Chun-hua Bai,Xing-yu Zhao,Jian Yao,Bin-feng Sun

State Key Laboratory of Explosion Science and Technology,Beijing Institute of Technology,Beijing,100081,China

Keywords: Fuel-air explosive Numerical simulation Multi-sources explosion Shockwave overpressure field

ABSTRACT Shockwaves from fuel-air explosive(FAE)cloud explosions may cause signi ficant casualties.The ground overpressure field is usually used to evaluate the damage range of explosion shockwaves.In this paper,a finite element model of multi-sources FAE explosion is established to simulate the process of multiple shockwaves propagation and interaction.The model is veri fied with the experimental data of a fourfoldsource FAE explosion,with the total fuel mass of 340 kg.Simulation results show that the overpressure fields of multi-sources FAE explosions are different from that of the single-source.In the case of multisources,the overpressure fields are in fluenced signi ficantly by source scattering distance and source number.Subsequently,damage ranges of overpressure under three different levels are calculated.Within a suitable source scattering distance,the damage range of multi-sources situation is greater than that of the single-source,under the same amount of total fuel mass.This research provides a basis for personnel shockwave protection from multi-sources FAE explosion.

1.Introduction

Liquid fuel,or solid dust can mix with air to form a flammable cloud,which is usually called Fuel-Air Explosive(FAE).The shockwave from an FAE explosion is one of the main human hazards,as the body may suffer damage if the shockwave overpressure reaches to 30 KPa[1].Furthermore,the shockwave intensity of an FAE explosion attenuates more slowly than that of a dynamite explosion.This means that at far-field,unprotected personnel runs a higher risk of shockwave damage from FAE than from conventional explosive.The ground overpressure field of FAE explosion is usually applied to evaluate the damage range of FAE,based on the blast overpressure criterion.Therefore,studies related to the shockwave overpressure fields of FAE explosions are important for the shockwave protection of personnel.

A series of studies have been conducted to reveal the explosion characteristics of FAEs in con fined environments.With the help of explosion vessels or detonation tubes,various kinds of FAEs have been studied,such asn-hexane-air mixtures[2],diethyl ether-air mixtures[3],nitro methane mist-aluminum dust-air mixtures[4],and diethyl ether mist-aluminum dust-air mixtures[5].However,these experiments have been carried out on small scale and in restricted equipment spaces,which are not comparable to the actual situation.An FAE canister is one way to obtain large-scale uncon fined FAE cloud[6].Fuel is dispersed by central explosives and mixed with the air to form a large scale FAE cloud.A charge is subsequently applied to trigger the explosion of the cloud.Some scholars have paid attention to the factors that in fluence the overpressure fields of FAE explosions,such as fuel type[7-9],cloud shape[10],and charge motion[11].Furthermore,with advances in computational technology,numerical simulation is now capable of providing complementary details and reducing costs for large-scale cloud explosion experiments.For example,Zhang et al.[12]established a numerical model of fuel dispersion by means of the dynamic analytical software ANSYS/LS-DYNA.Chen et al.[13]modelled the shockwave propagation process for a non-cylindrical cloud explosion by using ANSYS/LS-DYNA.

From the aforementioned studies,one may conclude that the coverage range of FAE cloud has a great impact on its damage ability.However,when fuel payload reaches to the order of 102kg,or even 103kg,it’s quite dif ficult to disperse fuel to the desired range by simply increasing the central charge amount.To solve this problem,a feasible method is to use multiple FAE sources,which detonate synchronously to produce an overall destructive effect,so as to reduce the dispersion dif ficulty of the single FAE canister.Murray et al.[14]realized the simultaneous detonation of three FAE sources,and observed shockwave focusing phenomenon,which enhanced the shockwave overpressure in speci fic regions.Chen et al.[15]studied the shockwave focusing intensity of fourfoldsource FAE.Currently,research related to multi-sources FAE mainly focuses on the local region of shockwave focusing,and few involves the overall overpressure field.

Therefore,this study pays attention to the overpressure field characteristics of multi-sources FAE,and demonstrates the in fluence of the two main factors,scattering distance and source number,on its overpressure field.The present study is under the ideal condition that all FAEs are ignited simultaneously.In this paper,a finite element model of multi-sources FAE explosion is established by using ANSYS/LS-DYNA.The model is veri fied by the results from a fourfold-source FAE experiment.Based on the numerical model,the overpressure fields of various multi-sources conditions are obtained,and the damage ranges are calculated.The findings will be applied to improve the effectiveness of FAE device and provide theoretical guidance for personnel shockwave protection from multi-sources FAE.

2.Numerical simulation method

The shape and size of the FAE clouds are referred from a fourfold-source FAE experiment,in our previous study[15].Accordingly,the experimental layout and results are brie fly introduced to this paper.Then,the numerical simulation model of multisources FAE explosion is established and veri fied by peak overpressure values from the experiment.

2.1.The fourfold-source FAE experiment

The experimental system consists of four FAE canisters,four initiators with control system,a video subsystem,and a pressure measurement subsystem,as shown in Fig.1.The aforementioned FAE canisters,the cross sections of which are fan-shaped(Fig.2),are each filled with 85 kg of solid-liquid mixed fuel.A 0.9 kg cylinder of high explosive TNT is used as the center charge in each FAE canister.The source scattering distance of each FAE canister is 22 m.Moreover,a 7.9 kg of TNT charge is used as the initiator.Each initiator is located 1.5 m behind the corresponding FAE canister and 2.5 m above the ground.The fuses of canisters and initiators are connected to the control system circuit to achieve precise detonating time control.The primary fuses are set to be ignited simultaneously,and the four FAE canisters disperse fuel into air.The fuel mixes with ambient air for a moment to form large-scale fuel-air clouds,wrapping the initiators inside the clouds.The predicted dispersion range of each FAE cloud(grey area)is marked on Fig.1.Then,at a 220 ms delay time after the ignition of the primary fuses,the secondary fuses are initiated to trigger the detonation of the clouds simultaneously.The video subsystem includes a RedLake HG-100 K high-speed camera(shooting rate of 2000 fps)and a Canon-EOS-30D DV(shooting rate of 25 fps).The pressure measuring subsystem contains 32 Kilster-ICP piezoelectric pressure sensors.Each sensor is mounted on a steel plate with a 0.3 m diameter.These plates are laid on the ground annularly from the site center along the four directions:0°,90°,135°and 180°.The distances between the site center and the sensor rings are 5 m,8 m,10 m,15 m,20 m,30 m,40 m and 50 m,respectively.The effect of TNT initiator is negligible,because the shockwave it generated has decayed to 0.15 MPa at the nearest sensor,while the overpressure inside the cloud can reach to 3-4 MPa.

Fig.3 displays images before and after detonation,which are captured by the high-speed camera.According to Fig.3(a),the shape of each cloud is similar to a quadrangular prism,the cross section of which is an isosceles trapezoid.By using image processing technology,the approximate size of the cloud is calculated to be as follows:the cloud height is 3.0 m;the length of the short side of the trapezoid is 11.8 m and long side is 25.7 m;and the trapezoidal height is 25.0 m.Thus,the total volume of the fourfoldsource clouds are 5625 m3and the apparent concentration of fuel is 60 g/m3.The sensor results of 0°,90°and 180°are used to calculate the detonation parameters of FAE.In the three directions,the sensors at 10 m,15 m,20 m and 30 m are shrouded in the cloud detonation zones,according to the location of the clouds.The results of peak overpressure are averaged to be the detonation pressure.Further,the result of arrival time of detonation wave at each sensor is obtained.And the distance between initiator and sensor is the propagation distance of the detonation wave.The detonation velocity is calculated by dividing distance by arrival time.Thus,the average detonation pressure is calculated to be 3.32 MPa and the average detonation velocity is 1280 m/s.

Fig.3.Experimental image.(a)Fourfold-source FAE cloud(Dashed lines are cloud outlines).(b)Detonation.

2.2.Finite element model

A numerical model for multi-sources FAE explosion is established based on ANSYS/LS-DYNA software.Considering the distance for shockwave propagation,the computational domain is a cylinder with a radius of 150 m and a height of 20 m.The computational domain includes two components,the FAE clouds and the air,as shown in Fig.4.The shape and size of fourfold-source clouds are determined by experimental data in the Chapter 2.1.Variabledin Fig.4 describes the distance between the geometric center of the trapezoid and the center of the cylindrical domain,which presents as the source scattering distance.Here,two assumptions are used.First,the height-to-diameter ratio of the FAE cloud stays constant under different source situations.Second,fuel concentration of the FAE cloud stays constant.Thus,under the condition that the total fuel mass is 340 kg,cloud volume and detailed geometric size for different FAE clouds are calculated for different source situations,as shown in Table 1.A comparison model for a single-source FAE cloud is shown in Fig.4(d).The shape of the single-source FAE cloud is a cylinder,as the FAE canisters are generally cylindrically shaped,which produces cylindrical clouds[9-12].

Table 1 FAE cloud volume and geometric size for different source situations.

Fig.4.Computational domain under different source numbers(Length unit:meter).(a)Double-source cloud.(b)Fourfold-source cloud.(c)Sixfold-source cloud.(d)Single-source cylindrical cloud.

Considering the symmetry,a quarter of the domain is meshed with hexahedral grid to establish a finite element model.Fig.5 displays the grids and boundary condition of the model,taking the double-source situation as an example.The yellow region represents the FAE cloud,and the blue region represents the air.The grid size is 0.6 m,and the total element number is approximately 1.6 million.On the two symmetry planes,the normal displacement for the nodes is constrained in order to follow a symmetric boundary condition.The ground plane adopts a rigid wall boundary condition.The other surfaces use a non-re flecting boundary condition with a 105Pa pressure along the inner normal direction to simulate the in finite air domain.Fourfoldsource,sixfold-source and single-source situations are modelled with the same way,as shown in Fig.6.The finite element model is calculated by using the arbitrary Lagrange-Euler(ALE)multimaterial algorithm.The material group IDs are defined by the keyword “*ALE_MULTI-MATERIAL_GROUP” [16].Additionally,the two components are coupled by co-nodes on the connection surface to achieve better interaction between the shockwave and the material[17].The detonation points,which are located at the vertical axis through the geometric center of the cloud and are 0.83 times that of the height of the cloud(according to the experimental arrangement), are defined by the keyword “*INITIAL_DETONATION” [16].All detonation points are triggered simultaneously.

Fig.5.The finite element model for double-source FAE explosion.(a)Bottom view(Mirrored).(b)Meshed FAE cloud(Mirrored).(c)Meshed quarter model.

Fig.6.The finite element model for fourfold-source,sixfold-source and single-source FAE explosion(Mirrored bottom view).

The material model for the fuel-air explosive cloud is described by the keyword “*MAT_HIGH_EXPLOSIVE_BURN” ,with a Jones-Wilkins-Lee(JWL)equation of state(keyword “*EOS_JWL” )[16].The expression for the shockwave pressure in the JWL equation is given as follows:

where,Pis the pressure;Eis the initial internal energy per unit volume;Vis the relative volume;andA,B,R1,R2andωare empirical parameters related to the explosive properties.Moreover,the High Explosive Burn Model also needs to determine the initial densityρ,the detonation velocityD,and the detonation pressurePCJfor the explosive.The values ofDandPCJwere calculated from the experiment in Chapter 2.1.The other parameters are obtained and modi fied from Ref.[13],as presented in Table 2.

Table 3 Parameters for the air component.

The material model for the air is applied by using the keyword “*MAT_NULL” with an equation of state “*EOS_LINEAR_POLYNOMIAL” ,which is given as follows:

where,Pis the pressure;μ=ρ/ρ0-1,whereρ/ρ0is the relative density;Eis the initial internal energy per unit volume;andC0~C6are constants related to the material properties.The material parameters of the air are obtained from Ref.[18],as presented in Table 3.

2.3.Validation of the numerical model

2.3.1.Validation results for single-source FAE

The numerical results of single-source FAE are calculated and compared with published articles,as shown in Table 4.The peak overpressures under three different scaled distances are selected for comparison.The scaled distance is defined asHere,Rrepresents the distance from burst andMfuelis the weight of fuel.Among the articles,Ref.[18]is mainly studied by numerical simulation method,while the others are experimental.Both numerical methods simplify the actual ground to rigid wall boundary condition,leading to a slightly higher prediction on overpressure values.However,comparing with Ref.[18],the present result in this work is closer to the experimental result.This is because the present model considers the shape and size of real FAE cloud,while Ref.[18]neglects the cloud by TNT equivalent method.In terms of computational cost,this work is a little higher than Ref.[18].With the increase of the scaled distance,the numerical results of this work gradually agree with the experimental values.

Table 4 Comparison of the peak overpressure values.

2.3.2.Validation results for multi-sources FAE

The validation model for multi-sources FAE is under the condition of a fourfold-source FAE with a 22 m source scattering distance,which is same as the experimental condition described above.In order to verify the model’s ability to predict shockwave interaction and explosion far-field overpressure,the numerical results of the ground peak overpressure in the 135°direction are compared with the experimental values,as shown in Fig.7.The black line represents the numerical simulation results fitted by using the B-spline method,while the red point represents the experimental values measured by the pressure sensor.The simulation results are consistent with the experimental data in terms of trends because both demonstrate the high pressure(near the 15 m position)generated by shockwave focusing.And then the pressure decreased by propagation in two directions.When the propagation distance is small,there are some deviations between the simulated results and the experimental data.The reason is that FAE clouds under the aforementioned experimental conditions differ in shape,size and concentration distribution because of the randomness caused by explosion dispersion.However,as shockwave propagation distance increases,the simulation results gradually approach the experimental data.When the propagation distance exceeds 30 m,in the explosion far-field,the numerical results are close to the experimental values.Thus,the numerical model from this paper can be applied to future research.

Fig.7.Comparison of the simulation results and the experimental values in the 135°direction.

3.Results and discussion

3.1.Shockwaves propagation characteristics

Fig.8 displays the pressure contour of the ground for multisources FAE explosions,under a source scattering distance of 40 m.The unit of pressure is the Pascal.The detonation moment,at which each cloud source detonates simultaneously,is defined as the initial moment.To illustrate the multiple shockwaves propagation characteristics,moments when shockwave focusing is apparent are selected for display.For a double-source cloud(Fig.8(a)),the blast wave propagates outward from the explosive sources,leading to a signi ficant negative pressure in the source area(t=40 ms).The two waves interact in the y-z plane,causing an obvious high-pressure zone on the ground(t=60 ms).For the fourfold-source cloud(Fig.8(b)),the spherical shockwaves generated by adjacent FAE sources converge to form four high-pressure zones(t=36 ms).Then,65 ms after detonation,the four shockwaves coalesce at the center,producing a pressure 2.1 times higher than that of the non-focusing area at the same distance.Similar to the fourfold-source cloud,the pressure distribution of the sixfoldsource FAE explosion(Fig.8(c))also obviously shows shockwave focusing phenomena,at 22 ms after ignition.Subsequently,the six high-pressure zones move towards the center as the shockwaves propagate.Eventually,all the shockwaves converge at the center(t=68 ms).In summary,the blast wave propagation characteristics are more and more complicated as the source number increases,and they are different from those of the single-source cloud explosion.

Fig.8.The pressure contour for multi-sources FAE explosions.(a)Double-source cloud(d=40 m).(b)Fourfold-source cloud(d=40 m).(c)Sixfold-source cloud(d=40 m).

3.2.Overpressure fields under various source scattering distances

Reference[19]gives the level of damage to unprotected personnel from shockwave overpressure,as presented in Table 5.Therefore,this paper selects three overpressure values,0.03 MPa,0.05 MPa and 0.1 MPa,as the thresholds for medium damage,serious damage and fatal damage,respectively.According to these overpressure thresholds,the overpressure fields are calculated by the peak overpressure values of all nodes on the ground and drawn in Fig.9.Peak overpressure represents the maximum value of a nodal overpressure curve can reach.The red region in Fig.9 represents an overpressure value greater than 0.1 MPa.The orange region indicates that the overpressure value is between 0.05 MPa and 0.1 MPa.The yellow region shows that the overpressure value is between 0.03 MPa and 0.05 MPa.An overpressure value below 0.03 MPa is drawn in grey(see Fig.9).

Table 5Shockwave overpressure damage to unprotected personnel[19].

Table 6 The TNT equivalency values under various source numbers.

Fig.9.Overpressure fields under various source scattering distances.(a)Doublesource cloud.(b)Fourfold-source cloud.(c)Sixfold-source cloud.

Fig.9 illustrates the overpressure fields under various source scattering distances.In cases with the same source number,the overpressure field changes obviously with variation in source scattering distance.There are three stages in the effect of source scattering distance on overpressure field.When the source scattering distance is relatively short,the high overpressure zones produced by different FAE cloud sources partially overlap with each other and lead to relatively small damage ranges.At this situation,the energy from the FAE explosion is concentrated in the central region,and most of the damage is created by the overpressure exceeding 0.1 MPa,causing a higher overpressure intensity but smaller damage range.As the source scattering distance increases,the outlines of multiple sources gradually emerge,which means that the overlapped region decreases.The range of each damage level has increased signi ficantly,especially for the medium damage and serious damage where the overpressures are between 0.03 MPa and 0.1 MPa.In addition,due to shockwave focusing,fractional regions of relative high pressure are formed between adjacent cloud sources.These fractional regions further expand the damage range of multi-source FAE explosion.When the scattering distance continues to increase,shockwave focusing ability is weakened.The fractional high-pressure regions shrink and disappear gradually,leading to a slightly reducing of the damage range.According to the analysis,there is a speci fic scattering distance at which the damage range reaches a maximum value.

3.3.Overpressure fields under various source numbers

A comparison of the overpressure fields under different source numbers is shown in Fig.10,at a source scattering distance of 50 m.The overpressure field structure for a single-source cylindrical cloud(Fig.10(a))resembles relatively regular geometry.A highpressure zone is at the center,and when propagating outward,the peak overpressure drops rapidly.The damage range is same in all directions.While the multi-sources FAE explosions have irregular overpressure field structures(Fig.10(b)~10(d)),under the combined action of multiple shockwaves.Multiple shockwaves propagate from each FAE source to the surroundings,and connect together into a ring-like overpressure field structure.Because of shockwave focusing,many local high-pressure zones are formed,for example,the central high-pressure region.Therefore,highpressure zones and low-pressure zones appear alternately in the overpressure fields of multi-sources FAE explosions.The damage ranges are different along different directions.For unprotected personnel,if they are located in the regions covered by clouds or converged by shockwaves,they are very likely to suffer severe or even fatal shockwave damage.

As the source number increases,the damage range obviously increases.This is because the energy from a single-source FAE explosion is concentrated to generate relative high pressure near the FAE cloud,and consumed by the attenuation of the shockwave with propagation distance.When far away from the cloud,the shockwave overpressure drops rapidly and finally below the damage threshold.While for multi-sources situations,the energy distributes among multiple smaller FAE clouds,and utilizes more ef ficiently compared with the single-source FAE cloud.Moreover,shockwaves interaction enhances the overpressure in some regions to a higher damage level.Thus,a multi-sources FAE explosion can have a greater damage range than a single-source condition,under the same weight of fuel.

Fig.10.Overpressure fields under various source numbers.(a)Single-source cloud.(b)Double-source cloud(d=50 m).(c)Fourfold-source cloud(d=50 m).(d)Sixfold-source cloud(d=50 m).

3.4.Damage ranges under various source scattering distances

The damage ranges of the three overpressure thresholds are calculated from the overpressure fields.The variation in damage range with source scattering distance is shown in Fig.11.The damage range is evaluated by coverage area and equivalent damage radius.The three colors,black,red and blue,represent the overpressure thresholds of 0.1 MPa,0.05 MPa and 0.03 MPa,respectively.Solid points indicate the damage areas for the multi-sources FAE explosions under different scattering distances.And they are fitted to curves by using the B-spline method.The dotted lines denote the damage areas for a single-source cylindrical FAE explosion.Because a single-source explosion is not affected by the scattering distance,in order to draw a comparison with a multisources explosion,the results for a single-source explosion are drawn as horizontal lines.

For a double-source cloud explosion(Fig.11(a)),when the scattering distance is small,the damage ranges grow as the distance increases.At a scattering distance of 35 m,the 0.1 MPa damage area reaches its maximum value,which increases by 30%compared to a single-source explosion.At a scattering distance of 45 m,the 0.05 MPa damage area reaches its maximum value,which is 32%higher than that of a single-source explosion.In addition,at a scattering distance of 65 m,the maximum value of 0.03 MPa damage area improves 37%over a single-source explosion.After the damage range for each threshold overpressure reaches its maximum,the value slowly decreases as the scattering distance increases.Eventually,the damage area tends towards a certain value,which is still higher than that obtained under single-source explosion conditions.

Fig.11(b)and(c)demonstrate the damage range curves for fourfold-source and sixfold-source explosions.The curves increase first and then slightly decrease as the scattering distance increases,which is similar to the double-source condition.Due to severe overlapping of the overpressure fields produced by different sources,the damage areas of multi-source for 0.05 MPa and 0.03 MPa are smaller than those for a single-source at a scope of scattering distances from 20 m to 25 m.When the scattering distance is as high as 30 m,the damage ranges exceed that of the single-source condition.For a fourfold-source cloud explosion,the optimal source scattering distance for the 0.1 MPa,0.05 MPa and 0.03 MPa threshold is 35 m,45 m,and 60 m,respectively.At the optimal distances,the improvement in damage area can reach to 60%,64%and 80%over the single-source condition.In the case of sixfoldsource cloud explosions,the maximum damage area improvement can reach to 81%,91%and 113%,under the optimal scattering distance of 40 m,55 m,and 70 m,respectively.

In summary,the damage ranges of multi-sources FAE explosion are considerably affected by source scattering distance.When the scattering distance is relatively small,the damage range of multisources clouds are lack of advantage compared with single-source cloud.However,within a suitable source scattering distance,the damage ranges of the multi-sources reach the maximum,and is obviously greater than that of the single-source condition.It is worth mentioning that for different overpressure thresholds,different scattering distances are required for the damage range to reach its maximum value.

Fig.11.Damage ranges under various source scattering distances.(a)Double-source cloud.(b)Fourfold-source cloud.(c)Sixfold-source cloud.

3.5.Damage range enhancement under various source numbers

In order to study the effect of source number,a ratio is defined,which re flects the enhancement of damage ranges.The ratio is fitted by using allometric functions and is displayed in Fig.12.The fitting functions are shown as follows:

Fig.12.Damage range ratio of multi-sources to single-source under various source numbers.

For a 0.1 MPa threshold:

For a 0.03 MPa threshold:

where,Smulti-sourcesis the maximum damage range of multi-sources conditions;Ssingle-sourceis the damage range of single-source condition;Δprepresents the overpressure;Nrepresents the source number,andN>1;

With the same source number,the damage range enhancement of 0.03 MPa threshold is the largest,that of 0.05 MPa is the second largest,and that of 0.1 MPa is the smallest.This trend is due to shockwave interaction,which more easily exceeds the lower overpressure threshold.So the contribution of shock focusing to the damage range is the 0.03 MPa threshold,0.05 MPa threshold,and 0.1 MPa threshold,in descending order.As the source number increases,the ratio of each threshold is greater than 1.0,and this value gradually increases.The highest growth rate is obtained for the 0.03 MPa threshold,followed by the 0.05 MPa threshold,and then the 0.1 MPa threshold.It implies a growing trend of the contribution of the shockwave focusing effect as source number increases.For a double-source FAE,the average damage range enhance to 1.3 times compared with a single-source FAE.In case of the fourfoldsource and sixfold-source conditions,the improvement of average damage range is to 1.7 times and 1.9 times,respectively.

The power of fuel-air explosives is generally estimated by TNT equivalency values(MTNT/Mfuel)[20].Here,Mfuelis the weight of fuel.AndMTNTrepresents the weight of TNT required for getting equivalent blast intensity at the same scaled distance.Considering the shockwave reflection on the actual ground,the overpressure from a TNT charge[8]can be expressed by:

where,Δprepresents the overpressure;is the scaled distance,which is defined as.

The damage radii under the three thresholds(0.1 MPa,0.05 MPa,0.03 MPa)are adopted to calculate the TNT equivalent for multisources FAE.The values are shown in Table 6 and compared with other single-source FAE from references.In case of single-source FAE,the TNT equivalent from this work is between 2.65 and 3.91,which is similar to other research.However,the values increase with the FAE source numbers.When the source number reaches to fourfold-source,it shows a clear advantage on the TNT equivalency value.

These results indicate that,by using multiple FAE sources method,the damage ranges increase without changing the weight or type of the fuel.For personnel shockwave protection,the injury distance should consider not only the weight and type of the fuel but also the FAE source numbers.

4.Conclusions

In this paper,the shockwave overpressure fields of multisources FAE explosions by using numerical simulation are investigated and the research demonstrates the following:

(1)Shockwaves from the different sources interact with each other and produce shockwave focusing,which increases the pressure in some regions.Regions of signi ficant shockwave focusing phenomenon appear between the two adjacent FAE sources.At the field center,shockwaves converge together to form a central high-pressure zone.

(2)The overpressure fields of the multi-sources FAE are quite different compared with the single-source condition.The overpressure fields for multi-sources explosions change signi ficantly versus source scattering distances.For a certain overpressure threshold,there exists an optimum source scattering distance such that the damage range reaches its maximum value.The optimum scattering distance for the 0.1 MPa,0.05 MPa,and 0.03 MPa threshold is 35 m,50 m,and 65 m,respectively,for a double-source FAE.In case of a fourfold-source FAE,the optimum distances are 35 m,45 m,and 60 m.And for a sixfold-source FAE,the optimum distances change to 40 m,55 m,and 70 m.

(3)The damage ranges improve with the increasing of source number.The average enhancement of damage range is 1.3 times,1.7 times and 1.9 times for double-source,fourfoldsource,and sixfold-source FAE cloud explosion,respectively.

(4)The TNT equivalency values enhance with the increasing of source number.When source number exceeds 4,the TNT equivalent of multi-sources FAE shows obvious advantage.The method of scattering single FAE source to multiple FAE sources can effectively improve the damage ability.

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.

Acknowledgements

The authors would like to acknowledge the China Postdoctoral Science Foundation(Grant No.2019M660488)to provide fund for this work.