Investigating Different Grounds Effects on Shock Wave Propagation

applied sciences

Article Investigating Diﬀerent Grounds Eﬀects on Shock Wave Propagation Resulting from Near-Ground Explosion

Yan Wang *, Hua Wang, Cunyan Cui and Beilei Zhao

Space Engineering University, Beijing 101416, China * Correspondence: [email protected]; Tel.: +86-010-66365330

Received: 19 July 2019; Accepted: 28 August 2019; Published: 3 September 2019

Abstract: A massive explosion of a liquid-propellant rocket in the course of an accident can lead to a truly catastrophic event, which would threaten the safety of personnel and facilities around the launch site. In order to study the propagation of near-ground shock wave and quantify the enhancement effect on the overpressure, models with different grounds have been established based on an explicit nonlinear dynamic ANSYS/LS-DYNA 970 program. Results show that the existence of the ground will change the propagation law and conform to the reflection law of the shock wave. Rigid ground absorbs no energy and reflects all of it, while concrete ground absorbs and reflects some of the energy, respectively. Ground may influence the pressure-time curve of the shock wave. When the gauge is close to the explosive, the pressure-time curve presents a bimodal feature, while when the gauge reaches a certain distance to the explosive, it presents a single-peak feature. For gauges at different heights, different grounds may have different effects on the peak overpressure. For gauges of height not greater than 4 m, the impact on the shock wave is obvious when the radial to the explosive is small. On the contrary, as for the gauges of height greater than 4 m, the impact on the shock wave is obvious when the radial to the explosive is big. Ground has the enhancement effect on peak overpressure, but different grounds have different ways. For rigid ground, the peak overpressure factor is about 2. However, for the concrete and soil ground, peak overpressure factor is from 1.43 to 2.1.

Keywords: near-ground explosion; shock wave; peak overpressure; LS-DYNA; peak overpressure factor

1. Introduction The launch vehicle is a complex and large-scale technology-intensive system and it plays an important role in the deep space exploration mission. Once an explosion occurs, it would threaten the safety of personnel and facilities around the launch site. Owing to its very low boiling point, liquid propellant is extremely hazardous and its leakage/sudden release from a pressurized tank would lead to vapor cloud formation resulting in subsequent fire and explosion. This point was amply demonstrated by a series of explosion accidents over decades [1]. The study of the rocket shock environment is an ongoing effort to characterize the environment resulting from catastrophic rocket explosion. The purpose is to develop the data and information required to allow launch vehicle designers to develop safer launch systems. Many workers have studied liquid propellant explosion characteristics by conducting experiments and analyzing experimental data [2–9]. The liquid propellant explosion test has high risk and high cost, and the experimental conditions are strict and the repeatability is poor, which makes it difficult. With the development of computer simulation technology, numerical simulation has become the main means to solve such problems [10–17].

Appl. Sci. 2019, 9, 3639; doi:10.3390/app9173639 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 3639 2 of 17

Explosion hazard modeling is a challenging problem since it involves several physical, often complex, interacting aspects. In combustion explosions, the available quantity of energy changes rapidly and is constantly redistributed among heat, the kinetic and chemical energy forms. The problem becomes even more complex when the subsequent shock wave propagation is assumed to interact with the ground, which has a more destructive effect on the personnel and facilities at the launch site. Ground and shock waves interact closely since the shock wave impulses affect the ground, while the ground in turn modifies the shock wave’s propagation patterns [18]. Therefore, it is of great significance to investigate different grounds effects on shock wave propagation resulting from near-ground explosion. It is difficult to deduce and calculate the shock wave parameters in theory, so most of the current research adopts the empirical formula based on a large number of tests for prediction. However, due to the limitation of experimental conditions, the results calculated by different empirical formulas often differ greatly, which will bring hidden dangers to the safety analysis of the launch site [19]. However, the explosion test has the characteristics of short time, poor repeatability, and high cost, so it is not convenient for research. Therefore, research on the propagation law of shock wave in complex environment has been investigated by many researchers [20–28]. When a rocket explodes on the launch pad, the shock wave does not travel through the infinite air. The ground has certain influence on the propagation of shock wave. The effect of the ground on the shock wave is reflected in two aspects. On the one hand, the ground will reflect and enhance the shock wave. On the other hand, the earth may absorb some energy of the shock wave [29]. Common types of launching site ground include concrete ground and soil ground. Concrete ground reflects more energy of the shock wave, while soil absorbs more energy of the shock wave and reflects less energy due to its looseness. In order to study the influence of different ground properties on the propagation law of shock wave, this paper considers two extreme cases, one is air material, the other is rigid ground. The air material simulates the shock wave propagation in free space without any obstruction. Rigid ground, on the other hand, absorbs no energy and reflects it all. In this paper, in order to simulate the propagation of near-ground shock wave and quantify the enhancement effect on peak overpressure, an explicit nonlinear dynamic ANSYS/LS-DYNA Version 970 (Livermore Software Technology Corporation, Livermore, CA, America) program is employed to setup near-ground explosion models with different grounds. The enhancement effect on peak overpressures of different grounds was carried out, which will provide a reliable reference for design and safety evaluation of the launch site.

2. Deﬁnition of Explosive Yield TNT equivalent model is to convert the explosive materials into equivalent TNT according to the equal energy, and then the explosive law of TNT is applied to predict the effect of liquid propellant explosion [30]. It is widely used in the field of explosive characteristics of liquid propellant [31]. Given that the mass of liquid propellant M0 and explosive yield Y, the TNT MT can be calculated by Equation (1).

MT = Y M (1) · 0 where, MT represents mass equivalent to TNT, kg. Y is explosive yield, dimensionless. M0 is mass of liquid propellant, kg. The object of this paper is a two-stage rocket with four boosters, whose propellant quality of each stage is shown in Table1. It is very di fficult to obtain an accurate explosive yield theoretically under different explosion modes, because the chemical reaction mechanism of liquid propellant is different from that of solid explosive, and is affected by various factors such as the type, mass, detonation time of the propellant, falling speed, and ground property, etc. Through statistical analysis of test data [2–4], the explosive yield of liquid propellant is estimated. This method has certain authority and has been widely used abroad. Explosive yield estimation is mainly calculated by two methods, that is, Appl. Sci. 2019, 9, 3639 3 of 17 the table-lookup-method and chart-reading-method. Therefore, explosive yield under the two methods is calculated respectively and it is finally determined according to the minimum principle [32]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 17 Table 1. Equivalent yields at different stages of a certain rocket. by two methods, that is, the table-lookup-method and chart-reading-method. Therefore, explosive yield under theItems two methods is calculated Boosters respectively and it Stage-1is finally determined according Stage-2 to the minimum principle [32]. LOX RP-1 LOX LH2 LOX LH2 The Propellanttable-lookup mass- (kg)method is 4to 104,000obtain the4 40,000explosive 133,300 yield by reading 24,360 Table 22,200 2, on the 3800 basis of × × determiningTotal the mass propellant (kg) type and the 576,000 explosion mode. W 157,660hile as for the chart-reading 26,000 -method, explosive powerTable-lookup-method can be obtained from Figure 0.1 1 according to propellant 0.6 mass, and the final 0.6 explosive Y Chart-reading-method 0.0576 125% = 0.072 0.0644 370% = 0.238 0.0713 370% = 0.264 × × × yield is obtained byChosen multiplying the specific 0.072 coefficients corresponding 0.238 to different 0.264propellants in Table 2. The explosive yields are shown in Table 1. The table-lookup-method is to obtain the explosive yield by reading Table2, on the basis of Table 1. Equivalent yields at different stages of a certain rocket. determining the propellant type and the explosion mode. While as for the chart-reading-method, explosive powerItems can be obtained from FigureBoosters1 according to propellantStage mass,-1 and the finalStage explosive-2 yield is obtained by multiplying the specificLOX coefficientsRP-1 correspondingLOX LH to2 differentLOX propellants LH2 in Propellant mass (kg) 4 × 104,000 4 × 40,000 133,300 24,360 22,200 3800 Table2. The explosiveTotal mass (kg) yields are shown in Table5761,000. 157,660 26,000 Table-lookup-method 0.1 0.6 0.6 YTable 2. ExplosiveChart-reading yields-method and specific coe0.0576fficients × 125% of = 0.072 different liquid0.0644 × propellant 370% = 0.238 combinations 0.0713 × 370% under = 0.264 different explosionChosen modes. 0.072 0.238 0.264

Explosion Mode TablePropellant 2. Explosive yields and specific coefficients of different liquid propellantMix combinationsSpeciﬁc under Combination Outside Launch Proportion Coeﬃcient different explosion modes.On Launch Pad Pad Propellant Explosion Mode Mix Specific LOX/LH2 60% 60% 1:5 370% Combination On Launch Pad Outside Launch Pad Proportion Coefficient Within 226.7995t is 20% plus 10% LOX/LOXLH/2RP-1 60% 10%60% 1:1.251:5 125%370% for rest part LOX/RP-1 Within 226.7995t is 20% plus 10% for rest part 10% 1:1.25 125% N2O4/UDMH 10% 5% 1:2 240% N2O4/UDMH 10% 5% 1:2 240%

1.0

0.8

0.6

Y 0.4

0.2

0.0

100 101 102 103 104 105 106 107 M [lb] 0 Figure 1. The curve between the explosive yields and the mass of propellant.

Combined explosive yields and combined explosive center coordinate coordinatess are acquire acquiredd by Eq Equationuation (2)(2) and (3).(3). TheThe height of the combined explosive center is 21.968 m. Given the height of the launch pad, the height of the true explosiveexplosive centercenter isis 30.66830.668 m.m. X X yzhyzh = mi y im/ i yi m/ i mi (2)(2) where, yzh is the combined explosive yield, yi is the explosive yield of stage i, mi is total propellant where, yzh is the combined explosive yield, yi is the explosive yield of stage i, mi is total propellant mass of stage i, whosewhose unitunit isis kg. X X x = m x / m (3) xzz  m i x i/ i i m i i (3) wwhere,here, xi is thethe distancedistance between between the the middle middle of of tanks tanks of of stage stagei and i and the th explosivee explosive center, center, whose whose unit isunit m. is m.

Appl. Sci. 2019, 9, 3639 4 of 17

3. Finite Element Model of Near-Ground Explosion

3.1. Governing Equations and Computation Code In the continuum mechanics, the momentum equation is

.. σij,j + ρ fi = ρxi (4) where, σij,j is one order partial derivative of Cauchy stress component σij with respect to the coordinate .. variable in the jth direction, fi is the volume force vector of the unit mass, xi is the acceleration vector, and ρ is the density of air. The conversation equation of mass is

ρ = Jρ0 (5)

= ∂xi where, ρ0 is the initial density of air, J ∂xj , xi is the space coordinate in the ith direction, xj is the matter coordinate in the jth direction, and ∂xi/∂xj is the strain gradient with respect to xj. The energy equation is . . . E = VSijεij (p + q)V (6) − . where, V is volume, εij is strain rate tensor, and q is artificial viscidity. Deviatoric stress tensor is given by

Sij = σij +(p + q)εij (7) where, εij is Kronecher symbol and p is hydrodynamic pressure, given by

1 p = σkk q. (8) −3 − ANSYS/LS-DYNA 970 is a finite element computation code for dynamics suitable for impact, explosion, and other instantaneous problems, and is used to carry out the numerical simulation process. In LS-DYNA, there are three classes of methods available for generating the grids and mesh, including Eulerian, Lagrangian, and arbitrary Lagrangian–Eulerian (ALE) method. In this study, the ALE method is employed to simulate the process of the TNT explosion, the propagation of air shock waves, and the interaction of explosion shock waves with the grounds. The ALE algorithm consists of a classical Lagrangian step in which the mesh moves along with the modeled material, a rezone step in which the mesh is modified to preserve good quality through the computation of the Eulerain time step, and a remapping step in which the solution is conservatively transferred from the old mesh to the new rezoned one, and thereby performs the fluid–solid coupling transient analysis [33].

3.2. Modeling Geometry and Meshing In order to investigate the impact of diﬀerent grounds on the shock wave propagation law when the rocket exploded on the launch pad, the whole rocket is selected to carry out numerical simulation. Numerical models of four diﬀerent materials are chosen, including concrete ground, soil ground, rigid ground, and air. Model 1 to Model 3 are the TNT that explodes above the rigid ground, the concrete ground and the soil ground, respectively, while Model 4 is the TNT that explodes in the air. In Model 1, air domain is a Φ 258 m 40 m cylinder and the ground is rigid constraint. In Model 2 and Model × 3, air domain which is a Φ 258 m 50 m cylinder and ground domain which is a Φ 258 m 10 m × × cylinder are established. Model 4 only includes air domain which is a Φ 258 m 50 m cylinder. After × considering the symmetry of the simulation model, a 1/4 symmetrical geometrical model is adopted. In order to reduce the calculation amount, the air domain is divided into three parts according to the radial, which are 0~30 m, 30~70 m, 70~129 m. The mesh size of the air domain is 0.58 m, 1.16 m, and 2.32 m, respectively. The TNT and the ground are divided into mesh sizes of 0.20 m and 2.32 m. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 17

and 2.32 m. The height of the detonation center to the ground surface is 30.668 m. The finite element model and gauges distribution are shown in Figure 2. It is assumed that the radius of the TNT cylinder is r0 (m) and the height is h0 (m). Therefore, the length-diameter ratio of TNT is h0/2r0. Given the mass of TNT, r0, and h0 of TNT cylinder can be calculated by using Equation (10) which is derived from Equation (9). Unit system of cm-g-ms is adopted in the numerical model. The explosive source is located at the center of mass and a point- detonation-method is used [34].

MVT   2 V  r00 h (9)  hr00/ 2 1

M r  3 T (10) 2

Eight-node element of Solid 164 is adopted for the 3D explicit analysis. In order to prevent the element distortion in large deformation and nonlinear structural analyses, ALE algorism is used. TNT and air are modeled with ALE multi-material grids, but different grounds with Lagrangian grids. Fluid–solid coupling algorithm is used between two grids [35]. Appl. Sci. 2019Furthermore,, 9, 3639 the transitional displacement of the nodes normal to the symmetry planes is5 of 17 constrained, in order to simulate the propagation effect of shock wave in the symmetric plane. Non- reflecting boundary condition is applied to top and lateral surfaces, which allows the fluid medium TheAppl.to height flow Sci. out2019 ofthe ,in 9, orderxdetonation FOR PEERto simulate REVIEW center the toeffect the of ground infinite surface space [36 is]. 30.668 The displacement m. The ﬁnite constraint element is applied model5 of 17 and gaugesin the distribution z direction areof the shown ground in, Figurewhile r2igid. constraint is applied on the rigid ground. and 2.32 m. The height of the detonation center to the ground surface is 30.668 m. The finite element model and gauges distribution are shown in Figure 2. It is assumed that the radius of the TNT cylinder is r0 (m) and the height is h0 (m). Therefore, the Air length-diameter ratio of TNT is h0/2r0. Given the mass of TNT, r0, and h0 of TNT cylinder can be calculatedTNT by using Equation (10) which is derived from Equation (9). Unit system of cm-g-ms is adopted in the numerical model. The explosive source is located at the center of mass and a point- detonation-method is used [34]. Groud

MVT   2 V  r00 h (9)  hr00/ 2 1 (a) (b)

Figure 2. Physical model and finite element modelM of near-ground explosion. (a) Physical model; (b) Figure 2. Physical model and ﬁnite elementr  model3 T of near-ground explosion. (a) Physical model;(10) (b) FiniteFinite element element model. model. 2

EightAs are-node shown element in Figure of Solid 3, gauge 164 sis in adopted the model for are the set 3D at explicit five different analysis. heights, In order which to prevent are from the 2 It is assumed that the radius of the TNT cylinder is r0 (m) and the height is h0 (m). Therefore, elementm to 10 mdistortion with an ininterval large deformation of 2 m. Twelve and gauge nonlinears are structural set at each analyses, height, whoseALE algorism x is from is used0 m to. TNT 110 the length-diameter ratio of TNT is h0/2r0. Given the mass of TNT, r0, and h0 of TNT cylinder can andm, with air arean intervalmodeled of with 10 m. ALE multi-material grids, but different grounds with Lagrangian grids. be calculatedFluid–solid by coupling using Equationalgorithm is (10) used which between is derived two grids from [35] Equation. (9). Unit system of cm-g-ms is adoptedFurthermore, in the numerical the transitional model. displacement The explosive of the source nodes is normal located to at the the symmetry center of planes mass and is a point-detonation-methodconstrained, in order to issimulate used [34 the]. propagation effect of shock wave in the symmetric plane. Non- reflecting boundary condition is applied to top and lateral surfaces, which allows the fluid medium  to flow out in order to simulate the effect of infiniteMT = spaceρ V [36]. The displacement constraint is applied  2· in the z direction ofE1 the ground, while rigid constraintV = πr his0 applied on the rigidE12 ground. (9)  0 E1  h0/2r0 = 1

A1 s 3 MT A1 Air r = A12 (10) 2πρ TNT Eight-node element of Solid 164 is adopted for the 3D explicit analysis. In order to prevent the element distortion in large deformation and nonlinear structural analyses, ALE algorism is used. TNT and air Groud are modeled with ALE multi-material grids, but different grounds with Lagrangian grids. Fluid–solid coupling algorithm is used between two grids [35]. Furthermore, the transitional displacement of the nodes normal to the symmetry planes is constrained, in order to simulate(a) the propagation effect of shock wave(b) in the symmetric plane. Non-reflecting boundary condition is applied to top and lateral surfaces, which allows the fluid Figure 2. Physical model and finite element model of near-ground explosion. (a) Physical model; (b) medium to flow out in order to simulate the effect of infinite space [36]. The displacement constraint is Finite element model. applied in the z direction of the ground, while rigid constraint is applied on the rigid ground. As areAs are shown shown in Figurein Figure3, gauges3, gauge ins i then the model model are are set set at at five five di differentfferent heights, which which are are from from 2 2 m to 10m m to with 10 man with interval an interval of 2 m.of 2 Twelve m. Twelve gauges gauge ares are set set at at each each height, height, whose whose xx isis fromfrom 0 0 m m to to 110 110 m, withm, an with interval an interval of 10 m.of 10 m.

E1 E12 E1

A1 A12

Figure 3. Distribution of gauges at diﬀerent heights. Appl. Sci. 2019, 9, 3639 6 of 17

3.3. Material Models and Equation of State Air is commonly modeled by *MAT_NULL with a linear polynomial equation of state, EOS_LNIEAR_POLYNOMIAL. *MAT_NULL is material type 9 in LS-DYNA 970 code. This material allows equations of state to be considered without computing deviatoric stresses. Optionally, a viscosity can be defined. Also, erosion in tension and compression is possible. Air and water usually use this model. This model avoids the calculation of stress and strain. EOS_LNIEAR_POLYNOMIA defines the pressure by 2 3 2 P = C0 + C1µ + C2µ + C3µ + C4 + C5µ + C6µ E0 (11) where, the parameter µ is defined as µ =ρ/ρ0, ρ is the current density, and ρ0 is a nominal or reference density; C0–C6 are the equation coefficients; and the parameter E0 is the initial internal energy of reference specific volume per unit. The parameters used are shown in Table3[37,38].

Table 3. Parameters of the air.

3 3 ρ (g/cm ) C0 C1 C2 C3 C4 C5 C6 E0 (J/m ) V0 1.29 1.0 10 6 0 0 0 0.4 0.4 0 25 1.0 − × −

TNT is modeled by the high explosive material model, MAT-HIGH-EXPLOSLVE-BURN, it allows the modeling of the detonation of a high explosive. In addition, an equation of state must be defined. Burn fractions, F, which multiply the equations of states for high explosives, control the release of chemical energy for simulating detonations. The Jones–Wilkins–Lee (JWL) equation of state defines pressure by ! ω R V ω R V ωE P = A 1 e− 1 + B 1 e− 2 + (12) − R1V − R2V V where, A, B, R1, R2, $ are the equation coefficients and V is the initial relative volume. Parameters selected are shown in Table4[39].

Table 4. Parameters of TNT charge.

3 3 ρ (g/cm ) vD (km/s) PCJ (GPa) A (GPa) B (GPa) R1 R2 $ E0 (J/m ) V0 1.63 6.93 27 371 7.43 4.15 0.95 0.3 7 103 1.0 ×

The concrete ground is modeled using MAT_PLASTIC_KINEMATIC [40]. This model can describe the isotropic hardening and follow-up hardening plastic models, and can also consider the influence of strain rate. It is applicable to beam, shell, and solid units, and the calculation efficiency is very high. In the plastic kinematic material model, the mechanical properties are characterized by the material yield strength (σ0), the Young’s modulus (E), and the tangent modulus (Et: The slope of the stress–strain curve in the plastic region). The yielding is defined by von Mises yield criterion. In the plastic kinematic plasticity algorithm, the flow stress (σ) is given as

 . ! 1   ε P  e f f σy = 1 +  σ + βE ε (13)  C  0 P p

. where, ε is the strain rate, C and P are the strain rate parameters, β is the hardening coeﬃcient and e f f εp and EP are the eﬀective plastic strain and plastic hardening modulus, respectively. The plastic hardening modulus is calculated by using the following relation

EEt EP = (14) E Et − The parameters of the concrete used are shown in Table5. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 17

EEt EP  (14) EE t

The parameters of the concrete used are shown in Table 5.

Table 5. Parameters of the concrete [41].

ρ/(g/cm3) E/GPa μ σ0/GPa EP/GPa β C P FS VP 2.65 4.0e × 10−1 0.3 1.0e × 10−3 4.0e × 100 0.50 0 0 0.8 0.8

The soil ground is modeled using MAT_SOIL_AND_FOAM [42]. Due to its simplicity and added flexibility, this model has a shear failure surface that is pressure dependent, which is a basic property of geo-material. It is more fluid-like under many conditions, which is an idea for soft soil. Appl. Sci. 2019, 9, 3639 7 of 17 The pressure dependent shear strength envelope is written in terms of a quadratic in pressure by the equation: Table 5. Parameters of the concrete [41].

3 ρ/(g/cm ) E/GPa µ σ0/GPa EP/GPa β CP FS VP 4.0e  J21.0e () a 0 4.0e a 1 p  a 2 p (15) 2.65 1× 0.3 3× 0× 0.50 0 0 0.8 0.8 10− 10− 10 where, J2=SijSji/2,The Sij soiland ground Sji are is modeled the stress using MAT_SOIL_AND_FOAMtensors, p is the mean [42]. Duepressure, to its simplicity a0, a1, and and added a2are constants which are determinedﬂexibility, this from model hasthe a triaxial shear failure compression surface that is pressure tests. dependent,The parameters which is a basic of the property soil of are shown in geo-material. It is more ﬂuid-like under many conditions, which is an idea for soft soil. The pressure Table 6. dependent shear strength envelope is written in terms of a quadratic in pressure by the equation:

φ = J (a + a p + a p) (15) Table 6. Parameters2 − 0 of1 the2 soil [43].

P P 3 where, J2 = SijSji/2, Sij and Sji are the stress tensors, p is the mean pressure, a0, a1, and a2 are constants ρ(g/cm ) G/GPa B a0/GPa a1/GPa a2/GPa  e2  e3 which are determined from the triaxial compression tests. The parameters of the soil are shown in 1.8 Table1.606 ×. 10−6 1.382 3.3 × 10−5 1.31 × 10−9 1.23 × 10−3 0.05 × 10−2 9 × 10−4 P P P P P P P  e4  e5  e6 Table 6.e7Parameters of the soile8 [43].  e9  e10 P2/GPa −3 −3 −3 −3 −3 −3 −3 −4 1.1 × 10 1.5 × 103 1.9 × 10 2.1 × 10 2.2×10 2.5×10 P 3.0 × P10 3.42 × 10 ρ (g/cm ) G/GPa B a0/GPa a1/GPa a2/GPa εe2 εe3 P3/GPa P4/GPa P5/GPa6 P6/GPa 5 P7/GPa9 P38/GPa 2 P9/GPa4 P10/GPa 1.8 1.60 10− 1.382 3.3 10− 1.31 10− 1.23 10− 0.05 10− 9 10− −4 −4 × −4 × −4 × −4 × −×4 × −4 −4 4.53 × 10 6.76 P× 10 1.27P × 10 P 2.08 × 10P 2.71P × 10 P 3.92 × 10 P 5.66 × 10 1.23 × 10 εe4 εe5 εe6 εe7 εe8 εe9 εe10 P2/GPa 1.1 10 3 1.5 10 3 1.9 10 3 2.1 10 3 2.2 10 3 2.5 10 3 3.0 10 3 3.42 10 4 × − × − × − × − × − × − × − × − 4. Propagation PLaw3/GPa of NearP4/GPa-GroundP5/GPa ExplosionP6/GPa ShockP7/GPa Wave P8/GPa P9/GPa P10/GPa 4.53 10 4 6.76 10 4 1.27 10 4 2.08 10 4 2.71 10 4 3.92 10 4 5.66 10 4 1.23 10 4 × − × − × − × − × − × − × − × − 4.1. Verification of Blast Model 4. Propagation Law of Near-Ground Explosion Shock Wave Figure 4 shows the empirical data and numerical data about the peak overpressure along x at 4.1. Verification of Blast Model the height of 2 m. As can be seen, the prediction values of each empirical formula in the near-field of the explosion areFigure quite4 shows different, the empirical which data is andmainly numerical because data about the thedetonation peak overpressure products along inx at the near-field the height of 2 m. As can be seen, the prediction values of each empirical formula in the near-field of are quite differentthe explosion from are the quite test di ffvalueserent, which. In isthe mainly far-field because of the the detonation explosion, products the in influe the near-fieldnce of detonation products decreasesare quite and different the from predicted the test values. values In theof far-fieldeach e ofmpirical the explosion, formula the influence tend ofto detonation be consistent. When products decreases and the predicted values of each empirical formula tend to be consistent. When scaled distancescaled is small, distance different is small, diff empiricalerent empirical formulas formulas havehave a largea large deviation deviation because ofbecause the test error. of the test error. While when theWhile scaled when the distance scaled distance is big, is big,different different empirical empirical formulas formulas have a have large consistency a large consistency [44–46]. [44–46].

0.8 Sadovski [42] 0.7 Brode [43] Henrych [44] 0.6 Xinhua Chen [31] 0.5 Simulation 0.4 0.3

Pressure Pressure [MPa] 0.2 0.1 0.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 -1/3 Scaled distance [mkg ] Figure 4. ComparisoFigure 4. Comparisonn of theoretical of theoretical and and numerical numerical data data in airin explosion air explosion at the height at the of 2 m.height of 2 m. Appl. Sci. 2019, 9, 3639 8 of 17

By comparison, simulation data is almost the same as the empirical data, and the deviation is within the permissible scope of the project. Certain deviation between the simulation result and the empirical formula is mainly because that the empirical formula is obtained by ﬁtting according to the test results, while the simulation model is a relatively ideal result obtained by the calculation of the aerodynamic equation, and there are certain diﬀerences between the test conditions and the simulation models.

4.2. Reflection of Shock Wave above Ground When the explosive detonates at the height h above the ground, the air shock wave generated spreads in all directions in a spherical shape. When the shock wave reaches the ground, it will be reflected above it When the incident angle is approximately equal to zero, the front of the incident shock wave is parallel to the ground, so a positive reflection occurs. When the incident is greater than zero, an oblique reflection of the shock wave occurs. When the incident angle continues to increase to the limit, the Mach reflection occurs, and the incident wave and the reflected wave become Mach waves. Reference [47] presents the functional relationship between the limit incidence angle and the reciprocal of the proportional height measured by the test. According to law, the limit incidence angle of the models in this paper is about 40◦.

4.3. Propagation Law of the Shock Wave above Rigid Ground Visualization of pressure contours available during the post-processing stage allows for a better understanding of complex process of shock wave interaction with the ground. Figure5 shows the pressure contours at different moments, which reappears in the interaction process between the shock wave and the rigid ground [48]. Different images represent typical propagation moments of shock wave, and different pressure values are represented by different colors. A strong chemical reaction takes place after detonation, resulting in a sharp expansion and diffusion in all directions of the high temperature and high pressure explosive products, and shock wave propagates outward in the form of a spherical wave. As shown in Figure5a, the air shock wave encounters the rigid ground and a local high pressure zone near the ground is formed. ‘x’ is defined as the radial from the gauge to the explosive center. When x is equal to zero, the incident angle of the shock wave is equal to zero, and a positive reflection will occur. This is because the air particle at the shock wave front is hindered and its velocity is reduced, which is superimposed with the later wave to form the enhanced wave near the ground. As shown in Figure5b, the shock wave overlaps near the ground and a high pressure zone is formed. This is because when x is less than 44.116 m, the incident angle is less than the limit incident angle of 40◦, and regular reflection occurs. As shown in Figure5c, the area of high pressure gradually begins to rise off the ground. This is because when x is greater than 44.116 m, the shock wave incident angle is greater than the limit incident angle of 40◦, and Mach reflection will occur. As shown in Figure5d, the height of the Mach bar gradually increases. This is because the pressure of the shock wave decreases when x increases, and gradually approaches to the plane wave. Appl.Appl. Sci.Sci. 20192019,, 99,, 3639x FOR PEER REVIEW 99 of 1717 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 17

(a) t=18ms h φl=40° (a) t=18ms x=44.116m h φl=40° x=44.116m

(b) t=36ms

(d) t=130ms

Figure 5. Reflected pressure contours images of the shock wave explosion above the rigid FiguregroundFigure 5. 5.. RReflectedeflected pressurepressure contours contours images images of theof shockthe shock wave wave explosion explosion above the above rigid the ground. rigid ground. 5.5. ResultsResults andand AnalysisAnalysis 5. Results and Analysis 5.1.5.1. TheThe EEffectffect onon thethe PropagationPropagation ofof Near-GroundNear-Ground ShockShock WaveWave 5.1. TheAsAs Effect shownshown onin inthe FigureFigure Propagation6 6,, the the pressure pressureof Near-Ground contour contour Shock images images Wave of of the the shock shock wave wave above above the the concrete concrete ground ground are given. When t = 120 ms, the air shock wave will be stratified when it encounters the ground. are given.As shown When in Figuret = 120 ms6, the, the pressure air shock contour wave will images be stratified of the shock when wave it encounters above the theconcrete ground. ground Some Some energy of the shock wave will be transmitted into the ground, and some will be reflected, areenergy given. of When the t shock = 120 ms wave, the willair shock be transmitted wave will be into stratified the ground, when it encounters and some the will ground. be reflected, Some superimposing with the incident wave near the ground. When t = 255 ms, the shock wave under the energysuperimposing of the shockwith the wave incident will wave be transmitted near the ground. into the When ground, t = 255 and ms, the some shock will wave be reflected,under the ground precedes that in the air, this is because the shock wave travels much faster in the solid than in superimposingground precedes with that the in incident the air, wavethis is near because the ground.the shock When wave t =travels 255 ms, much the shockfaster wavein the unde solidr thanthe the gas. When t = 675 ms, a high pressure area appears near the ground due to Mach reflection. When groundin the gas.precedes When that t = 675in the ms, air, a highthis is pressure because area the shockappears wave near travels the ground much fasterdue to in Mach the solid reflection. than t = 1140 ms, the height of the Mach bar gradually rises, and the pressure near the ground is higher inWhen the gas. t = 1140When ms, t = the675 height ms, a highof the pressure Mach bar area gradu appearsally rises,near theand ground the pressure due to near Mach the reflection. ground is than other areas in the air. It indicates that the ground has an enhanced effect on the propagation of Whenhigher t = than 1140 other ms, the areas height in theof the air. Mach It indicates bar gradu thatally the rises, ground and the has pressure an enhanced near the effect ground on theis the shock wave, and the enhanced area is mainly concentrated near the ground. In the face of the higherpropagation than otherof the areas shock in wave, the air.and Itthe indicates enhanced that area the is groundmainly concentrated has an enhanced near the effect ground. on the In enhancement effect of the shock wave above different grounds, it is necessary to conduct in-depth propagationthe face of theof the enhancement shock wave, effect and the of the enhanced shock wavearea is above mainly different concentrated grounds, near it the is necessaryground. In to research through data analysis. theconduct face ofin - thedepth enhancement research through effect data of the analysis. shock wave above different grounds, it is necessary to conduct in-depth research through data analysis.

(a) t=120ms (b) t=255ms (a) t=120ms (b) t=255ms

(c) t=675ms (d) t=1140ms (c) t=675ms (d) t=1140ms FigureFigure 6. 6. PPressureressure contours contours images images of the shockshockwave wave above above the the concrete concrete ground. ground . Figure 6. Pressure contours images of the shock wave above the concrete ground. Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 17

5.2. The Effects on the Pressure-Time Curves of Shock Wave above Different Grounds Figures 7–10 respectively show the pressure-time curves of the shock wave of gauges A1 to A12 when the explosion occurs above different grounds. As shown in Figure 7, shock wave pressure of different gauges rises linearly and then decreases gradually with time. The pressure curve shows the single-peak characteristic, and the peak overpressure decreases gradually with the increase of x. This is mainly because the scaled distance to explosion center increases from A1 to A12. Different from the explosion pressure curves explosion in the air, the gauges in Figures 8–10 are divided into two groups according to the characteristics of the pressure-time curves. Group 1 is the gauges from A1 to A4, while Group 2 is the gauges from A5 to A12. The pressure-time curves of the two groups show different characteristics. The pressure-time curves in Group 1 have obvious bimodal characteristics. The first peak is the incident wave, and the second one is the reflected wave. As is shown in Figure 8, the smaller the x is, the higher reflected peak overpressure is. For gauges A1 and A2, the peak pressure of the reflected shock wave is larger than that of the incident one, while the result is opposite for gauges A3 and A4. This is mainly because the smaller x is, the smaller the incident angle is, and the higher the energyAppl. Sci.of2019 the, 9, 3639reflected shock wave will be. The pressure-time curve10 of 17 s in Group 2 show a single peak characteristic. This is because the Mach reflection is formed at this zone, and the peak 5.2. The Effects on the Pressure-Time Curves of Shock Wave above Different Grounds overpressure is the maximum pressure of the composite wave. As can be seen, the peak pressure of Figures7–10 respectively show the pressure-time curves of the shock wave of gauges A1 to A12 the gauge at the samewhen position the explosion is occurs higher above dithanfferent that grounds. in As the shown air. in FigureAs is7, shockshown wave in pressure Figure of 9, for gauges A1 different gauges rises linearly and then decreases gradually with time. The pressure curve shows the and A2, the bimodalsingle-peak characteristics characteristic, andof the peak pressure overpressure-time decreases curve graduallys are with not the increaseobvious. of x. This While for gauges A3 and A4, and the pressureis mainly because-time the curve scaled distances, have to explosionobvious center bimodal increases from characteristics. A1 to A12. Different fromThe peak pressure of the explosion pressure curves explosion in the air, the gauges in Figures8–10 are divided into two incident wave and groupsreflected according wave to the characteristicsis relatively of the pressure-timeclose, but curves. obviously Group 1 is thelower gauges than from A1 the to peak pressure of rigid ground. This A4,is whilemainly Group because 2 is the gauges when from A5 the to A12. inciden The pressure-timet angle curves is small, of the two more groups energy show is sent into the different characteristics. The pressure-time curves in Group 1 have obvious bimodal characteristics. ground, so the reflectedThe first peakwave is the is incident not wave,obvious. and the secondHowever, one is the reflectedwith the wave. increase As is shown inof Figure the8 ,incident angle, the the smaller the x is, the higher reflected peak overpressure is. For gauges A1 and A2, the peak pressure energy sent into theof the ground reflected shock by wave the is shock larger than wave that of decreases, the incident one, so while the the result reflected is opposite wave for is obvious. The pressure-time curvegaugess in A3G androup A4. This2 show is mainly a because single the peak smaller xcharacteristicis, the smaller the incident like angle the is, curves and the in Figure 8b, but higher the energy of the reflected shock wave will be. The pressure-time curves in Group 2 show a there is just a decreasesingle in peak the characteristic. overpressure This is because at the the gauge. Mach reflection The iscurve formed in at this Figure zone, and 10 the shows peak a similar pattern to that in Figure 9, overpressureexcept that is the there maximum are pressure some of the differences composite wave. in As canthe be pressure seen, the peak pressurevalues. of the gauge at the same position is higher than that in the air. As is shown in Figure9, for gauges A1 and A2, It can be seen thefrom bimodal Figure characteristicss 8–1 of0 the that, pressure-time the pressure curves are not curves obvious. Whileof gauges for gauges A3from and A4, A1 to A4 have two and the pressure-time curves, have obvious bimodal characteristics. The peak pressure of incident peaks. The first onewave is the and reflectedpeak wavepressure is relatively of close,the butincident obviously lowerwave, than and the peak the pressure second of rigid one ground. is the peak pressure of the shock wave reflectedThis is mainly becauseby the when ground. the incident However angle is small,, for more gauges energy is sent from into the A5 ground, to A12 so the, this phenomenon reflected wave is not obvious. However, with the increase of the incident angle, the energy sent into is not obvious. Onlythe ground one by obvious the shock wave peak decreases, pressure so the reflected appears, wave is obvious. indicating The pressure-time that curves Mach in reflection occurs. Therefore, for the shockGroup 2 wave show a single generated peak characteristic by the like theexplosion curves in Figure above8b, but therethe isground, just a decrease Mach in the reflection starts to overpressure at the gauge. The curve in Figure 10 shows a similar pattern to that in Figure9, except occur at the incidenthatt a therengle are of some about differences 40°, in thewhich pressure has values. nothing to do with the property of the ground.

1.8 A1 A7 1.6 A2 A8 1.4 A3 A9 1.2 A4 A10 1.0 A5 A11 0.8 A6 A12

[MPa] 0.6 P P 0.4 0.2 0.0 -0.2 0 50 100 150 200 250 300 t [ms] Figure 7. Pressure-time curves at 2 m gauges of shock wave in the air. Figure 7. Pressure-time curves at 2 m gauges of shock wave in the air. Appl. Sci. 2019, 9, 3639 11 of 17 Appl.Appl. Sci.Sci. 20192019,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 1111 ofof 1717 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 17

77 1.81.8 7 A1 1.8 A5A5 A9A9 6 A1 1.61.6 A5 A9 6 A2A1 1.6 A6A6 A10A10 6 A2 1.41.4 A6 A10 55 A3A2 1.4 A7A7 A11A11 5 A3 1.21.2 A7 A11 A3 1.2 A8A8 A12A12 44 A4A4 1.01.0 A8 A12 4 A4 1.0 33 0.80.8 0.8

[MPa] 3 [MPa] [MPa] 0.6 [MPa] 0.6

P 2

P P

[MPa] 2 P

[MPa] 0.6

P 2 0.4 P 0.4 11 0.4 1 0.20.2 0.2 00 0.00.0 0 0.0 -1-1 -0.2-0.2 -1 00 2020 4040 6060 8080 100100 -0.2-20-20 00 2020 4040 6060 8080 100100120120140140160160180180 0 20 40 60 80 100 -20 0 20 40 60 80 100120140160180 tt [ms][ms] tt [ms][ms] t [ms] t [ms] ((aa)) GaugesGauges ofof GGrouproup 11 ((bb)) GaugesGauges ofof GGrouproup 22 (a) Gauges of Group 1 (b) Gauges of Group 2 FigureFigure 88.. PPressureressure--ttimeime curvescurves atat 22 mm gaugesgauges ofof shockwaveshockwave aboveabove thethe rigidrigid ground.ground. FigureFigure 8 8.. PPressure-timeressure-time curves curves at 2 m gaugesgaugesof of shockwaveshockwave above above the the rigid rigid ground. ground.

2.0 1.61.6 2.0 1.6 A5 A9 2.0 A1A1 1.41.4 A5 A9 A1 A6A5 A10A9 A2A2 1.4 A6 A10 1.5 1.21.2 A6 A10 1.5 A3A2 1.2 A7A7 A11A11 1.5 A3 1.0 A7 A11 A4A3 1.0 A8A8 A12A12 A4 1.0 A8 A12 1.0 A4 0.80.8 1.0 0.8

1.0 0.60.6

[MPa] [MPa]

[MPa] 0.6

[MPa]

P [MPa]

P 0.4 P [MPa] 0.5 0.4

0.5 P P 0.5 0.4 0.20.2 0.2 0.00.0 0.00.0 0.0 0.0 -0.2-0.2 -20 0 20 40 60 80 100 120 140 -0.2 00 5050 100100 150150 200200 250250 300300 -20 0 20 40 60 80 100 120 140 0 50 100 150 200 250 300 -20 0 20 40 tt [ms]60[ms]80 100 120 140 t [ms] tt [ms][ms] t [ms] ((aa)) GaugesGauges ofof GGrouproup 11 ((bb)) GaugesGauges ofof GGrouproup 22 (a) Gauges of Group 1 (b) Gauges of Group 2 FigureFigure 99.. PPressureressure ttimeime curvescurves atat 22 mm gaugesgauges ofof shockshock wavewave aboveabove thethe concreteconcrete ground.ground. FigureFigure 9. 9. PPressureressure time time curves curves at 2 m gauges ofofshock shock wave wave above above the the concrete concrete ground. ground.

1.81.8 1.61.6 A1A1 1.6 1.61.8 A1 A5A5 A9A9 1.6 A2A2 1.41.4 A5 A9 1.41.6 A2 1.4 A6A6 A10A10 1.4 A3A3 1.21.2 A6 A10 1.4 1.2 A7A7 A11A11 1.21.2 A4A3 A7 A11 1.2 A4 1.01.0 A8A8 A12A12 1.01.0 A4 1.0 A8 A12 1.0 0.80.8 0.80.8 0.8 0.8

[MPa] 0.6 [MPa] 0.6 0.6

0.6 [MPa]

[MPa] P

[MPa] 0.6

0.6 P

P [MPa] P 0.40.4 0.40.4 0.4 P 0.4 0.20.2 0.20.2 0.2 0.2 0.00.0 0.0 0.0 0.0 -0.2-0.2 0.0 -0.2 -0.2-0.2 -20-20 00 2020 4040 6060 8080 100100 120120 140140 -0.2 -20 0 20 40 60 80 100 120 140 00 5050 100100 150150 200200 250250 300300 0 50 100 150 200 250 300 tt [ms][ms] t [ms] t [ms] t [ms] t [ms] ((aa)) GaugesGauges ofof GroupGroup 11 ((bb)) GaugesGauges ofof GGrouproup 22 (a) Gauges of Group 1 (b) Gauges of Group 2 FigureFigureFigure 1100 10... PPressureressurePressure-time--ttimeime curvescurves curves atat at 22 2 mm m gaugesgauges of of shockshock wavewavewave above aboveabove the thethe soil soilsoil ground. ground.ground. Figure 10. Pressure-time curves at 2 m gauges of shock wave above the soil ground. 5.3.5.3. TheTheIt can EffectsEffects be seen onon thethe from PeakPeak Figures OverpressureOverpressure8–10 that, ofof Shock theShock pressure WaveWave aboveabove curves DifferentDifferent of gauges GroundsGrounds from A1 to A4 have two peaks. 5.3. The Effects on the Peak Overpressure of Shock Wave above Different Grounds The first one is the peak pressure of the incident wave, and the second one is the peak pressure of the AsAs shownshown inin Figure Figure 1111aa,, the the peak peak overpressure overpressure curvescurves ofof gaugesgauges atat 22 mm heightheight for for different different shockAs wave shown reflected in Figure by the 11 ground.a, the peak However, overpressure for gauges curves from of A5gauges to A12, at 2 this m phenomenonheight for different is not groundsgrounds presentpresent differentdifferent rulesrules withwith xx.. ForFor rigidrigid ground,ground, whenwhen xx isis notnot greatergreater thanthan 5050 m,m, thethe peakpeak obvious.grounds Onlypresent one different obvious rules peak with pressure x. For appears, rigid ground, indicating when that x Machis not reflectiongreater than occurs. 50 m, Therefore, the peak overpressureoverpressure ofof thethe shockshock wavewave isis largerlarger thanthan thatthat ofof otherotherss.. However,However, whenwhen xx isis greatergreater thanthan 5050 m,m, foroverpressure the shock wave of the generated shock wave by theis larger explosion than above that of the other ground,s. However, Mach reflection when x is starts greater to occur than at50 the m, thethe peakpeak overpressureoverpressure graduallygradually approachesapproaches thatthat ofof thethe concreteconcrete groundground andand soilsoil ground.ground. ThisThis isis duedue incidentthe peak angle overpressure of about gradually 40 , which approaches has nothing that to doof the with concrete the property ground of and the ground.soil ground. This is due toto thethe factfact thatthat thethe rigidrigid groundground◦ doesdoes notnot absorbabsorb energyenergy ofof thethe shockshock wave,wave, butbut reflectsreflects thethe shockshock wavewave to the fact that the rigid ground does not absorb energy of the shock wave, but reflects the shock wave andand generatesgenerates regularregular andand irregularirregular reflectionsreflections wwithith thethe incidentincident waves,waves, enhancingenhancing thethe shockshock wave.wave. and generates regular and irregular reflections with the incident waves, enhancing the shock wave. Appl. Sci. 2019, 9, 3639 12 of 17

5.3. The Effects on the Peak Overpressure of Shock Wave above Different Grounds As shown in Figure 11a, the peak overpressure curves of gauges at 2 m height for different grounds present different rules with x. For rigid ground, when x is not greater than 50 m, the peak overpressure of the shock wave is larger than that of others. However, when x is greater than 50 m, the peak overpressure gradually approaches that of the concrete ground and soil ground. This is due to the fact that the rigid ground does not absorb energy of the shock wave, but reflects the shock wave and generates regular and irregular reflections with the incident waves, enhancing the shock wave. The enhancement effect is more obvious when x is smaller, and less when x increases. For the explosion above concrete ground and soil ground, when x is not greater than 10 m, the peak overpressure of shock wave is very close to that in the air. However, when x is greater than 10 m and less than 40 m, peak overpressure starts to be higher than that in the air, and gradually approaches to the peak overpressure of rigid ground as x increases. When x is greater than 40 m, the peak overpressures of the shock wave above different grounds gradually approach, but all of them are higher than the peak overpressure of explosion in the air. Figure 11b presents the peak overpressure curves of gauges at 4 m height. For rigid ground, when x is not greater than 70 m, the peak overpressure is higher than that of others. However, when x is greater than 70 m, the peak overpressure is gradually close to that of concrete and soil ground. For concrete and soil ground, when x is not greater than 20 m, the peak overpressure of shock wave is very close to that in the air. However, when x is greater than 20 m and less than 60 m, peak overpressure starts to be higher than that in the air, and gradually approaches to that of rigid ground. When x is greater than 60 m, the peak overpressures of the shock wave above different grounds gradually approach. Figure 11c presents the peak overpressure curves of gauges at 6 m height. For rigid ground, when x is not greater than 20 m, the peak overpressure of the shock wave above different grounds tends to be consistent. However, when x is greater than 20 m and not greater than 100 m, the peak overpressure is higher than that of concrete ground and soil ground. When x is greater than 100 m, the peak overpressure is gradually close to that of concrete ground and soil ground. For concrete ground and soil ground, when x is not greater than 30 m, the peak overpressure of the shock wave is very close to that in the air. However, when x is more than 30 m, peak overpressure starts to be higher than that in the air, and gradually approaches to the peak overpressure of rigid ground as x increases. Figure 11d presents the peak overpressure curves of gauges at 8 m height. For rigid ground, when x is not greater than 30 m, the peak overpressure of the shock wave above different grounds tends to be consistent. However, when x is greater than 30 m, the peak overpressure is higher than that of concrete ground and soil ground. For concrete ground and soil ground, when x is not greater than 30 m, the peak overpressure of the shock wave is very close to that in the air. However, when x is greater than 30 m, peak overpressure starts to be higher than that in the air, but is lower than that above the rigid ground. As for Figure 11e, the law of peak overpressure curves is generally similar to that in Figure 11d, except for the value of the peak overpressure. From what has been discussed above, when h is not greater than 4 m, the peak overpressure generated by the explosion above the rigid ground is higher when x is relatively small, and the enhancement effect on the overpressure is no longer obvious with the increase of x. When x is small, the peak overpressure of concrete ground and soil ground is the same as that in the air. However, when x increases to a certain extent (10 m or 20 m), the enhancement effect on the shock wave gradually becomes prominent, which is consistent with that of rigid ground. As for h greater than 4m and less than 10 m, the peak overpressure curves of different grounds are the same when x is small, while the differentiation begins when x reaches a certain value. The peak overpressure of rigid ground is higher than that of concrete and soil ground, and the peak overpressure of concrete and soil ground is higher than that of the air. This is mainly because when x is relatively small, the incident wave plays the main role, while when x is relatively big, it is the superimposed Mach wave that plays a major role. Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 17

The enhancement effect is more obvious when x is smaller, and less when x increases. For the explosion above concrete ground and soil ground, when x is not greater than 10 m, the peak overpressure of shock wave is very close to that in the air. However, when x is greater than 10m and less than 40 m, peak overpressure starts to be higher than that in the air, and gradually approaches to the peak overpressure of rigid ground as x increases. When x is greater than 40 m, the peak overpressures of the shock wave above different grounds gradually approach, but all of them are higher than the peak overpressure of explosion in the air. Figure 11b presents the peak overpressure curves of gauges at 4 m height. For rigid ground, when x is not greater than 70 m, the peak overpressure is higher than that of others. However, when x is greater than 70 m, the peak overpressure is gradually close to that of concrete and soil ground. For concrete and soil ground, when x is not greater than 20 m, the peak overpressure of shock wave is very close to that in the air. However, when x is greater than 20 m and less than 60 m, peak overpressure starts to be higher than that in the air, and gradually approaches to that of rigid ground. When x is greater than 60 m, the peak overpressures of the shock wave above different grounds gradually approach. Figure 11c presents the peak overpressure curves of gauges at 6 m height. For rigid ground, when x is not greater than 20 m, the peak overpressure of the shock wave above different grounds tends to be consistent. However, when x is greater than 20 m and not greater than 100 m, the peak overpressure is higher than that of concrete ground and soil ground. When x is greater than 100 m, the peak overpressure is gradually close to that of concrete ground and soil ground. For concrete ground and soil ground, when x is not greater than 30 m, the peak overpressure of the shock wave is very close to that in the air. However, when x is more than 30 m, peak overpressure starts to be higher than that in the air, and gradually approaches to the peak overpressure of rigid ground as x increases. Figure 11d presents the peak overpressure curves of gauges at 8 m height. For rigid ground, when x is not greater than 30 m, the peak overpressure of the shock wave above different grounds tends to be consistent. However, when x is greater than 30 m, the peak overpressure is higher than that of concrete ground and soil ground. For concrete ground and soil ground, when x is not greater than 30 m, the peak overpressure of the shock wave is very close to that in the air. However, when x is greater than 30 m, peak overpressure starts to be higher than that in the air, but is lower than that aboveAppl. Sci. the2019 rigid, 9, 3639 ground. As for Figure 11e, the law of peak overpressure curves is generally similar13 of to 17 that in Figure 11d, except for the value of the peak overpressure.

7 5 6 air air rigid ground 4 rigid ground 5 concret ground concrete ground soil ground 4 3 soil ground

/MPa

/MPa m

m 2 P 2 P 1 1 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 x/m x/m Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 17 (a) Height of 2 m (b) Height of 4 m 5 5 air air rigid ground 4 rigid ground 4 concrete ground concrete ground soil ground 3 3 soil ground

/MPa 2

2 /MPa

P P 1 1

0 0

0 20 40 60 80 100 120 0 20 40 60 80 100 120 x/m x/m

4 air rigid ground 3 concrete ground soil ground

/MPa m P 1

0 20 40 60 80 100 120 x/m (e) Height of 10 m

Figure 11.11. Distribution ofof peakpeak overpressureoverpressure of of gauges gauges at at di differentﬀerent height height produced produced by by explosion explosion on ondiﬀ differenterent ground. ground.

5.4. TheFrom Peak what Overpressure has been Factordiscussed of Di ffabove,erent Ground when h is not greater than 4 m, the peak overpressure generated by the explosion above the rigid ground is higher when x is relatively small, and the It is assumed that the peak overpressure of gauges in Model 1 to Model 3 is Pground, and the peak enhancement effect on the overpressure is no longer obvious with the increase of x. When x is small, overpressure of gauge in Model 4 is Pair, then “peak overpressure factor” is defined in this paper, the peak overpressure of concrete ground and soil ground is the same as that in the air. However, which is α = Pground/Pair. It reflects the enhancement effect for peak overpressure of different grounds. whenAs x shownincreases in Figure to a certain 12a, for extent rigid (10 ground, m or when 20 m),x theis small, enhancementα at different effect heights on the varies shock greatly. wave Thegradually lower becomes the height, prominent the greater, which the αisis, consistent and the maximumwith that of reaches rigid ground. 3.8. With As thefor increaseh greater ofthanx, α 4mof anddifferent less than heights 10 graduallym, the peak approaches, overpressure and curves eventually of different approaches ground 2. Ass are shown the same in Figure when 12 b,c,x is forsmall, the whileconcrete the and differentiation soil ground, begins with the when increase x reaches of height, a certainα increases value successively. The peak overpressure from 1, the lower of rigid the ground is higher than that of concrete and soil ground, and the peak overpressure of concrete and soil ground is higher than that of the air. This is mainly because when x is relatively small, the incident wave plays the main role, while when x is relatively big, it is the superimposed Mach wave that plays a major role.

5.4. The Peak Overpressure Factor of Different Ground

It is assumed that the peak overpressure of gauges in Model 1 to Model 3 is Pground, and the peak overpressure of gauge in Model 4 is Pair, then “peak overpressure factor” is defined in this paper, which is α = Pground/Pair. It reflects the enhancement effect for peak overpressure of different grounds. As shown in Figure 12a, for rigid ground, when x is small, α at different heights varies greatly. The lower the height, the greater theαis, and the maximum reaches 3.8. With the increase of x, α of different heights gradually approaches, and eventually approaches 2. As shown in Figure 12b, c, for the concrete and soil ground, with the increase of height, α increases successively from 1, the lower the height, the greater the rise will be, and then starts to stabilize at a specific value. The maximum αwith a height of 6 m is 2.1, and the minimum α with a height of 10 m is 1.43. Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 17

Appl.It Sci. can2019 be, 9 , 3639 seen from Figure 12a–c that, for rigid ground, the enhancement effect of peak14 of 17 overpressure with different height is roughly the same, and the peak overpressure factor is about 2. However, the concrete ground and soil ground have different effects on the peak overpressure factors atheight, different the heights. greater theAmong rise will them, be, peak and thenoverpressure starts to stabilize factor of at 6m a speciﬁc height value.is the largest, The maximum reachingα 2.1,with whilea height the factor of 6 m of is 10 2.1, m and height the minimumis the smallest,α with which a height is 1.43. of10 m is 1.43.

h=2m h=8m 4.0 4.0 h=2m h=8m h=4m h=10m h=4m h=10m 3.5 3.5 h=6m h=6m 3.0 3.0 2.5 2.5

 2.0  2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 x/m x/m (a) Rigid ground (b) Concrete ground 4.0 h=2m h=8m h=4m h=10m 3.5 h=6m 3.0 2.5  2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 x/m

FigureFigure 12. 12.TheThe peak peak over overpressurepressure factor factor of of the the explosion explosion above above different diﬀerent ground.

6. ConclusionIt can bes seen from Figure 12a–c that, for rigid ground, the enhancement effect of peak overpressure with different height is roughly the same, and the peak overpressure factor is about 2. However, A numerical method for simulating rocket explosion on the launch pad was developed in order the concrete ground and soil ground have different effects on the peak overpressure factors at different to study the propagation law of near-ground explosion and quantify the enhancement effect on peak heights. Among them, peak overpressure factor of 6m height is the largest, reaching 2.1, while the overpressure. From analysis and discussion of results, the following main conclusions were drawn: factor of 10 m height is the smallest, which is 1.43. (1) The existence of the ground will change the propagation law and conform to the reflection law6. Conclusionsof the shock wave. Rigid ground absorbs no energy and reflects all of it, while concrete ground absorbs and reflects some of energy, respectively. However, the shock waves are all enhanced at the near-groundA numerical of different method materials. for simulating rocket explosion on the launch pad was developed in order to study(2) theGround propagation may have law certain of near-ground effect on the explosion pressure- andtime quantify curve of thethe enhancementshock wave. The effect pressure on peak– timeoverpressure. curves of shock From wave analysis in andthe air discussion show a single of results,-peak the feature. following If the main ground conclusions exists, the were pressure drawn:– time(1) curveThe existenceof the gauge of the whose ground inciden will changet angle the is propagationless than 40° law presents and conform a bimodal to the feature, reflection namely, law of incidentthe wave shock and wave. reflected Rigid wave. ground However, absorbs when no energy the inciden andt reflects angle of all the of gauge it, while is greater concrete than ground 40°, the pressureabsorbs–time and curve reflects shows some a of single energy,-peak respectively. feature, and However, the peak the value shock is Mach waves reflected are all enhanced pressure. at (3)the For near-ground gauges at ofdifferent different heights, materials. different ground may have different effects on the peak overpressure.(2) Ground For may gauges have certainwhose eheightffect on is thenot pressure-timegreater than 4curve m, when of the x is shock small, wave. the peak The overpressure pressure–time above curvesthe rigid of ground shock wave is the in highest. the air show With a the single-peak increase feature.of x, the If peak the ground overpressure exists, theabove pressure–time the three kinds of ground gradually tends to be the same. As for h greater than 4 m, the peak overpressure curve of the gauge whose incident angle is less than 40◦ presents a bimodal feature, namely, curvesincident of different wave ground and reflectedare the same wave. when However, x is small when. With the the incident increase angle of x, t ofhe thepeak gauge overpressure is greater of rigid ground is higher than that of concrete and soil ground, and the peak overpressure of concrete than 40◦, the pressure–time curve shows a single-peak feature, and the peak value is Mach and soilreflected ground pressure. is higher than that of the air. Appl. Sci. 2019, 9, 3639 15 of 17

(3) For gauges at different heights, different ground may have different effects on the peak overpressure. For gauges whose height is not greater than 4 m, when x is small, the peak overpressure above the rigid ground is the highest. With the increase of x, the peak overpressure above the three kinds of ground gradually tends to be the same. As for h greater than 4 m, the peak overpressure curves of different ground are the same when x is small. With the increase of x, the peak overpressure of rigid ground is higher than that of concrete and soil ground, and the peak overpressure of concrete and soil ground is higher than that of the air. (4) Grounds have the enhancement effect on peak overpressures, but different grounds have different ways. For rigid ground, the enhancement effect of peak overpressure at different heights is nearly the same and the peak overpressure factor is about 2. However, for the concrete and soil ground, the enhancement effect of the peak overpressure at different heights is quite different, among which the peak overpressure factor of 6 m is about 2.1, and the peak overpressure factor of 10m is the smallest, about 1.43.

Author Contributions: Methodology, Y.W., H.W.; software, Y.W., B.Z.; formal analysis, C.C.; investigation, Y.W.; resources, Y.W., H.W., C.C.; data curation, Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, Y.W.; visualization, Y.W. Funding: This research received no external funding. Acknowledgments: The authors greatly appreciate the comments from the reviewers, whose comments helped to improve the quality of the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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