of the propellant gas in the barrel are determined by the result from the interior ballistics model.

Fig.2.Mesh model of the computational domain.

The mesh model is built with structured grids for computation.As shown in Fig.2,the size of the grids is restricted under 2 mm×2 mm for computational accuracy.The dynamic mesh is introduced to deal with the movement of the projectile.As shown in Figs.1 and 2,the rectangular zones,which the motion of the projectile covers,are solved by the dynamic layering method.In this method,the base and the head of the projectile are speci fied as “rigid body” ,as well as the dynamic mesh,and the velocity of the dynamic parts is defined by the interior ballistics model.Furthermore,a grid sensitivity investigation is implemented.In this investigation,a re fined mesh is built in the size of 1 mm×1 mm and a coarse mesh is built in the size larger than 3 mm×3 mm as a contrast.The simulations are performed by the same initial conditions and solving methods,the simulated value along the oblique sample line,as shown in Fig.5,are presented in Fig.3.The results of the re fined mesh and the mesh adopted in this study are nearly identical.Therefore,the simulation results in this study are expected to be mesh independent.

Fig.3.Simulated results of different mesh size.

4.Results and discussion

4.1.Veri fication of the model

In order to verify the reliability of the interior ballistics model,the simulated results are compared with the measured results[22].In the simulation,the propellant particle with seven perforations is adopted,the impetus of the propellant is 932 kJ/kg,and the covolume of the propellant gas is 0.001 m3/kg.The charge weight of the propellant is 116 g,the mass of the projectile is 389 g,the caliber is 30 mm,and the length of the projectile travel is 1502 mm.All parameters are consistent with the experiment.

The measured results and the simulated results are shown in Table 3,where vmis the muzzle velocity of the projectile,pmis the maximum breech pressure.Comparing the measured and the simulated results,the relative error in the muzzle velocity of the projectile is 0.61%,the relative error in the maximum breech pressure is 1.81%.This shows that the simulated results are reliable when the interior ballistics model is used to calculate the velocity of the projectile.

According to the methods illustrated above and the veri fications of the models,the reliable simulated results of the muzzle flow is obtained.In Fig.4,the results of the simulation in this present study and the simulation implemented by Qin and Zhang[29]are compared in order to verify the accuracy of the chemical reaction kinetic model.The contours of the muzzle flow near the projectile when the projectile has left the muzzle for 0.4 ms are presented in Fig.4(a)and(c).The temperature of the gases and the mass fraction of the species,which are in fluenced by the chemical reactions,are in good agreement as shown in Fig.4(b)and(d).The comparison indicates that the result of the secondary combustion simulated in this study is reliable.

Fig.4.Veri fication of the chemical reaction kinetic model(a and b is the results of the simulation implemented by Qin and Zhang[29],c and d is the results of the simulation in this study.).

Fig.5.Distribution of the muzzle flow velocity and pressure:(a)t′=0.05ms,(b)t′=0.12ms,(c)t′=0.20 ms(d)t′=0.35 ms.

4.2.Development of muzzle flow

Based on the models,numerical methods and initial conditions illustrated above,the entire developing process and the data of the muzzle flow are obtained.The developing process is presented in Fig.5,the results are shown in order of the relative timet’which indicates the flying time of the projectile since the projectile base leaves the barrel.It is obvious to distinguish the precursor flow and the propellant gas flow.The precursor flow is generated by the compressed air in front of the projectile.As shown in Fig.5(a),t′=0.05 ms,the first shock wave of the precursor flow diffuses at the speed about 1 Mach(the speed of sound is 340 m/s),but the propellant gas flow diffuses at a higher velocity.The maximum pressure of the shock wave in the outer space reaches 1.1 MPa.It can be concluded from Fig.5(b)and(c)that the shock waves caused by the precursor flow and the propellant gas flow merge gradually.And,the flows from the lateral slots of the muzzle brake interact with each other.Because of the interactions of the jet flows,the structure of the muzzle flow becomes more complicated,and vortexes are born near the interfaces between the jet flows.The conclusion is supported by the result shown in Fig.5(d),the disordered frontier of the muzzle flow is presented.The jet flows sprayed from the lateral slots carry enormous total energy,thus these flows are a threat to the objects in the rear of the muzzle.

4.3.Secondary combustion of propellant gas

During the developing process of the muzzle flow,the underexpended propellant gas interacts with the ambient air.A secondary combustion phenomenon comes up because the temperature of the gases increases with the shock waves.The flash of the secondary combustion is clearly to be observed as the temperature of the gases rises again.

Before the projectile leaves the barrel,the propellant burns in the limited space and produces the propellant gas.The state of the propellant gas in the interior ballistic process is a negative oxygen balance.The combustion occurs only when the combustible components,COand H2,contact with O2in the ambient air.In Figs.6-9,the contours of the strain rate,the temperature,the mass fraction of oxygen and the heat of chemical reactions are shown in order of the relative timet’.For a better understanding of the secondary combustion,the zone,where the most violent chemical reactions occurred,is marked with arrows in every figure.The arrow marks are coordinated with the dash line.The contour of the strain rate is helpful to identify the whole region of the muzzle flow and the expansion region of the propellant gas.The contour of the mass fraction of oxygen shows the frontier between the propellant gas and ambient air.The contour of the heat of reaction shows the strength of the secondary combustion.The contour of the temperature shows the result of the secondary combustion and the flash region of the muzzle flow.

Fig.6.Distribution of fluid parameters at(t′=0.05 ms).

Fig.7.Distribution of fluid parameters(t′=0.12 ms).

Fig.8.Distribution of fluid parameters(t′=0.20 ms).

Fig.9.Distribution of fluid parameters(t′=0.35 ms).

Fig.10.Results distributed along the oblique sample line(t′=0.05 ms).

Fig.11.Results distributed along the oblique sample line(t′=0.12 ms).

Fig.12.Results distributed along the oblique sample line(t′=0.20 ms).

Fig.6 shows that the reactions start since the gas is released into the outer space.In Figs.7-9,it is indicated that the temperature of the propellant gas decreases in the expansion region and then increases after the gas crosses the boundary of the expansion region.The secondary increase of the temperature comes up where the propellant gas interacts with air,correlating with the chemical reactions.Another conclusion drawn from Figs.7-9 is that the flow sprayed from the middle slot is compressed by the adjacent jet flows according to the contours of the strain rate and the contours of the pressure in Fig.5.As shown in Figs.7-9,the expansion region of the flow from the middle slot is much smaller than other flows.Furthermore,the combustion of the middle flow is more violent,as the contour of the heat of reaction shown in Fig.7.The same conclusion is presented in Fig.8,the combustion is more violent when the flow sprayed from the first slot interacts with the flow from the central vent.The gases are compressed because of the interaction of the two flows,which is indicated by the contour of the mass fraction of oxygen.

Fig.13.Results distributed along the oblique sample line(t′=0.35 ms).

In this study,the secondary combustion phenomenon is illustrated in detail by the chemical mechanism.As shown in Figs.10-13,the mass fraction of the species and the heat of chemical reactions distributed along the oblique sample line are presented.The position of the oblique sample line is shown in Fig.6.In the graphs of the heat of reaction,each peak value of the curves indicates the strength of the combustion of the propellant gas because the heat of the reaction represents the rate of the energy generated by the chemical reactions.According to the chemical reaction kinetic model,Eq.(27)~(33),the rate of the reactions is affected by the temperature.Therefore,the maximum value of the heat of the reaction at different times is different,as shown in Figs.10-13.

Another conclusion that can be drawn from the figures is that the mass fraction of CO and H2decrease and the mass fraction of CO2increases in the same position.Comparing the maximum heat of the reaction and the mass fraction of CO,the conclusion is that the strength of the combustion is also related to the mass fraction of CO.In Figs.10 and 11,the maximum heat of reaction is about 240 kW,where the mass fraction of CO is about 0.48.In Fig.12,the maximum heat of reaction is about 74 kW,where the mass fraction of CO is about 0.26.The same conclusion can be drawn from Fig.13.

Fig.14.Pressure of the flows.

Fig.15.Heat of the reaction of the flows.

Fig.16.Temperature of the flows.

Fig.17.Analysis of the sample data.

4.4.Interaction of the flows from lateral slots

The conclusion drawn from Fig.6~9 is that the flows sprayed from the lateral slots interact with each other and the secondary combustion is more violent because of the effect of the interactions.To analyze the interactions of the flows,the magni fied contours are presented in Figs.14-16.Because of the shocks of the high-pressure flows,the gases between the core regions of the flows are compressed.The compressed regions are marked by the rectangles in Fig.14,the pressure of the compressed gases increases abnormally.As a result,chemical reactions occur in these regions.

In Fig.17,the data on the horizontal sample line is presented.The change of the temperature and the pressure roughly conforms to the law of ordinary jet.The positions of the jet flow core regions are marked as c1,c2 and c3.The positions labeled by line#1~#6 indicate the locations where the secondary combustion occurs.Especially,the compressed regions are labeled by line#2 and line#4.It implies that the heat of reactions generated in the compressed region is much higher than the other burning regions.As a consequence,the temperature of the gas rises sharply.With the development of the muzzle flow,a large flash region is formed as shown in Figs.8 and 9.

5.Conclusion

Because of the speci fic structure of the muzzle brake,the jet flows sprayed from the lateral slots generate hazardous impingements.The interactions of the jet flow from the lateral slots result in more violent combustion in the primary stage of the muzzle flow.Therefore,a more distinct muzzle flash is observed.Based on this study,the conclusion is that the distance and the orientation angle between the lateral slots need to be considered for the suppression of the muzzle flash and impingement.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.