Research on simulation of gun muzzle flow field empowered by artificial intelligence

Mengdi Zhou , Linfng Qin ,b,*, Congyong Co , Gungsong Chen , Jin Kong ,Ming-ho Tong

a School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

b Northwest Institute of Mechanical and Electrical Engineering, Xianyang, 712099, Shaanxi, China

c School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China

Keywords:Muzzle flow field Artificial intelligence Deep learning Data-physical fusion driven Shock wave

ABSTRACT Artificial intelligence technology is introduced into the simulation of muzzle flow field to improve its simulation efficiency in this paper.A data-physical fusion driven framework is proposed.First,the known flow field data is used to initialize the model parameters,so that the parameters to be trained are close to the optimal value.Then physical prior knowledge is introduced into the training process so that the prediction results not only meet the known flow field information but also meet the physical conservation laws.Through two examples, it is proved that the model under the fusion driven framework can solve the strongly nonlinear flow field problems, and has stronger generalization and expansion.The proposed model is used to solve a muzzle flow field, and the safety clearance behind the barrel side is divided.It is pointed out that the shape of the safety clearance under different launch speeds is roughly the same, and the pressure disturbance in the area within 9.2 m behind the muzzle section exceeds the safety threshold,which is a dangerous area.Comparison with the CFD results shows that the calculation efficiency of the proposed model is greatly improved under the condition of the same calculation accuracy.The proposed model can quickly and accurately simulate the muzzle flow field under various launch conditions.

1.Introduction

As a powerful,long-range barrel weapon, artillery is one of the important weapons in the army's conventional operations.During the launching process, the propellant gas with high temperature and pressure is rapidly released from the bore [1].A muzzle flow field with a complex wave system and severe disturbance is formed,impacting and even damaging the surrounding equipment and personnel[2].It is a crucial step in gun research[3]and design[4] to clarify the disturbance mechanism of the muzzle flow field.Due to the strong nonlinearity and discontinuity of muzzle flow field, it is difficult to solve it accurately by analytical method.At present, time and space variation characteristics of muzzle flow field parameters are most obtained by employing numerical simulation [5].Gao et al.[6,7] dynamic simulated flow field with secondary arc in the muzzle area based on the magneto hydrodynamic equations.Li [8,9] simulated the secondary combustion of muzzle based on the chemical reaction model, and analyzed the formation mechanism of muzzle flash.When looking at the literature,we can see that a lot of manpower is consumed to complete pre-processing.The numerical simulation takes a long time, and one single calculation can only obtain a muzzle flow field under specific launching conditions, which is inefficient.It is not conducive to the further development of artillery research and development.

As one of the three high technologies in the world since the 21st century,Artificial Intelligence(AI)has been widely used in various disciplines in recent years[10].In the field of fluid mechanics,there are many types of research based on machine learning and deep learning methods, trying to solve the flow field with AI [11].Numerous researchers use AI to complement numerical simulations performed using computational fluid dynamics (CFD).For example, Liu et al.[12] used AI to select the key time step in numerical calculation by establishing a Siamese deep neural network which has a symmetry structure with two Convolutional Neural Networks (CNN) to balance the calculation efficiency and accuracy.Hanna et al.[13] used two machine learning methods,neural network and random forest, to establish a surrogate model that can predict and correct the calculation error on the coarse grid,and achieve accurate numerical simulation on the coarse grid.Ray et al.[14] established a Deep Multilayer Perceptron (MLP) using neural networks to construct a problem unit detector, which is combined with the minmod limiter to effectively suppress numerical oscillations in the calculation process.Wu et al.[15]improved the Reynolds-averaged Navier-Stokes (RANS) equations in turbulence simulation,and used machine learning to predict the linear and nonlinear parts of Reynolds stress tensor respectively,to overcome the ill-conditioning RANS equations.

In recent years, with the explosion of machine learning, researchers began to try to completely replace CFD with AI methods.The early research mainly adopted a data driven approach[16].Big data is naturally generated in fluid numerical simulation and testing,which can be used to build data sets.Black box models for flow field prediction are established by machine learning these data using neural networks, random forests, and other methods.For example, CNN and MLP are used to mine the flow field pressure data obtained from numerical simulation [17].An AI with airfoils and angles of attack as input and flow field pressure as output can quickly solve the flow field around the wing [18].In multiphase flow,Ma et al.[19]established a Model Averaging Neural Network(MANN)to find the relationships between unknown closure terms in a simple model equation for the average flow and the resolved variables.The prediction of bubbly up-flow in a periodic vertical channel is realized according to different initial conditions.Flow field inversion and reconstruction is also one of the research focuses [20].By taking the geometric structure, characteristic parameters and boundary loads of the flow field as inputs, the unsteady reduced order model of the flow around cylinders is established using a convolution neural network and full connection neural network,and the inversion of the flow field around a single/multiple cylinders is realized [21].This kind of data driven AI is a surrogate model for the numerical simulation of a specific flow field.It has potential advantages in computational efficiency but lacks of generalization.The accuracy of the model depends badly on the data accuracy in the training set.The datasets of the above research are mostly generated by direct numerical simulations(DNS),as such,CFD is still indispensable.In addition,the black box model lacks practical physical significance [22].Especially for the strong discontinuity problems such as shock waves,it is difficult to completely avoid the appearance of non-physical results such as negative density.

After seeing the good performance of neural networks for complex behaviors[23],researchers began to try to embed physical knowledge into deep fully connected neural networks.With the help of Automatic Derivation (AD) technology [24], a neural network driven by physics is established, which is called Physics Informed Neural Network (PINN) [25].Successfully solved various classes of Partial Differential Equations (PDEs), such as stochastic differential equations [26], fractional equations [27] and phase transitions[28],etc.Maziar et al.[29]introduced PINN into the field of flow field solution.The solving process of differential equations is transformed into the training process of a neural network to simulate the flow field, by taking flow field governing equations,initial conditions (ICs), and boundary conditions (BCs) into loss functions.At present, PINN has successfully solved classical flow field problems such as cylinder flow[30]and blood flow[31].It has done well in turbulence [32], free boundary problems [33] and phase-field fracture problems [34].Jagtap et al.[35] proposed a generalized space-time domain decomposition approach for the PINNs,which is called eXtended PINN(XPINN).XPINN has stronger parallelization capability and can be extended to any type of PDEs,further promoting the development of PINN.The most attractive advantage of PINN is that compared with the traditional CFD methods, it voids complicated mesh generation work and greatly saves manpower.In addition,it can also solve the inverse problem[36]of flow field which is difficult to traditional CFD.However,for the forward problem, most researches focus on incompressible problems.The strong nonlinearity of compressible Navier-Stokes(NS) equations poses a great challenge to network training.The training process will become extremely complex if the shock wave and other discontinuities are considered.For compressible problems, PINNs are often difficult to converge after a long time of training.Mao [37] pointed out that for shock flow fields such as shock tubes, the results of PINN are not even as good as those of traditional CFD models, and there are defects that the initial conditions and boundary conditions cannot be strictly met.However,the idea of introducing governing equations into loss functions in PINN makes the calculation results more consistent with the physical laws and provides a way for solving muzzle flow fields.

Based on previous research, we will give full play to the advantages of data driven framework and physical driven framework,discard the disadvantages, and build a data physical fusion driven framework.In this paper, we take both advantages of data driven frameworks and physical driven frameworks and discard the disadvantages.A data-physical fusion driven framework is established,under which a simulation model of muzzle flow field empowered by AI is proposed.

This paper is structured as follows.In Section 2, the physical model and mathematical model of the muzzle flow field are established, and the multi-dimensional characteristic parameters are extracted.Section 3 discusses how to fuse physical equations and data information according to the characteristics of our research object and the shortcomings of traditional CFD models.A deep neural network under the fusion driven framework is built,and the input parameters are preprocessed.A model empowered by AI for solving the muzzle flow field is established.Section 4 presents several numerical examples to verify the proposed model.Taking a gun as an example,the disturbance characteristics of the flow field behind the muzzle are discussed.

2.Description of muzzle flow field

In this section, the physical model and mathematical model of muzzle flow field are established.Governing equations are given and muzzle characteristic parameters are selected.It lays a foundation for the establishment of solution models empowered by AI.

2.1.Physical model of muzzle flow field

Considering that we aim to establish a new method to quickly solve the muzzle flow field, the physical model of the flow field is greatly simplified.As shown in Fig.1,the reflection of the ground on the shock wave is not considered.The barrel is simplified as a cylinder,retaining only its shape.The muzzle section is regarded as the source term of the flow field,and the flow field in the tube is not modelled.The impact of propellant gas on the flow field is reflected by giving the form of the change rules of flow field parameters at the muzzle.Finally, taking the middle of the muzzle as the center and 6 m as the radius, a two-dimensional axisymmetric muzzle model is established.The area on the side of the muzzle is a key area,considering plenty of equipment such as turrets and cabs are located in this area.

Fig.1.A two-dimensional axisymmetric muzzle model.

2.2.Governing equations of muzzle flow field

Two-dimensional axisymmetric compressible unsteady NS equations are employed, which can be written in the following form [38]:

Here t is the time,x and y are positions,ρ is the fluid density,u is the velocity in the x direction,v is the velocity in the y direction,E is the energy, p is the pressure.

The ideal gas equation of state(EOS)is employed to describe the relationship between pressure and energy, namely

where γ is the specific heat ratio of gas.

The starting time of the aftereffect period is taken as the starting time.The initial flow field developed when the projectile moves in the bore are ignored.The initial flow field is regarded as a static field.A solid wall boundary condition is given for the barrel wall,and an inlet boundary condition is given for the muzzle.Parameters of the flow field at the free outlet boundary are not limited.

2.3.Multidimensional feature parameter extraction of muzzle flow field

The muzzle flow field is a typical jet flow field.High temperature and high-pressure gunpowder gas flow out of the muzzle,causing an impact on the external flow field.As a source term of the flow field,laws of the parameters at the muzzle play a crucial role in the development of the flow field.Based on the one-dimensional quasisteady theory and similarity assumption, previous studies considered that gas parameters on the muzzle under different launching conditions follow similar attenuation laws [39].The muzzle fluid pressure and density at the beginning of the aftereffect period,the launching velocity and the length of the barrel are key parameters as shown in Eq.(3).According to the parameter category,it can be divided into structural parameters and launch parameters.For a specific type of artillery, its structural parameters are determined values,so they are not used as input parameters of the model.The muzzle fluid pressure, density, and launching velocity are determined by charge conditions and are not independent of each other.Therefore, one parameter can be selected to represent the launching conditions.The launching velocity is selected as a representative parameter to describe the launching conditions due to its intuitiveness.

where ρmis the muzzle fluid density, pmis the muzzle fluid pressure,umis the muzzle fluid normal velocity,L is the length of barrel,and subscript 0 represents the parameter at the beginning of the after-effect period.

3.A flow field solution model empowered by AI under a dataphysical fusion driven framework

Through the above analysis,it can be seen that the muzzle flow field is a strong nonlinear compressible flow field,and the flow field is severely disturbed in a short time with strong transient.These characteristics lead to the defect of time-consuming solving the muzzle flow field by traditional numerical methods.In this paper,a new method for solving muzzle flow field is proposed by using a model empowered by AI to improve calculation efficiency.

3.1.Construction of a data-physical fusion driven framework

As shown in Fig.2, A flow field solution model under the data driven framework extracts information features from a large number of known flow field data through the depth learning method and establishes the mapping relationship between the physical parameters of the flow field and the network inputs.The inversion and prediction of the flow field are realized by building an agent model.This kind of model transforms the solving process of the flow field into a supervised learning process.The numerical solution of flow field parameters can be obtained efficiently because the iterative process of solving discrete governing equations in traditional numerical methods is avoided.Under the datadriven framework,the credibility of the model greatly depends on the number and accuracy of sample data in the training set,and the accuracy of prediction outside the range of sample data is often poor.For muzzle flow fields, whether through CFD or sensor testing,it is extremely expensive to obtain flow field data in a wide space-time range under different launching conditions.In addition,the lack of physical significance of the complete black box model is not conducive to the rational research of flow motivation.

The solution process of the model under the physical driving framework is also given in Fig.2.Independent variables of flow field governing equations are taken as the input,and basic parameters of the flow field are taken as the output.The loss function is composed of control Eq.(1) initial conditions and boundary conditions.The partial derivatives of the output parameters concerning time and space are obtained by using the chain derivative technique.The residuals are obtained by bringing them back to governing equations.To minimize the residual error, combined with the penalty term consisting of initial conditions and boundary conditions, an optimizer is used to iteratively optimize the network parameters,to obtain the numerical solution of the partial differential equations and solve the flow field.By fusing governing equations into machine learning as physical prior information,the prediction results are highly consistent with physical laws.However, with high accuracy comes a huge training cost.The model cannot be converged for complex flow field problems after a long time of training.

Fig.2.Processes of flow field solution model under different frameworks (left: data driven, right: physical driven).

After analyzing the advantages and disadvantages of the data driven framework and the physical driven framework, we can't help thinking about whether we can take both superiorities of them to establish a model that can efficiently and accurately solve the compressible flow field.A neural network method is used as the black box of the model, and the training of the network is divided into two stages.In the first stage,the known flow field information is used to form a training set, to minimize the average error of predicted flow field parameters and sample values.The loss function can be written as

where N is the number of samples, U is the vector of flow field parameters predicted by the model,U is the vector of known flow field parameters.

The first stage can be regarded as an initialization process of model parameters such as weight and bias.By learning the known flow field information in the dataset, the model has preliminary intelligence.Flow field parameters predicted by the model can be consistent with the test data after the first stage.In the second stage, governing equations, initial conditions and boundary conditions are taken as loss functions,as shown in Eq.(5).At this time,values of the model parameters to be trained are close to optimal values, which greatly reduces the difficulty of optimization in the second stage.

where NFis the number of points in the calculation domain, NBis the number of points on boundaries of the calculation domain,NIis the number of points at the initial time, F(U) is the flux term in governing equations.

The training process of the model under the data-physical fusion driven framework is shown in Fig.3.The first stage is regarded as the initialization process of the second stage by constructing the loss function in sections.The model is trained by using the known flow field information in the data set so that the values of model parameters at the initial time of the second stage are close to the optimal values.In this way, the initial training error and the optimization difficulty are greatly reduced.The second stage is the enhancement and generalization of the first stage.By introducing physical prior knowledge, the limitation of the training range on the accuracy of the model is broken,and the consistency between the prediction results and the physical laws is improved.The two training stages are integrated to realize the complementary advantages of data driven and physical driven,which makes it feasible to enable the model empowered by AI to solve the compressible flow field.

Fig.3.A Data-physical fusion driven framework.

3.2.Deep neural network construction under fusion driven framework

Based on deep learning methods,a black box model for solving the muzzle flow field is constructed.A fully connected neural network is adopted,composed of input,hidden,and output layers.Each hidden layer contains 20 neurons, a total of 7 hidden layers.According to the analysis in Section 2, the space coordinates x,y,time t and launching velocity um0are taken as the input, and the flow field parameters are taken as the output.A deep neural network with the structure shown in Fig.4 is established.The tangent function is the activation function among neurons, as:

Finally,the model empowered by AI for muzzle flow field can be written as

where loss functions Loss are given in Eqs.(4) and (5).

3.3.Preprocessing of flow field data

After the establishment of a model empowered by AI, the muzzle flow field can be solved ideally.However,the solution may be affected due to the large difference in the numerical values of parameters of muzzle flow field.For example, if the SI system is adopted, the value of pressure may reach 106while the value of density is less than 101.To eliminate this adverse effect,the launch speed um0is normalized, and the dimensionless NS equation is introduced.The form of the dimensionless NS equation remains unchanged, and the dimensionless parameters are defined as Ref.[38]

Fig.4.The deep neural network of the muzzle flow field.

where the superscript*denotes dimensionless parameters,values without superscript are dimensioned parameters, the subscript ∞represents the undisturbed fluid parameters in the far field,tcis the total simulation time,L is the characteristic length of the flow field,ρ is the fluid density.u is the fluid velocity in the x direction,v is the fluid velocity in the y direction, c is the speed of sound.

The data set for model training can be established according to the known flow field information.Both test methods and CFD methods can be used to obtain flow field information.By adding test data to the training set,the simulation can be guided by reality,making the simulation results closer to the actual situation.

4.Numerical results

This section will test the performance of the proposed model.First, two theoretical examples are used to discuss the advantages of the data-physical fusion driven framework over the single driven framework and verify the feasibility of the model empowered by AI under the framework to solve the shock wave flow field.Then,Taking the case of 900 m/s launching velocity as an example, the advantages of the proposed model in muzzle flow field are discussed by comparing the solution results with the traditional CFD model.Finally, based on the model empowered by AI, the characteristics of the flow field behind the muzzle of a gun are analyzed.

4.1.Example 1: One-dimensional flow field with periodic boundary conditions

Considering a one-dimensional flow field subject to the following initial condition:

where x∊[- 1,1].All parameters are dimensionless.

With periodic boundary conditions, the analytical solution to the problem can be written as

Models under different driven frameworks are used to solve the problem.The simulation time is t∊[0,1].The data set required for training is constructed by using the analytic solution.201 sampling points are uniformly arranged in time and space, and the training set contains 40,401 samples in total(Figs.6 and 7).Figs.5-7 show that the model under a physical driven framework has not converged after a long time of training.It can be seen that the model under a simple physical driven framework cannot solve this problem quickly and accurately.Models under data driven and data-physical fusion driven frameworks both achieve the solution of this problem well,and the prediction results are very close to the analytical solution.However,when the predicted time exceeds the given simulation time,the accuracy of the model under data driven framework becomes worse over time.As shown in Fig.8.

4.2.Example 2: One-dimensional flow field with a moving contact discontinuity

Consider a flow field with a moving contact discontinuity.The computation domain is x∊[0,1].An initial shock is given at the center of the domain, and the left and right states are given as follows:

Fig.5.Density distribution of the flow field when t = 0.5.

Fig.6.Density distribution of the flow field when t = 0.5.

Fig.7.Density distribution of the flow field when t = 0.75.

Fig.8.Difference between the MSE of the model under data driven framework and that of the model under data-physical fusion driven.

Similarly,in the case of the same neural network and calculation parameter settings, models under three different driven frameworks are used to solve the problem.The simulation time is t∊[0,1].101 sampling points are uniformly arranged in time and space,and the training set contains 10,201 samples in total.Fig.9(a)and 9(b) show that models under three driven frameworks can accurately predict the flow field within the time range covered by the data set (interpolation).When we want to get the flow field parameters at the time other than the training set(extrapolation),as shown in Fig.9(c) and 9(d), the results of the model under the data driven framework appear to be unsatisfactory.The predicted flow field at t =1.35 is still in good agreement with the analytical solution of models under the framework of physical driven and fusion driven.When the prediction time is further away from the training set, as shown in Fig.10, the error of the model under a physical driven framework also starts to rise, but the error of the model under data-physical fusion driven is always small.It is proved that the model under the fusion driven framework is more generalized and extensible.

4.3.Example 3: Two dimensional muzzle flow field without brake

After verifying that the model empowered by AI under a dataphysical fusion driven framework can solve the compressible flow field with shock waves, we will check the performance of the proposed model in the muzzle flow field.Based on the model established in Section 2,and considering the operating condition of 900 m/s launching velocity, the results of the proposed model are compared with those of the CFD model.

Different from the first two examples,the muzzle flow field is an engineering example,and there is no accurate solution to form the data set required for model training.In this regard,we use the CFD results to build the data set.On the ray at 90°,105°,120°,135°and 150°to the barrel axis, one monitoring point is set every 0.5 m,as shown in Fig.11.Fifty monitoring points shall be set for each launching condition.According to the charge condition of this type of gun, numerical simulations are carried out for ten working conditions with launching velocities of 850-940 m/s.The simulation time is set to 10 ms.The working condition with the launching velocity of 900 m/s is taken as the test set, and the other nine working conditions are taken as the training set.The number of samples in the training set is 243,450 finally.

The CFD results of muzzle flow field are obtained by using commercial software Fluent.The setting of relevant calculation parameters is referred to Ref.[39].Taking the muzzle flow field problem in Ref.[40] as an example, the CFD results are compared with the test results in Ref.[40]to verify the validation of our CFD model for muzzle flow field simulation.It can be seen from Fig.12 that the CFD model can accurately capture the complex wave system in the muzzle flow field, and the simulation results are consistent with the test results, which proves the accuracy of the data set in this paper.

Fig.9.Density distribution of the flow field: (a) t = 0.25; (b) t = 0.55; (c) t = 1.35; (d) t = 1.5.

Fig.10.MSE of the flow field.

The development of the muzzle flow is crucial information to check the accuracy of the CFD results of muzzle flow.The evolution of muzzle Mach field obtained by CFD is shown in Fig.13.It can be seen that in the early after-effect period,spherical and bottle shock wave form at the muzzle and continue to grow.Then a typical under-expanded jet structure is presented.The muzzle flow field is divided into three well-defined regions:the free expansion region in the bottle shock wave, the supersonic region between the intersecting shock wave and the jet boundary, and the subsonic region in the Mach disk.After that,the muzzle flow field entered a stable stage, and the bottle shock wave grew slowly.After about 0.3 ms, the flow field attenuates and the bottle shock shrinks continuously.

Fig.11.Layout of monitoring points.

Fig.12.Comparison of experimental shadowgraphs (left) [40] and numerical schlieren images (right): (a) t = 17.5 μs; (b) t = 95.57 μs.

For the convenience of expression,a polar coordinate system is established with the muzzle center as the origin,the barrel axis as the polar axis, and the firing direction as the positive direction.It can be seen from Figs.14 and 15 that the results of the two models are very close, and the solution accuracy of the proposed model is equivalent to that of the CFD model.The results show that the flow field behind the muzzle side has a high temporal and spatial transient characteristic in the process of propellant gas emptying.The pressure and density near the shock wave change dramatically in a short time.The rising time is significantly less than the falling time.The flow field disturbance law at each position behind the muzzle side is similar.When the shock wave passes, the pressure and density of the flow field rapidly rise to the maximum value and then fall below the initial value.As the measuring point is far away from the muzzle, the rising time and falling time of the pressure and density curve increase gradually,and the peak value decreases,indicating that the intensity of the flow field disturbance decreases with the increase of the distance from the muzzle.

However, the computational efficiency of proposed model is much higher than that of CFD model.As shown in Table 1, on a desktop computer with a 3.90 GHz CPU and 16 GB memory,it takes about 40 min(5099 grid cells)for the CFD model to solve the twodimensional axisymmetric muzzle flow field under one single launching condition.It only takes 4 ms for the proposed model to solve the flow field with the same amount of data,and it only takes an additional 255 s even considering the training time.In addition,the CFD model can only obtain the muzzle flow field of a specific launching condition at one time.It must be calculated from time t =0 ms,and the time advance would be carried out in a very small time step until the required time is reached.The proposed model can obtain the muzzle flow field information at any time of any launching conditions with very little time, without the need for a time advancement process, and reduces the generation of redundant information.It can be seen that the model empowered by AI can greatly improve the calculation efficiency when it has the same accuracy as the CFD model.The proposed model can quickly and accurately predict the muzzle flow field.

4.4.Characteristic analysis of the flow field behind muzzle of a gun

With the help of the proposed model, the simulation of the muzzle flow field under arbitrary launching conditions can be completed quickly.Taking a 155 mm caliber gun with a reducer as an example,its launch velocity range is 850-940 m/s.Fig.16 shows relationships between peak values of fluid overpressure and distances from the muzzle.At various angles, the peak value of fluid pressure tends to decrease with the increase of the distance from the muzzle.The higher the chamber pressure is, the higher the overpressure peak value at the same position behind the side is.The smaller the angle, the more obvious the change of pressure.Compared with the area behind the barrel, the pressure disturbance in the area on the side of the barrel is more intense.Compared with the muzzle flow field without the brake, the pressure disturbance behind the muzzle flow field side with the brake is more severe.The reason for this phenomenon is that, in order to counteract the recoil force, part of the propellant gas is sprayed to the rear area under the guidance of side holes of the brake,which will exert a more severe impact on the side area of the barrel while exerting a forward impact on the barrel.

Fig.13.Contours of Mach number in muzzle flow field.

Fig.14.Computed of the muzzle flow field of pressure using two different models.

Fig.17 shows the pressure contour of the flow field behind the muzzle side when the launch velocity is 940 m/s.The changing trend of the flow field behind the muzzle side can be clearly observed.It can be seen that when the projectile leaves the muzzle,the high-pressure gas in the chamber will be ejected rapidly.The gas forms a violent pressure disturbance on the rear area of the muzzle side through holes of reducer, forming a shock wave that increases continuously.As the propellant gas is pushed forward and continuously replenished,each side hole of the brake forms its own shock wave in turn, accompanied by chasing, intersection and superposition, forming a clear shock wave group.After about 2 ms,the shock wave from each side hole merged into a large shock wave on the side and increased continuously.With the limited energy supplement of side hole jet, the farther the shock wave ball propagates,the smaller the disturbance to the position near the muzzle.In the middle of the aftereffect period,the energy supply of the side hole has been far behind the pressure drop in the inner area of the impact ball.At the same time, the jet of the side hole has accelerated its expansion, and the gas pressure in the shock wave ball has dropped sharply below atmospheric pressure.3 ms later,the gas in the bore flows empty, and the flow field enters the attenuation stage.Losing the energy supplement of high-pressure gas in the bore, the shock wave diffuses and attenuates at the same time.After the jets intersect, they flow to the rear of the muzzle side and are squeezed by the static gas in the flow direction,forming a small bottled shock wave.The peak pressure decreases with the diffusion of the spherical wave.

Fig.15.Computed of the muzzle flow field of density using two different models.

Table 1 Comparison between CFD model and model empowered by AI.

Fig.16.Relationships between peak values of flow field pressure and muzzle distances at different launching velocities: (a) 940 m/s; (b) 850 m/s.

Fig.17.Contours of fluid pressure (radius: 5 m).

The most noticeable is the impact damage to personnel caused by the muzzle blast wave.It shows that people who are located at the place where the peak overpressure exceeds 41.2 kPa need to take protective measures to prevent physical injury.Fig.18 draws the safe range of the flow field behind the muzzle side under 4 launching velocities with 41.2 kPa as the limit.We can see that shape of the safety clearance are roughly the same.When the launch velocity is 940 m/s,the peak overpressure in the area about 9.2 m behind the muzzle exceeds the threshold value.With the rise of chamber pressure, the position of safety clearance gradually moves backward.

Fig.18.Schematic diagram of safety area (radius: 10 m, red: non-safety, green: safety).

5.Conclusions and future prospects

In this paper,artificial intelligence technology is introduced into the solution of the muzzle flow field.Aiming at the characteristics of high transient and strong nonlinearity of muzzle flow field, a data-physical fusion driven framework is proposed.Under this framework, a model empowered by AI is established, which can quickly and accurately simulate muzzle flow field under various launch conditions.

Relevant conclusions are as follows:

(1) The wave system of the muzzle flow field is complex and highly nonlinear.Traditional numerical methods require a large number of grids and a very small time step to achieve high-precision simulation,which takes a long time.Artificial intelligence provides a new technology for solving the flow field.The simulation efficiency of the muzzle flow field can be greatly improved by using a depth neural network.

(2) PINN has insufficient ability to solve compressible fluid problems.The data driven model performs well in interpolation,but not well in extrapolation.Neither of them has the potential to solve the muzzle flow field rapidly and accurately.

(3) A data-physical fusion driven framework is proposed,under which a model empowered by AI for compressible flow field is established.Results of numerical examples show that the proposed model can solve the flow field with the shock wave,and has better generalization and expansion.Compared with the CFD results,the calculation time of the proposed model is greatly reduced when the calculation accuracy is equivalent.

(4) The proposed model is used to solve the muzzle flow field of a gun, and the muzzle flow field under various charge conditions is obtained.The results show that the flow field characteristics are similar at different launching velocities.Through numerical simulations,we found that in the process of gun firing,the shock wave passes through the rear area of the muzzle, causing severe disturbance.

(5) In the case of no brake, the muzzle shock wave is approximately spherical,and the pressure and density disturbances in the area behind the side are similar.The closer to the muzzle, the more intense the disturbance of the flow field.

(6) When the brake is considered,the flow field wave system in the area behind the muzzle side will be more complex due to the diversion effect of the side hole.The pressure disturbance at the side of the barrel is more severe than that at the rear,and the peak overpressure of the flow field is positively correlated with the distance between the measuring point and the muzzle.The distribution characteristics of muzzle flow field are similar under different launch speeds.The higher the bore pressure,the more severe the disturbance of flow field.

(7) Based on the simulation results,with the limit that the peak value of overpressure shall not exceed 41.2kPa, the safety zone is determined to be 9.2 m away from the rear of the muzzle.

By using the proposed model,we successfully solved the muzzle flow field quickly and accurately.The future research is prospected as follows：

(1) In future work, we plan to consider more neural network structures,such as long and short memory neural networks,and convolution neural networks, to reduce the number of parameters of the black box model and reduce the training difficulty.

(2) On the other hand,we will consider more factors that affect the muzzle shock wave,such as the reflection of the ground,cab, and other structures on the shock wave.In future research, we plan to take these factors into account one by one and use artificial intelligence technology to research the interaction between muzzle shock waves and structure, to provide technical means for the design of gun cab structures against shock waves.

Declaration of competing interest

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

Acknowledgments

Supported by the Natural Science Foundation of Jiangsu Province of China (Grant No.BK20210347).Supported by the National Natural Science Foundation of China (Grant No.U2141246).