Recoil Reduction Method of Gun with Side to Rear Jet Controlled by Piston Motion

S S symmetry

Article Recoil Reduction Method of Gun with Side to Rear Jet Controlled by Piston Motion

Ming Qiu 1,*, Peng Si 1, Jie Song 1 and Zhenqiang Liao 1,2

1 School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China; [email protected] (P.S.); [email protected] (J.S.); [email protected] (Z.L.) 2 School of Mechanical and Electrical Engineering, Global Institute of Software Technology, Suzhou 215163, China * Correspondence: [email protected]

Abstract: Excessive recoil severely restricts the loading of high-power traditional guns on modern vehicles. To reduce the recoil without breaking the continuous firing mode and reducing the projectile velocity, a recoil reduction method that controls the lateral ejecting of propellant gas by a piston was proposed. The recoil reduction device is symmetric about the barrel axis. First, a one-dimensional two-phase flow model of interior ballistic during the gun firing cycle was established. Next, the MacCormack scheme was used to simulate, and the piston motion was gained. Then the propagation of the rarefaction wave in the barrel was presented. Finally, the propulsion difference between the piston-controlled gun and the traditional gun was discussed. The results showed that the recoil momentum was reduced by 31.80%, and the muzzle velocity was decreased by just 1.30% under the reasonable matching of structural parameters.

Keywords: two-phase ﬂow; recoil reduction; rarefaction wave; piston motion; ﬂuid-solid coupling

Citation: Qiu, M.; Si, P.; Song, J.; Liao, Z. Recoil Reduction Method of Gun with Side to Rear Jet Controlled 1. Introduction by Piston Motion. Symmetry 2021, 13, In modern war, power increase and maneuverability improvement are the develop- 396. https://doi.org/10.3390/ ment trend for traditional guns. However, they conflict with each other. The propellant sym13030396 gas produces recoil force on the gun system while pushing the projectile to accelerate. With the projectile velocity increasing, heavier mount is needed to absorb the recoil force. Academic Editor: Jan Awrejcewicz Especially for guns on vehicles and aircrafts, excessive recoil will cause these carrying platforms to vibrate violently. Therefore, recoil reduction is the key to solve the conflict. Received: 22 January 2021 To reduce the recoil, many techniques have been developed, including soft recoil [1] and Accepted: 24 February 2021 Published: 28 February 2021 magnetorheological fluid technology [2,3]. However, these techniques lead to problems, such as low reliability and complex structure. Moreover, the recoil reduction efficiency

Publisher’s Note: MDPI stays neutral is limited. with regard to jurisdictional claims in Considering that the energy of the propellant gas venting from the muzzle usually published maps and institutional affil- accounts for over 40% of the total propellant energy, numerous studies have been done iations. on reducing the recoil by using propellant gas energy. The muzzle brake [4,5] is a typical example. However, its efficiency is limited. Kathe [6,7] proposed a concept of rarefaction wave gun to reduce the recoil by venting propellant gas through the breech. Nevertheless, this method breaks the continuous firing mode of a traditional gun, which restricts its development in engineering application. Liao [8], Chen [9], and Cheng [10] studied the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. reversely jet low recoil gun, which vents propellant gas through the orifice in the side wall This article is an open access article of the barrel. This gun can achieve continuous firing, but the projectile base pressure drops distributed under the terms and once propellant gas vents through the orifice, resulting in a decrease of the muzzle velocity. conditions of the Creative Commons Zhang [11], Wang [12], Zhang [13–15], and Coffee [16] studied the propulsion process of Attribution (CC BY) license (https:// the rarefaction wave gun based on two-phase flow theory. Zhang [15] proposed a front creativecommons.org/licenses/by/ orifice rarefaction wave gun. However, he ignored the control device movement, so the 4.0/). exact recoil reduction efficiency could not be obtained. Xiao [17] studied the launching

Symmetry 2021, 13, 396. https://doi.org/10.3390/sym13030396 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 396 2 of 14

process of a time-delay nozzle device. Due to the use of a lumped-parameter model [18], it is difficult to capture the rarefaction wave movement, so the accurate opening time of the orifice could not be obtained. At present, the propulsion process of guns is often simulated based on two-phase flow theory [19–23]. Monreal [24] established a one-dimensional two-phase interior ballistics model for a gun based on the theoretical and experimental study of porous propellant combustion. Menshov [25] adopted a two-dimensional axisymmetric nonequilibrium gas– solid two-phase interior ballistics model to analyze the flow field in the barrel, considering the combustion of inhomogeneously distributed propellant. Hu [26] proposed a method based on the Riemann problem to improve the simulation of propellant combustion. Cheng [27,28] and Cao [29,30] proposed a two-fluid model with two moving boundaries to study the propulsion process of a two-stage combustion gun. However, the above studies focus on the flow characteristics of the propellant gas in a straight tube, excluding the analysis of the flow in a tube with porous channels. In this work, a recoil reduction method, which controls the lateral ejecting of propellant gas by a piston-spring device, is proposed for a 30-mm gun. As the gun barrel presented here is equipped with a piston cavity and an exhaust pipe, the propulsion process of the gun is quite different from that of a traditional gun. First, the propulsion process of the gun is numerically simulated based on two-phase interior ballistics theory. Next, the piston motion is analyzed. Then the flow characteristics of the propellant gas in the barrel and the additional channels are studied. Finally, the changes in the muzzle velocity and the recoil are discussed.

2. Principle of the Piston-Controlled Gun The schematic of the gun with side to rear jet controlled by piston motion is shown in Figure1. To avoid the imbalance of lateral force on the barrel, the recoil reduction device is symmetric about the barrel axis, and just one side is displayed in Figure1. In the side wall of the barrel, there is a vent which is near the breech and in front of the chamber. Initially, the barrel vent is blocked by the piston under the action of spring force, as shown in Figure1a . After the projectile reaches the barrel vent, the propellant gas in the barrel flows into the piston cavity and pushes the piston to move upwards. Only when the piston travels a specified distance can the rear spray channel be opened, as shown in Figure1b. Then the propellant gas flows through the exhaust pipe and finally ejects from the divergent nozzle at high speed, producing a forward thrust to reduce the recoil momentum. When the rear spray channel is opened, the rarefaction wave is generated due to the pressure drop at the barrel vent, and travels towards the projectile base and the breech. The pressure reduces where the rarefaction wave reaches. However, the projectile base pressure does not drop immediately because the projectile has arrived at a position far from the barrel vent. Thus, it is possible to make the rarefaction wave unable to catch the projectile until the muzzle under the reasonable matching of the parameters, such as the barrel vent size and the spring stiffness. The piston will be reset under the action of the spring force after the propulsion process is over, so the continuous firing of the gun is not affected. This gun is called the piston-controlled gun in the following paragraphs. Symmetry 2021, 13, x FOR SymmetryPEER REVIEW2021, 13, 396 3 of 15 3 of 14

(a)

(b)

Figure 1. Structure diagramFigure of 1.theStructure piston-controlled diagram of gun. the piston-controlled(a) Gas port in closed gun. position (a) Gas port; (b) inGas closed position; (b) Gas port port in opened position.in opened position.

3. Theoretical Models3. Theoretical Models 3.1. Assumptions of the3.1. Model Assumptions of the Model To simplify the problem,To simplify the following the problem, assumptions the following [5] are made. assumptions The assumptions [5] are made. The assumptions have been practiced andhave proven been practiced in theoretical and provenstudies inand theoretical the numerical studies calculations and the numerical of calculations of internal ballistics for weaponsinternal ballistics [5,14]. They for weaponswill bring [ 5errors,14]. They to the will calculation bring errors in th tois thepaper, calculation in this paper, but these errors couldbut be theseaccepted. errors could be accepted. (1) Each phase is a continuum(1) Each in phase line with is a continuumthe assumption in line of withtwo- thefluid assumption model. of two-fluid model. (2) One-dimensional (2)unsteadyOne-dimensional flow is considered unsteady in the flow barrel, is considered the piston in cavity, the barrel, and thethe piston cavity, and the exhaust pipe. exhaust pipe. (3) The viscous dissipation(3) The of propellant viscous dissipation gas and its of heat propellant loss to gasthe wall and itsare heat ignored. loss to the wall are ignored. (4) The propellant gas(4) flow The is treated propellant as an gas isentropic flow is treated flow when as an the isentropic propellant flow gas when flows the propellant gas flows through a bifurcated tubethrough or elbow. a bifurcated tube or elbow. 3.2. The Governing Equations of the Two-Phase Flow in the Barrel 3.2. The Governing Equations of the Two-Phase Flow in the Barrel The governing equations of the two-phase flow model in the barrel include the mass The governing equations of the two-phase flow model in the barrel include the mass conservation equations of the gas and solid phase, the momentum conservation equations conservation equations of the gas and solid phase, the momentum conservation equations of the gas and solid phase, and the energy conservation equation of the gas phase. The of the gas and solid phase, and the energy conservation equation of the gas phase. The energy transfer for the solid phase was considered in the solid temperature equation [14]. energy transfer for theAccording solid phase to thewas change considered of the in propellant the solid gastemperature flow characteristics equation after[14]. the projectile reaches According to the changethe barrel of the vent, propellant the equations gas flow are as characteristics follows in a conservative after the projectile vector form. reaches the barrel vent, the equations are as follows in a conservative vector form. ∂U ∂F H Before barrel vent is enabled UFH Before+ barrel= vent is enabled (1)  ∂t ∂x H − J After barrel vent is enabled(1) txHJ After barrel vent is enabled Among these formulas, Among these formulas,

Symmetry 2021, 13, 396 4 of 14

 Aϕρ u    g g  A(m + m )  Aϕρg c ign  A(1 − ϕ)ρpup     −Amc   A(1 − ϕ)ρp   2 p      Aϕρg ug + ∂Aϕ  Aϕρ u   ρg   A(mcup + mignuign − fs) + p  U =  g g , F =  , H =  ∂x  (2)  A(1 − ϕ)ρ u   A(1 − ϕ) ρ u2 + p + τ   ∂A(1−ϕ) ∂A  p p  p p p   A( fs − mcup) + p ∂x + τp(1 − ϕ) ∂x   2  2 ug  2   u   ug p  p p ∂Aϕ Aϕρg(eg + 2 ) Aϕρ u e + + A(mc(ep + + ) + mign Hign − fsup − Qs) − p g g g 2 ρg ρp 2 ∂x

  Amgb1  Am   pb1   Am u  J =  gb1 gb  (3)    Ampb1upb   u2  Am (e + pb + gb ) gb1 gb ρgb 2

where A is the cross-section area of the barrel, ρg and ρp are the gas and solid density, ug and up are the gas and solid velocity, eg and ep are the gas and solid internal energy, ϕ is the porosity, p is the pressure, τp and f s are the intergranular stress and interphase drag, Qs is the interphase heat transfer, mc is the gas generation rate of the solid propellant per unit volume, mign is the gas generate rate of the ignition powder per unit volume, uign is the velocity of the ignition powder, Hign is the enthalpy of the ignition powder, mgb1 and mpb1 are the mass flow rate of the gas and solid phase out of the barrel from the barrel vent per unit volume, ugb1 and upb1 are the velocity of the gas and solid phase flowing through the barrel vent, ρgb, pb, and egb are the density, pressure, and internal energy of the gas phase flowing through the barrel vent. The J term in Equation (1) only works in the domain corresponding to the barrel vent. The constitutive equations needed to close the above governing equations, such as the interphase drag and the interphase heat transfer, can be found in [14,18].

3.3. Dynamic Model of Gas–Solid Coupling in the Piston Cavity After propellant gas ﬂows into the piston cavity through the barrel vent, the piston moves under the action of the propellant gas pressure, the spring resistance, and the friction resistance. The equations about the piston motion are as follows. ( m dvh = (p − p )S − (F + k x ) − F h dt hd a h 0 h h v (4) dxh dt = vh

where vh, xh, and mh are the velocity, displacement, and mass of the piston, phd is the pressure at the piston base, Sh is the cross-section area of the piston cavity, F0 is the spring preload, kh is the spring stiffness, Fv is the friction force between the piston and the piston cavity. The governing equations of the two-phase ﬂow model in the piston cavity are similar to those in the barrel. According to the change of the propellant gas ﬂow characteristics after the rear spray channel is enabled, the equations are as follows in a conservative vector form.

∂U ∂F H + J Before rear spray channel is enabled h + h = h h (5) ∂t ∂x Hh + Jh − Kh After rear spray channel is enabled

Among these formulas, the terms in Uh, Fh, Hh, and Jh are similar to those in Equation (1). Symmetry 2021, 13, 396 5 of 14

3.4. The Governing Equations of Two-Phase Flow in the Exhaust Pipe The exhaust pipe is simpliﬁed as a straight pipe starting from the rear spray channel. The governing equations of the two-phase ﬂow model in the exhaust pipe is similar to those in the piston cavity. ∂U ∂F e + e = H + K (6) ∂t ∂x e e Among these formulas, the terms in Equation (6) are similar to those in Equation (5).

3.5. Gas Flow Model at the Barrel Vent and Rear Spray Channel According to the above assumptions, the flow of the propellant gas through the barrel vent can be regarded as an isentropic flow [5]. Based on the ratio of the pressure at the barrel vent to the pressure at the piston cavity entrance, there are four flow states through the barrel vent, namely, forward critical flow, forward noncritical flow, reverse noncritical flow, and reverse critical flow. The related equations can be found in [5]. As the nozzle exit is connected to the atmosphere, the gas flow state of the propellant gas through the rear spray channel can be divided into two types, namely, forward critical flow and forward noncritical flow.

3.6. Calculation of the Recoil Reduction Efficiency The recoil reduction effect of the buffer devices, such as the muzzle brake, was not considered in this work. The formulas for calculating the resultant force and the recoil momentum of the gun body can be found in [5]. The recoil reduction efficiency is defined as follows. Io − (Ir − Ip) η = × 100% (7) Io

where Io is the recoil momentum of the gun without nozzle, Ir is the recoil momentum of the gun with nozzle, Ip is the momentum imparted to the gun system by the gas ejecting from the nozzle.

4. Numerical Method 4.1. Difference Scheme The governing equations above were discretized by the MacCormack scheme [16], which is carried out by two steps, namely the predictor and corrector step.

4.2. Boundary Conditions The boundary conditions were different at different stages of the propulsion process. The nozzle exit and muzzle exit were taken as the free outlet boundary, which means a fixed outlet pressure was held for the subsonic flow, and the extrapolated condition for the supersonic flow. The reflective boundary was used at all walls of the domain, even if the projectile began to move. Once the projectile moves, the computational domain expands, and the length of the cell near the projectile base increases. When the cell length reached a given limit, the cell near the projectile base was split, and a new cell was added. The equation about the projectile movement can be found in [14]. The method that dealt with the piston movement was similar. In summary, the solution procedure is in Figure2. Symmetry 2021, 13, x FOR PEER REVIEW 6 of 15

Symmetry 2021, 13, x FOR PEER REVIEWprojectile base increases. When the cell length reached a given limit, the cell 6near of 15 the projectile base was split, and a new cell was added. The equation about the projectile movement can be found in [14]. The method that dealt with the piston projectile base increases. When the cell length reached a given limit, the cell near movement was similar. the projectile base was split, and a new cell was added. The equation about the In summary, the solution procedure is in Figure. 2. Symmetry 2021, 13, 396 projectile movement can be found in [14]. The method that dealt with the piston6 of 14 movement was similar. In summary, the solution procedure is in Figure. 2.

Figure 2. Solution procedure.

where xd is the projectile displacement, Lb and Lr are the location of the barrel vent Figure 2. SolutionFigure procedure. 2. Solution procedure. and the rear spray channel, pd is the projectile base pressure, Δx2 is the length of the cell near the projectile base, ts is setting time, mgb2 and mpb2 are the mass flow wWherehere xdx isis the the projectile projectile displacement, displacement, Lb Landand Lr areLr are the thelocation location of the of thebarrel barrel vent vent rate of thed gas and solid phase into the pistonb cavity from the barrel vent per unit andand the rearthe rear spray spray channel, channel,pd is p thed is projectilethe projectile base base pressure, pressure,∆x2 Δisx the2 is lengththe length of the of cell volume, mgr1 and mpr1 are the mass flow rate of the gas and solid phase out of the nearthe the cell projectile near the base, projectilets is setting base, time,ts is settingmgb2 and time,m pb2mgb2are and the m masspb2 are flow the ratemass of flow the gas piston cavity from the rear spray channel per unit volume, mgr2 and mpr2 are the andrate solid of phase the gas into and the solid piston phase cavity into from the thepiston barrel cavity vent from per unitthe barrel volume, ventm gr1perand unitm pr1 mass flow rate of the gas and solid phase into the exhaust pipe from the rear spray are thevolume, mass m flowgr1 and rate m ofpr1 theare the gas mass and solidflow phaserate of outthe ofgas the and piston solid cavityphase out from of the the rear channel per unit volume. spraypiston channel cavity per from unit volume,the rear spraymgr2 and channelmpr2 areper theunit mass volume, flow m rategr2 and of the mpr2 gas are and the solid phasemass into flow the exhaustrate of the pipe gas from and solid the rear phase spray into channel the exhaust per unit pipe volume. from the rear spray 5. Results and discussion channel per unit volume. 5. Results5.1. Validation and Discussion 5.1. Validation5. ResultsThe virtual and discussion AGARD (Advisory Group for Aerospace Research and Develop- ment5.1.The Validation) virtual gun model AGARD was (Advisory used to test Group our forcodes, Aerospace and its Researchparameters and can Development) be found in gun model[7, 15]. was The used results to test from our codes,our codes and are its parametersin Figure 3. canThe beresults found based in [7 ,on15]. the The pub- results The virtual AGARD (Advisory Group for Aerospace Research and Develop- fromlished our codes codes arecan in be Figure found3 .in The [7, 15, results 18]. basedThe data on are the in published Table 1. The codes results can be from found ment) gun model was used to test our codes, and its parameters can be found in in [7our,15, 18codes]. The are data in good are in agreement Table1. The with results the expected from our ones, codes as are shown in good in Table agreement 1. The with [7, 15]. The results from our codes are in Figure 3. The results based on the pub- the expecteddifference ones, between as shown these codes in Table is 1caused. The differenceby the submodels, between such these as codes the intergran- is caused by lished codes can be found in [7, 15, 18]. The data are in Table 1. The results from the submodels,ular stress and such interphase as the intergranular drag. stress and interphase drag. our codes are in good agreement with the expected ones, as shown in Table 1. The difference between these codes is caused by the submodels, such as the intergranular stress and interphase drag.

Figure 3. Simulation of theFigure AGARD 3. Simulation gun: (a) pressureof the AGARD at projectile gun: (a) base pressure and breech, at projectile (b) projectile base and velocity. breech, (b) projectile velocity. Table 1. Numerical results. Figure 3. Simulation of the AGARD gun: (a) pressure at projectile base and breech, (b) Symmetry 2021, 13, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/symmetry projectile velocity. Computed Value Acceptable Range Present Maximum breech pressure (MPa) 355–400 367 Symmetry 2021, 13, x. https://doi.org/10.3390/xxxxxMaximum projectile base pressure (MPa) 325–360www.mdpi.com/journal/ 336symmetry Muzzle velocity (m·s−1) 660–705 673 Exit time (ms) 14.66–16.58 15.57 Symmetry 2021, 13, 396 7 of 14

5.2. Decision of the Opening Time of the Rear Spray Channel Based on a traditional 30-mm gun, the piston-controlled gun was transformed by designing a new barrel, as shown in Figure1. The known parameters are in Table2. The opening time of the rear spray channel is controlled by the piston. It can be seen from Equation (4) that the piston movement is mainly determined by the parameters, such as the barrel vent size and the spring stiffness. However, these parameters are mutually restrictive. Besides, the flow fields in the barrel, piston cavity, and exhaust pipe are coupled after the rear spray channel is opened, so the relationship between these parameters is complicated and nonlinear. Therefore, it is difficult to derive an analytical expression to solve this relationship. The reasonable match of these parameters could only be obtained by numerous calculations. Taking the criterion that muzzle velocity loss is within 2%, a set of reasonable parameter matches was obtained, as shown in Table3. The values of the above parameters refer to only one recoil reduction device.

Table 2. Known parameters.

Parameter Value Barrel diameter (m) 0.030 Length of barrel (m) 2.457 Propellant mass (kg) 0.119 Propellant density (kg·m−3) 1600 Projectile mass (kg) 0.389

Table 3. Calculation parameters.

Parameter Value Position of barrel vent (m) 0.45 Diameter of the piston cavity (m) 0.021 Piston mass (kg) 0.3588 Spring stiffness (N·m−1) 10000 Length of side nozzle (m) 0.6 Sectional area of the barrel vent (mm2) 314 Sectional area of the rear spray channel (mm2) 346

5.3. Flow Field in the Piston Cavity The high-pressure propellant gas flowed into the piston cavity after the projectile reached the barrel vent, so the pressure at the piston base increased sharply at 3.23 ms, as shown in Figure4. However, the piston started to move, pushed by the propellant gas, so the pressure at the piston base reached the peak at 3.38ms and then decreased. At 4.34 ms, the rear spray was conducted due to the piston motion, as shown in Figure5. Then the propellant gas flowed into the exhaust pipe and was ejected from the nozzle. Therefore, the pressure at the piston base dropped sharply. At 5.38 ms, the piston reached the top and stopped here, so the pressure increased slightly. At 7.42 ms, the pressure at piston base became lower than the spring resistance, so the piston began to reset. The rear spray channel began to be closed at 13.45 ms and became fully closed at 13.96 ms. The piston returned to its initial position within 16 ms. Therefore, the proposed method does not break the continuous firing mode, because the firing rate of a traditional single-barrel gun does not exceed 1000 r/min and it takes at least 60 ms to finish the firing cycle. SymmetrySymmetry 2021 2021, 13, ,13 x ,FOR x FOR PEER PEER REVIEW REVIEW 8 of8 15of 15

TheThe high high-pressure-pressure propellant propellant gas gas flowed flowed into into the the piston piston cavity cavity after after the the pro- pro- jectilejectile reached reached the the barrel barrel vent, vent, so so the the pressure pressure at atthe the piston piston base base increased increased sharply sharply at at3.23ms, 3.23ms, as asshown shown in inFig Figureure 4. 4However,. However, the the piston piston started started to tomove move, pushed, pushed by by thethe propellan propellant gas,t gas, so so the the pressure pressure at atthe the piston piston base base reached reached the the peak peak at at3.38ms 3.38ms andand then then decreased. decreased. At At 4.34ms, 4.34ms, the the rear rear spray spray was was conducted conducted due due to tothe the piston piston motion,motion, as asshow shown inn inFig Figureure 5. 5.Then Then the the propellant propellant gas gas flowed flowed into into the the exhaust exhaust pipepipe and and was was ejected ejected from from the the nozzle. nozzle. Therefore, Therefore, the the pressure pressure at atthe the piston piston base base droppeddropped sharply. sharply. At At 5.38ms, 5.38ms, the the piston piston reached reached the the top top and and stopped stopped here, here, so so the the pressurepressure increased increased slightly. slightly. At At 7.42ms, 7.42ms, the the pressure pressure at atpiston piston base base became became lower lower thanthan the the spring spring resistance, resistance, so so the the piston piston be begangan to toreset. reset. The The rear rear spray spray channel channel beganbegan to tobe be closed closed at at13.45 13.45 ms ms and and became became fully fully closed closed at at13.96 13.96 ms. ms. TheThe piston piston returned returned to toits its initial initial position position within within 16 16 ms. ms. Therefore, Therefore, the the pro- pro- posed method does not break the continuous firing mode, because the firing rate Symmetry 2021, 13, 396 posed method does not break the continuous firing mode, because the firing rate8 of 14 of ofa traditionala traditional single single-barrel-barrel gun gun does does not not exceed exceed 1000 1000 r/min r/min and and it takesit takes at atleast least 6060 ms ms to tofinish finish the the firing firing cycle. cycle.

Figure 4. Pressure at piston base.FigureFigure 4. Pressure4. Pressure at atpiston piston base. base.

FigureFigure 5.FigureTime 5. Time histories5. Time histories histories of piston of ofpiston location piston location location and conducting and and conducting conducting area of area rear area of spray ofrear rear spray channel. spray channel channel. . 5.4. Rarefaction Wave Front Propagation 5.4.5.4. Rarefaction Rarefaction wave wave front front propagation propagation When the rear spray channel was opened, a large amount of propellant gas in the SymmetrySymmetry 2021 2021, 13, ,13 x., https://doi.org/10.3390/xxxxxx. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/www.mdpi.com/journal/symmetrysymmetry barrel ﬂowed into the piston cavity, resulting in a sudden decrease of pressure at the

barrel vent. Then the effect of the pressure drop travelled towards the breech and the projectile base in the form of continuous rarefaction waves. The movement characteristics of the projectile and the rarefaction wave front towards the projectile base are shown in Figures6 and7. The rarefaction wave was produced at the barrel vent at 4.46 ms, and then travelled towards the projectile base and the breech. Meanwhile, the projectile was 0.86 m away from the barrel vent. The velocity of the projectile and the rarefaction wave both increased monotonically. However, the velocity of the rarefaction wave was higher than that of the projectile. Therefore, the rarefaction wave caught the projectile at 1.28 ms after the rarefaction wave was produced. In the meantime, the projectile just reached the muzzle exit. SymmetrySymmetry 20212021,, 1313,, xx FORFOR PEERPEER REVIEWREVIEW 99 ofof 1515

When thethe rearrear sprayspray channelchannel waswas opened,opened, aa largelarge amountamount ofof propellantpropellant gasgas inin thethe barrelbarrel flowedflowed intointo thethe piston cavity,, resultingresulting inin aa suddensudden decreasedecrease ofof pres-pres- suresure atat thethe barrelbarrel vent.vent. ThenThen thethe effecteffect ofof thethe pressure drop travelled towards the breech and the projectile base in the form of continuous rarefaction waves. The movement characteristics of the projectile and the rarefaction wave front towards thethe projectileprojectile basebase areare shownshown inin FigFiguress 6 and 7. The rarefaction wave was produced at the barrel vent at 4.46.46 ms, and then travelled towards the projectile base and the breech. Meanwhile, the projectile was 0.86 m away from the barrel vent. The velocity of the projectile and the rarefaction wave both increased monotonically.ically. HoHowever, the velocity of the rarefaction wave was higher than that of the Symmetry 2021, 13, 396 projectile. Therefore, the rarefaction wave caught the projectile at 1.28 ms after the 9 of 14 rarefactionrarefaction wavewave waswas produced.produced. InIn thethe meantime,meantime, thethe projectileprojectile justjust reachedreached thethe muzzle exit.

Figure 6. VelocityFigure of 6. projectile Velocity andof projectile rarefaction and wave rarefaction front. wave front..

Figure 7. PositionFigure of 7. projectile Position of and projectile rarefaction and wave rarefaction front. wave front.. 5.5. Flow Field in the Barrel 5.5. Flow field in the barrel The comparisons of the pressure in the barrel of the two guns are shown in Figures8 and9. The comparisons of the pressure in the barrel of the two guns are shown in Figure9 displays the projectile base pressure during the interior ballistics period, and displays Figuress 8 and 9. Figure 9 displays the projectile base pressure during the interior the muzzle pressure during the after-effect period. The rarefaction wave reached the breech and ballistics period, and displays the muzzle pressure during the after--effecteffect period.period. the projectile base at 5.08 and 5.74 ms, respectively. However, the projectile base pressure of the The rarefaction wave reached the breech and the projectile base at 5.08 and 5.74 piston-controlled gun declined at 3.23 ms because the propellant gas in the barrel immediately ms,, respectiverespectively.ly. However,However, thethe projectileprojectile basebase pressurepressure ofof thethe pistonpiston--controlledcontrolled flowed into the piston cavity when the projectile reached the barrel vent, as shown in Figure9. gun declined at 3.23 ms because thethe propellant gas in the barrel immediately Besides, it dropped a little and oscillated slightly because the piston cavity was closed from flowedflowed intointo thethe pistonpiston cavitycavity whenwhen thethe projectileprojectile reachedreached thethe barrelbarrel vent,vent, asas shownshown 3.23 to 4.34 ms and there was both positive and reverse flow between the barrel and the piston inin Fig.Fig. 9.. Besides, it droppeded aa littlelittle andand oscillatedoscillated slightlyslightly becausebecause thethe pistonpiston cav-cav- cavity during this period. After the rarefaction wave caught the projectile at the muzzle exit, ityity waswas closedclosed fromfrom 3.233.23 toto 4.344.34 msms andand therethere wwas both positive and reverse flow the muzzle pressure of the piston-controlled gun reduced significantly compared with that of SymmetrySymmetry 20212021,, 1313,, x.x. https://doi.org/10.3390/xxxxxhttps://doi.org/10.3390/xxxxxthe traditional gun. www.mdpi.com/journal/symmetrysymmetry Similarly, the breech pressure of the piston-controlled gun started to reduce at 3.62 ms instead of 5.08 ms, as shown in Figure8. The projectile reached the barrel vent at 3.23 ms, and then a rarefaction wave traveled towards the breech and arrived at the breech at 3.62 ms. However, the breech pressure of the piston-controlled gun dropped slightly at this time, but dropped signiﬁcantly at 5.08 ms. The reasons are as follows. The pressure drop at the barrel vent at 3.23 ms was low, so the rarefaction wave intensity was also low. Thus, the pressure drop at the breech at 3.62 ms was small. The rear spray channel was opened at 4.34 ms, so the pressure drop at the barrel vent was higher, and the rarefaction wave intensity was higher. Therefore, the pressure drop at the breech was higher. Symmetry 2021, 13, x FOR PEER REVIEW 10 of 15

between the barrel and the piston cavity during this period. After the rarefaction Symmetry 2021, 13, 396 between the barrel and the piston cavity during this period. After the rarefaction 10 of 14 wave caught the projectile at the muzzle exit, the muzzle pressure of the piston- controlled gun reduced significantly compared with that of the traditional gun.

1 2 Figure 8. BreechFigure pressure 8. Breech comparison pressure of two comparison guns. of two guns.

3 4 Figure 9. Projectile/muzzleFigure 9. Projectile/muzzle pressure comparisonpressure comparison of two guns. of two guns.

The projectile base pressure of the piston-controlled gun was almost the same as that 5 Similarly, the breech pressure of the piston-controlled gun started to reduce at 3.62 of the traditional gun, although some propellant gas was ejected from the barrel during the 6 ms instead of 5.08 ms, as shown in Fig. 8. The projectile reached the barrel vent at 3.23ms, propulsion process. Therefore, the velocity–time curve of the projectile of the two guns was 7 and then a rarefaction wave traveled towards the breech and arrived at the breech at 3.62 not much different, as shown in Figure 10. Nevertheless, the inﬂuence of the barrel vent on 8 ms. However, the breech pressure of the piston-controlled gun dropped slightly at this the projectile velocity could still be seen. Compared with the traditional gun, the muzzle 9 time, but dropped significantly at 5.08 ms. The reasons are as follows. The pressure drop velocity of the piston-controlled gun reduced from 960.0 to 947.5 m/s, a decrease of 1.30%. 10 Symmetry 2021, 13, x FOR PEER REVIEWat the barrel vent at 3.23 ms was low, so the rarefaction wave intensity was also low. Thus2 of 15, 11 the pressure drop at the breech at 3.62 ms was small. The rear spray channel was opened 12 at 4.34 ms, so the pressure drop at the barrel vent was higher, and the rarefaction wave 13 intensity was higher. Therefore, the pressure drop at the breech was higher. 14 The projectile base pressure of the piston-controlled gun was almost the same as that 15 of the traditional gun, although some propellant gas was ejected from the barrel during 16 the propulsion process. Therefore, the velocity–time curve of the projectile of the two guns 17 was not much different, as shown in Figure 10. Nevertheless, the influence of the barrel 18 vent on the projectile velocity could still be seen. Compared with the traditional gun, the 19 muzzle velocity of the piston-controlled gun reduced from 960.0 to 947.5 m/s, a decrease 20 of 1.30%.

21 22 Figure 10. Velocity comparisonFigure 10. Velocity of two guns.comparison of two guns. Symmetry 2021, 13, x. https://doi.org/10.3390/xxxxx5.6. Flow Field in the Exhaust Pipe www.mdpi.com/journal/symmetry 23 5.6. Flow field in the exhaust pipe Figures 11 and 12 show the pressure and gas velocity distribution along the exhaust 24 Figures 11 and 12 show the pressure and gas velocity distribution along the exhaust pipe axis at four moments after igniting: 0.24 ms after the rear spray channel was opened 25 pipe axis at four moments after igniting: 0.24 ms after the rear spray channel was opened 26 (4.58 ms after igniting), 0.43 ms after the rear spray channel was fully opened (4.91 ms 27 after igniting), the moment when the projectile exited the muzzle (5.74 ms after igniting), 28 0.34 ms after the fifth moment (6.08 ms after igniting). Here we define that mark “0” of x- 29 coordinate denotes the rear spray channel, and the nozzle exit is rightward. As can be seen 30 from Figs. 11 and 12, a rapid increase of pressure and gas velocity occurred at the entrance 31 of the exhaust pipe at 4.58 ms because the rear spray channel was opened and the propel- 32 lant gas flowed into the exhaust pipe at high speed. Meanwhile, the propellant gas did 33 not fill the whole pipe. After 4.91 ms, the propellant gas filled the whole pipe and ex- 34 panded in the pipe, so the propellant gas velocity increased continuously. In the divergent 35 nozzle, the high-pressure propellant gas further expanded and accelerated, thus gaining 36 a greater impulse.

37 38 Figure 11. Pressure distribution along the exhaust pipe axis.

Symmetry 2021, 13, x FOR PEER REVIEW 2 of 15

21 22 Figure 10. Velocity comparison of two guns.

23 Symmetry 2021, 13, 396 5.6. Flow field in the exhaust pipe 11 of 14 24 Figures 11 and 12 show the pressure and gas velocity distribution along the exhaust 25 pipe axis at four moments after igniting: 0.24 ms after the rear spray channel was opened 26 (4.58 ms(4.58 after ms igniting), after igniting), 0.43 ms 0.43 after ms the after rear the spray rear channel spray channel was fully was opened fully opened(4.91 ms (4.91 ms 27 after igniting),after igniting), the moment the moment when the when projectile the projectile exited the exited muzzle the muzzle(5.74 ms (5.74 after ms igniting), after igniting), 28 0.34 ms 0.34after ms the after fifth themoment fifth moment (6.08 ms (6.08after msigniting). after igniting). Here we define Here we that define mark that “0” markof x- “0” of 29 coordinatex-coordinate denotes the denotes rear spray the rearchannel, spray and channel, the nozzle and exit the nozzleis rightward. exit is As rightward. can be seen As can be 30 from Figseens. 11 from and 1 Figures2, a rapid 11 increaseand 12, aof rapid pressure increase and gas of pressure velocity andoccur gasred velocity at the entrance occurred at the 31 of the exhaustentrance pipe of theat 4.58 exhaust ms because pipe at the 4.58 rear ms spray because channel the rear was spray opened channel and the was propel- opened and 32 lant gasthe flowed propellant into the gas exhaust flowed pipe into theat high exhaust speed. pipe Meanwhile, at high speed. the Meanwhile,propellant gas the did propellant 33 not fill gasthe didwhole not pipe fill. the After whole 4.91 pipe. ms, the After propellant 4.91 ms, gas the filled propellant the whole gas filled pipe theand whole ex- pipe 34 pandedand in the expanded pipe, so the in thepropellant pipe, so gas the velocity propellant increased gas velocity continuously. increased In the continuously. divergent In the 35 nozzle, divergentthe high-pressure nozzle, the propellant high-pressure gas further propellant expanded gas further and accelerated, expanded andthus accelerated, gaining thus 36 a greatergaining impulse. a greater impulse.

37 38 Figure 11. PressureFigure 11. distribution Pressure distribution along the exhaust along the pipe exhaust axis. pipe axis. Symmetry 2021, 13, x FOR PEER REVIEW 3 of 15

39 40 FigureFigure 12. Gas 12. phase Gas velocityphase velocity distribution distribution along thealong exhaust the exhaust pipe axis. pipe axis. 5.7. Analysis of the Recoil Reduction Efﬁciency 41 5.7. Analysis of the recoil reduction efficiency Here we deﬁne the force towards the muzzle as the positive. Compared with the recoil 42 Here we define the force towards the muzzle as the positive. Compared with the of the traditional gun, that of the piston-controlled gun declined slightly at 3.62 ms because 43 recoil of the traditional gun, that of the piston-controlled gun declined slightly at 3.62ms the breech pressure reduced, as shown in Figures8 and 13. At 4.79 ms, the propellant 44 because the breech pressure reduced, as shown in Figs.8 and 13. At 4.79 ms, the propellant gas was ejected from the nozzle, producing a strong reverse thrust, so the recoil of the 45 gas was ejected from the nozzle, producing a strong reverse thrust, so the recoil of the piston-controlled gun declined rapidly and became positive. However, the breech pressure 46 piston-controlled gun declined rapidly and became positive. However, the breech pres- peak occurred before the propellant gas was ejected from the barrel, so the recoil peak did 47 sure peak occurred before the propellant gas was ejected from the barrel, so the recoil not change. 48 peak did not change. 49 It can be seen from Figure 14 that the recoil momentum of the piston-controlled gun 50 also dropped at 3.62 ms and reached the minimum value of 372.77 N·s at 14.17 ms. From 51 then on, the recoil momentum of the piston-controlled gun began to increase. The reasons 52 are as follows. As the rear spray channel was fully closed at 13.96 ms, the propellant gas 53 in the exhaust pipe was completely ejected out, and then no reverse thrust would be pro- 54 duced. During the propulsion process, the recoil momentum of the traditional gun was 55 549.72 N·s, while that of the piston-controlled gun was 374.91 N·s, so the recoil reduction 56 efficiency was 31.80%. 57

59 Fig. 13. Recoil comparison of two guns.

Symmetry 2021, 13, x FOR PEER REVIEW 3 of 15

39 40 Figure 12. Gas phase velocity distribution along the exhaust pipe axis.

41 5.7. Analysis of the recoil reduction efficiency 42 Here we define the force towards the muzzle as the positive. Compared with the 43 recoil of the traditional gun, that of the piston-controlled gun declined slightly at 3.62ms 44 because the breech pressure reduced, as shown in Figs.8 and 13. At 4.79 ms, the propellant 45 gas was ejected from the nozzle, producing a strong reverse thrust, so the recoil of the 46 piston-controlled gun declined rapidly and became positive. However, the breech pres- 47 sure peak occurred before the propellant gas was ejected from the barrel, so the recoil 48 peak did not change. 49 It can be seen from Figure 14 that the recoil momentum of the piston-controlled gun 50 also dropped at 3.62 ms and reached the minimum value of 372.77 N·s at 14.17 ms. From 51 then on, the recoil momentum of the piston-controlled gun began to increase. The reasons 52 are as follows. As the rear spray channel was fully closed at 13.96 ms, the propellant gas 53 in the exhaust pipe was completely ejected out, and then no reverse thrust would be pro- 54 duced. During the propulsion process, the recoil momentum of the traditional gun was 55 Symmetry 549.722021, 13 ,N·s, 396 while that of the piston-controlled gun was 374.91 N·s, so the recoil reduction 12 of 14 56 efficiency was 31.80%. 57

58 Figure 13. Recoil comparison of two guns. 59 Fig. 13. Recoil comparison of two guns. It can be seen from Figure 14 that the recoil momentum of the piston-controlled gun also dropped at 3.62 ms and reached the minimum value of 372.77 N·s at 14.17 ms. From then on, the recoil momentum of the piston-controlled gun began to increase. The reasons are as follows. As the rear spray channel was fully closed at 13.96 ms, the propellant gas in the exhaust pipe was completely ejected out, and then no reverse thrust would be produced. During the propulsion process, the recoil momentum of the traditional gun was 549.72 N·s, while that of the piston-controlled gun was 374.91 N·s, so the recoil reduction

Symmetry 2021, 13, x FOR PEER REVIEW efﬁciency was 31.80%. 4 of 15

60 Figure 14. Recoil momentum comparison of two guns. 61 Figure 14. Recoil momentum comparison of two guns. There are many parameters which influence the recoil reduction efficiency, such as 62 There are many parametersthe size of the which barrel influence vent, the the length recoil of reduction the exhaust efficiency, pipe, and such the nozzleas expanding angle. 63 the size of the barrel vent,Thus, the the length piston-controlled of the exhaust gun pipe, needs andto the be nozzle optimized. expanding angle. 64 Thus, the piston-controlled gun needs to be optimized. 6. Conclusions 65 6. Conclusion Numerical simulations in the piston-controlled gun were carried out based on one- 66 Numerical simulationsdimensional in the two-phasepiston-controlled theory. Thegun flowwere field carried characteristics out based on in theone barrel,- the piston cavity, 67 dimensional two-phaseand theory. the exhaust The flow pipe field were characteristics obtained. The in mainthe barrel, conclusions the piston are ascav- follows. 68 ity, and the exhaust pipe(1) wereThe obtained. recoil reduction The main mechanism conclusions of theare piston-controlledas follows. gun was revealed. By con- 69 (1) The recoil reductiontrolling mechanism the lateral of the ejecting piston of-controlled propellant gun gas was with rev theealed. piston, By the rarefaction wave 70 controlling the lateral ejectingproduced of propellant at the gas barrel with vent the justpiston, caught the rarefaction the projectile wave at the pro- muzzle exit. Thus, the 71 duced at the barrel vent justprojectile caught the base projectile pressure at didthe notmuzzle reduce exit. too Thus much, the duringprojectile the propulsion process, 72 base pressure did not reduceminimally too much reducing during the the propulsion muzzle velocity. process, Meanwhile, minimally the reduc- recoil momentum was sig- 73 ing the muzzle velocity. Meanwhile,nificantly reducedthe recoil by momentum venting high-pressure was significantly propellant reduced gas by from the barrel as much 74 venting high-pressure propellantas possible. gas from the barrel as much as possible. 75 (2) The piston-controlled gun could be reset within the firing cycle, so the continuous 76 firing mode is not affected. However, the rear spray channel will be closed during the 77 piston reset motion, which is negative to the recoil reduction efficiency. 78 (3) After the projectile reached the barrel vent, some high-pressure propellant gas 79 flowed into the piston cavity, so the projectile base pressure dropped slightly. However, 80 the piston cavity was closed before the rear spray channel was opened, so the velocity loss 81 of the piston-controlled gun is similar to that of a traditional air-guided gun. 82 The recoil reduction method proposed in this work does not break the ammunition 83 structure of the existing weapons. Besides, the reset device is stable, and the continuous 84 firing mode is not affected. Thus, this method has a good application prospect. 85 86 Author Contributions: Conceptualization, M.Q.; methodology, M.Q. and P.S.; software, P.S.; val- 87 idation, M.Q. and P.S.; formal analysis, M.Q. and P.S.; investigation, M.Q. and P.S.; resources, 88 M.Q. and P.S.; data curation, M.Q. and P.S.; writing—original draft preparation, M.Q. and P.S.; 89 writing—review and editing, M.Q., P.S., J.S. and Z.L.; visualization, P.S. and J.S.; supervision, 90 M.Q. and Z.L.; project administration, M.Q. and Z.L.; funding acquisition, M.Q. All authors have 91 read and agreed to the published version of the manuscript.

92 Funding: This research was funded by the National Natural Science Foundation of China, grant 93 number 12072161 and 51376090.

94 Institutional Review Board Statement: Not applicable.

95 Informed Consent Statement: Not applicable.

96 Data Availability Statement: Not applicable.

97 Declaration of Conflicting Interests: The authors declare that there is no conflict of in- 98 terest.

Symmetry 2021, 13, 396 13 of 14

(2) The piston-controlled gun could be reset within the firing cycle, so the continuous firing mode is not affected. However, the rear spray channel will be closed during the piston reset motion, which is negative to the recoil reduction efficiency. (3) After the projectile reached the barrel vent, some high-pressure propellant gas flowed into the piston cavity, so the projectile base pressure dropped slightly. However, the piston cavity was closed before the rear spray channel was opened, so the velocity loss of the piston-controlled gun is similar to that of a traditional air-guided gun. The recoil reduction method proposed in this work does not break the ammunition structure of the existing weapons. Besides, the reset device is stable, and the continuous firing mode is not affected. Thus, this method has a good application prospect.

Author Contributions: Conceptualization, M.Q.; methodology, M.Q. and P.S.; software, P.S.; validation, M.Q. and P.S.; formal analysis, M.Q. and P.S.; investigation, M.Q. and P.S.; resources, M.Q. and P.S.; data curation, M.Q. and P.S.; writing—original draft preparation, M.Q. and P.S.; writing—review and editing, M.Q., P.S., J.S. and Z.L.; visualization, P.S. and J.S.; supervision, M.Q. and Z.L.; project administration, M.Q. and Z.L.; funding acquisition, M.Q. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China, grant number 12072161 and 51376090. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: This work was funded by the National Natural Science Foundation of China (Grant No. 12072161 and No. 51376090). Conﬂicts of Interest: The authors declare that there is no conﬂict of interest.

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