Three-dimensional high order large-scale numerical

investigation on the air explosion

Cheng Wanga, JianXu Dinga, Chi-Wang Shub,∗, Tao Lia

aState Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing

100081, ChinabDivision of Applied Mathematics, Brown University, Providence, RI 02912, USA

Abstract

Based on the double shockwave approximation procedure and combining the real ghost fluid

method (RGFM) with the level-set method, a local Riemann problem for strongly nonlinear

equations of state such as the JWL equation of state is constructed, which has suppressed

successfully the numerical oscillation caused by high-density ratio and high-pressure ratio

across the interface between the explosion products and the air. A fifth order finite difference

weighted essentially non-oscillatory (WENO) scheme and a third order TVD Runge-Kutta

method are utilized for spatial discretization and time advance, respectively. A fully enclosed

type MPI-based parallel methodology for the RGFM procedure on a uniform structured mesh

is presented to realize the parallelization of the three-dimensional RGFM-based code for the

air explosion. The overall process of the three-dimensional air explosions of both TNT and

aluminized explosives has been successfully simulated. The overpressure at different locations

of the three-dimensional air explosions for both explosives mentioned above is monitored and

analyzed for revealing the influence of the aluminum powder combustion on the overpressure

of the explosion wave. The numerical results indicate that, due to the aluminum powder

afterburning, the attenuation of the explosion wave formed by the aluminized explosives is

slower than that caused by TNT.

Keywords: Air explosion, WENO scheme, RGFM, local Riemann solver, parallelcomputation

∗Corresponding author: shu@dam.brown.edu

Preprint submitted to Elsevier November 4, 2015

1. Introduction

Air explosion is a typical multi-medium problem, in which the explosive flow field usually

consists of many media such as detonation products and air. It is obvious that the sharp

medium interface with high-density ratio and high-pressure ratio separates the detonation

products from the air. In the numerical simulation of the air explosion problem, because

of the abrupt change of density and pressure close to the interface, unphysical numerical

oscillations may easily appear in the neighborhood of the interface. Meanwhile, with the

continuous upgrade of the explosives, density ratio and pressure ratio increase constantly and

significantly. Therefore, tracking and treating the strong nonlinear discontinuous interface

in the air explosion have received considerable attention.

Many interface tracking techniques have been discussed over the past few decades. The

marker-and-cell (MAC) method proposed by Harlow and Welch [1] has often been used to

track the interface movement and the flow field evolution. However, the maintenance of

a sharp interface among multi-media is difficult. Hirt and Nichols [2] presented another

interface tracking algorithm known as the volume-of-fluid (VOF) method without tracking

the motion of the multi-medium interface, in which, as time increases, the volume of each

medium in each cell is changing continuously and the new interface is reconstructed from the

values of the volumes at the current time level. The main drawback of the VOF method is

that, for deciding the orientation of the multi-medium interface, massive non-local data must

be taken into consideration. To overcome such drawbacks of the VOF method, the moment of

fluid (MOF) method introduced by Ahn and Shashkov [3] and by Dyadechko and Shashkov

[4] with second order accuracy employed not only volume fractions but also the position

of the centroids of each medium, which allows the MOF method to utilize the cell where

reconstruction is performed. The well-known particle-in-cell (PIC) algorithm by Amsden [5]

is a Lagrangian description of flows in which particles are explicitly associated with different

media and thus the interface can be tracked easily. The principal drawbacks are a large

numerical diffusion and a numerical noise, due to the momentum transfer between grids and

particles and the employment of finite particle numbers. The level set method developed

2

by Osher and Sethian [6] is a simple and versatile method for tracking the evolution of a

locomotive interface. The level-set method [7, 8, 9] uses a Hamilton-Jacobi equation to

describe the moving interface, which is tracked as the zero level set of a continuous function

mimicking the signed distance function instead of an explicit function at the new time level.

Thus, cases with complex interfaces, such as crisscross, torsion and separation, etc., can be

easily handled by the level-set method.

Fedkiw et al. [10] presented the original ghost fluid method (OGFM), which can address

excellently the interaction between a weak shock wave and an interface. To further improve

the accuracy of the OGFM method, Liu et al. [11] discussed the modified ghost fluid method

(MGFM) procedure, which performed better than the OGFM due to a local Riemann solver,

where the status of the ghost fluid across the interface for each phase was defined by the

predicted interfacial status by this Riemann solver. The accuracy of the MGFM for the gas-

gas Riemann problem performed by Liu et al. in [12] showed that, compared with the exact

solution of a Riemann problem, the MGFM solution could reach second order accuracy

near the multi-medium interface. The key point of the real ghost fluid method (RGFM)

procedure, presented by Wang et al. [13] was that, according to the status of media across

the interface, a local Riemann problem was constructed at first and the interfacial status

obtained was then used to redefine the flow states for not only the real fluid grid points

next to the interface but also to the ghost fluid grid points, by which smaller errors were

introduced in the RGFM procedure in comparison to the MGFM procedure.

Since the OGFM, MGFM and RGFM were introduced, an increasing number of numerical

simulations combining these methods for multi-medium flows can be found in the literature.

However, most of these previous simulations with GFM-based procedures use the simple and

linear equations of state for describing the media of interest such as explosion products and

air in the air explosion, while the complex equations of state such as JWL have seldomly

been used. Meanwhile, numerical simulations based on the RGFM procedure by splitting

a multi-medium problem into single medium flows, usually require large parallel computer

resources. In general, serial computation cannot meet the requirements of multi-dimensional

large-scale engineering applications.

3

In this paper, based on a double shockwave approximation procedure, we first present

in detail a technique to construct and solve the local Riemann problem with the complex

equations of state such as the JWL equation of state used in the air explosion simulations.

Our numerical simulation demonstrates that this technique effectively eliminates unphysical

oscillations which often occur at the multi-medium interface in the explosive flow field. Com-

bining the RGFM method, formally transforming the multi-medium flow field into single flow

fields with an enclosed type parallelization module, the mechanisms of the three-dimensional

air explosion are studied by using the fifth order finite difference weighted essentially non-

oscillatory (WENO) scheme on a uniform structured mesh.

2. Governing equations

The governing equations in a conservative form without considering the viscous and

thermal diffusion effects for describing a three-dimensional air explosion can be written as

∂U

∂t+

∂F (U)

∂x+

∂G(U)

∂y+

∂H(U)

∂z= 0

U = (ρ, ρu, ρv, ρw, ρE)T

F (U) = (ρu, ρu2 + p, ρuv, ρuw, (ρE + p)u)T

G(U) = (ρv, ρuv, ρv2 + p, ρvw, (ρE + p)v)T

H(U) = (ρw, ρuw, ρvw, ρw2 + p, (ρE + p)w)T

(1)

where ρ and p denote the density and pressure, respectively. u, v and w are the velocity

components in the x ,y and z directions in the Euler coordinates, and E is the total energy,

generally consisting of internal energy and kinetic energy, is given as

E =(u2 + v2 + w2)

2+ e (2)

where e is the internal energy per unit mass.

For the aluminized explosive, the Miller model [14] describing the combustion and heat

release process of the aluminum powder can be used to describe the reaction process. By

coupling the above three-dimensional Euler equations with the Miller model, the complete

process of the air explosion for the aluminized explosive can be captured numerically. The

4

Tab. 1: JWL EOS parameters for the explosion products of the TNT charge.

ρ0(kg/m3) A(MPa) B(MPa) R1 R2 ω

1630.0 373800.0 3747.0 4.15 0.9 0.35

Miller model given in [14] and used in this paper is

∂λ

∂t=

1

4(1 − λ)

12 p

16 (3)

where λ is the chemical reaction process variable characterizing the aluminum powder reac-

tion degree. The reaction process parameters λ = 0 and λ = 1 are defined as the initially

unreacted and completely burned states, respectively.

To close the above governing equations, the respective equations of state for the air and

for the explosion products of TNT and aluminized explosive must be introduced. The ideal

gaseous equation of state for the air medium can be presented as

p = (γ − 1)ρe (4)

where e is the specific internal energy and γ = 1.4 for air is the gaseous constant which

depends on the particular gas of interest.

The explosion products of the TNT charge are usually described by the JWL equation

of state [15], which can be expressed in the following form

p = A(1 −ωρ

R1ρ0)e−

R1ρ0ρ + B(1 −

ωρ

R2ρ0)e−

R2ρ0ρ + ωρe (5)

where ρ, p and e denote the density, pressure and the specific internal energy in unit mass,

respectively. The parameters A, B, R1, R2, ω and ρ0 are material constants of the detonation

products, which are specifically shown in Table 1.

The equation of state of the detonation product of aluminized explosives should be able to

establish the essential relationship among pressure, density, internal energy and the reaction

process variable. The JWL-Miller equation of state [14] for describing the detonation product

of aluminized explosives are adopted as

p = A(1 −ωρ

R1ρ0)e−

R1ρ0ρ + B(1 −

ωρ

R2ρ0)e−

R2ρ0ρ +

ωρ(E + λQ)

ρ0

(6)

5

Tab. 2: JWL-Miller EOS parameters for the explosion products of aluminized explosives.

A(MPa) B(MPa) R1 R2 ω ρ0(kg/m3) Q(MPa)

695130.0 3130.0 5.4 1.4 0.4 1880 11000

where ρ, p and E denote the density, pressure and the energy content of the explosive which

keeps the CJ condition, respectively. The constant Q is defined as the afterburning energy

release after the CJ plane. The parameters A, B, R1, R2, ω and ρ0 are material constants

of the detonation products, which can be found in Table 2.

The semi-discrete approximation of equation (1) is given by

(∂U

∂t)i,j,k = −

1

∆x(Fi+ 1

2,j,k − Fi− 1

2,j,k) −

1

∆y(Gi,j+ 1

2,k − Gi,j− 1

2,k)

−1

∆z(Hi,j,k+ 1

2− Hi,j,k− 1

2)

(7)

where Fi+ 12,j,k ,Gi,j+ 1

2,k and Hi,j,k+ 1

2present the numerical fluxes, which are obtained by a fifth

order finite difference WENO scheme with the Lax-Friedrichs splitting [16]. A third order

TVD Runge-Kutta scheme [17] is used to integrate the system of the ordinary differential

equations (7) in time. The parallel adaptive code to solve three-dimensional detonation

problems is developed in [18].

3. The local Riemann solver for the air explosion

The level set technique provides an efficient method to describe multi-medium flows and

to track the interfaces, and is used in this paper. Besides the high order solution of the

Hamilton-Jacobi type equations with the methods presented in the literature [19, 20], the

reinitialization of the level set is a necessary step in the level set method to keep the level set

function close to the signed distance function and hence to maintain efficiency of the overall

solution method [21, 22, 23, 24]. After confirming the specific locations of the interface

between the explosion products and the air by advancing the implicit level-set function, the

RGFM-based multi-medium interface treatment should be utilized to change a multi-medium

problem into two single medium problems. In the RGFM method, a local Riemann problem

is constructed first at the interface and then solved. The predicted interface values obtained

6

by the Riemann solver are assigned to the real fluid nodes next to the interface in the real

fluid. Then, a normal constant extrapolation by solving the extension equation [25] is used

to acquire the values of the three ghost fluid nodes required for a fifth order WENO scheme

used for each pure medium.

Next, the solution to a local Riemann problem in the air explosion will be presented in

detail. Taking the one dimensional case as an example, with the help of the mass conservation

equation and the momentum conservation equation, the expression bridging the velocity and

pressure of the interface can be obtained as

u∗ = F1(p∗) = uL −

√(p∗ − pL)(

1

ρL−

1

ρ∗L) (8)

u∗ = F2(p∗) = uR +

√(p∗ − pR)(

1

ρR−

1

ρ∗R) (9)

here ρ∗L and ρ∗R are the densities on both sides of the interface, which are unknown quantities

for the time being and also need to be solved.

The nonlinear function relationship with respect to the interfacial pressure is obtained

by the above two equations, which can be easily written as

F (p∗) = F1(p∗) − F2(p∗) = 0 (10)

Combining the energy conservation equation with the equations of state, the implicit

function in regard to ρ∗L, ρ∗R and p∗ can be deduced as

eL(pL, ρL) − e∗L(p∗, ρ∗L) −1

2(pL + p∗)

ρL − ρ∗L

ρLρ∗L

= 0 (11)

e∗R(p∗, ρ∗R) − eR(pR, ρR) −1

2(p∗ + pR)

ρ∗R − ρR

ρ∗RρR= 0 (12)

For the local Riemann problem comprising of the JWL equation of state and gas equation

of state, the process of solving equation (10) by a Newton iterative method is described below.

It is first supposed that, for the one dimensional Riemann problem, the gaseous explosion

products are located at the left side of the interface and the air is on the right side. Obviously,

(10), (11) and (12) are now a system of nonlinear equations consisting of interfacial pressure

and densities on both sides of the interface as unknown variables. Thus the classical Newton

7

iterative method could be used to solve it:

p∗(n+1) = p

∗(n) −F1(p∗(n)) − F2(p∗(n))

F′

1(p∗(n)) − F′

2(p∗(n))(13)

The initial guess of p∗ is necessary for the Newton iteration, and should be selected

properly. Then the JWL equation of state is plugged into equation (11), and the following

implicit relation can be given

(pL − A(1 −ωρL

R1ρ0

)e−

R1ρ0ρL − B(1 −

ωρL

R2ρ0

)e−

R2ρ0ρL )ρ∗L

− (p∗ − A(1 −ωρ∗L

R1ρ0)e

−R1ρ0ρ∗L − B(1 −

ωρ∗L

R2ρ0)e

−R2ρ0ρ∗L )ρL −

1

2ω(pL + p∗)(ρL − ρ∗L) = 0

(14)

With an appropriate initial guess p∗, equation (14) becomes a nonlinear equation for the

dependent variable ρ∗L, which also needs to be solved using the Newton iterative method.

During the iterative process, the derivative of the equation (14) with respect to the density

can be written as

∂f

∂ρ∗L=pL + ρL(

Ae−

R1ρ0ρ∗L (R1ρ0 − ωρ∗L)

ρ2∗L

+Be

−R2ρ0ρ∗L (R2ρ0 − ωρ∗L)

ρ2∗L

)

− ρL(Aωe

−R1ρ0ρ∗L

R1ρ0

+Bωe

−R2ρ0ρ∗L

R2ρ0

) + ω(pL

2+

p∗2

)

−Ae

−R1ρ0

ρL (R1ρ0 − ωρL)

R1ρ0

−Be

−R2ρ0

ρL (R2ρ0 − ωρL)

R2ρ0

(15)

Similarly, for the ideal gaseous equation of state, we have

p∗γ − 1

−pRρ∗R

(γ − 1)ρR−

1

2(p∗ + pR)

ρ∗R − ρR

ρR= 0 (16)

The derivative of equation (16) with respect to the density can be given easily as

∂f

∂ρ∗R= −

pR + p∗2ρR

−pR

ρR(γ − 1)(17)

Now we substitute both the pressure p∗ and the density ρ∗L obtained by the iterative

method into equation (8) to produce the expression F1(p∗). In the same way, substitution of

the given p∗ and the iteration density ρ∗R into equation (9) yields F2(p∗). Apparently, the

derivatives of the functions F1(p∗) and F2(p∗) with respect to the interface pressure p∗ are

dF1(p∗)

dp∗= −

√

−

1ρ∗L

− 1ρL

p∗ − pL−

ρ′

∗L(p∗−pL)

ρ2∗L

+ 1ρ∗L

−1

ρL

p∗−pL

2

√−

1ρ∗L

−1

ρL

p∗−pL

(18)

8

dF2(p∗)

dp∗=

√

−

1ρ∗R

− 1ρR

p∗ − pR

+

ρ′

∗R(p∗−pR)

ρ2∗R

+ 1ρ∗R

−1

ρR

p∗−pR

2

√−

1ρ∗R

−1

ρR

p∗−pR

(19)

By a direct calculation of the derivatives of the density ρ∗L and ρ∗R at both sides of the

interface with respect to the interface pressure, we arrive at the desired results as follows

dρ∗L

dp∗=

ρL + ω(ρL−ρ∗L)2

pL + ρL(Ae−

R1ρ0ρ∗L (R1ρ0−ωρ∗L)

ρ2∗L

+ Be−

R2ρ0ρ∗L (R2ρ0−ωρ∗L)

ρ2∗L

− Aωe−

R1ρ0ρ∗L

R1ρ0− Bωe

−R2ρ0ρ∗L

R2ρ0)

+ ω(pL

2+

p∗2

) −Ae

−R1ρ0

ρL (R1ρ0 − ωρL)

R1ρ0

−Be

−R2ρ0

ρL (R2ρ0 − ωρL)

R2ρ0

(20)

dρ∗R

dp∗= −

1γ−1

− 12(ρ∗R

ρR− 1)

pR+p∗2ρR

+ pR

ρR(γ−1)

(21)

By substituting F1(p∗), F2(p∗), F′

1(p∗), F′

2(p∗) obtained above into equation (13), the new

pressure p∗(n+1) is obtained, which will be used as the starting value for the next iteration.

If the difference |p∗(n+1) − p∗(n)| is smaller than a threshold, the Newton iteration will be

stopped, and the final result p∗(n+1) is recorded as the pressure p∗. Then, the velocity and

the two values of density can be obtained by simultaneously solving (8), (11) and (12), and

the solution to the local Riemann problem in the air explosion is completed.

4. Parallelization for the RGFM procedure

For parallel implementation, the computational domain should first be divided into many

small subdomains. That is to say, each processor is only responsible for computing the

corresponding subdomain, as shown in Fig. 1. In this section we sometimes use the two-

dimensional case for illustration.

When calculating the spatial derivatives at any grid point, it is necessary to obtain the

values of its adjacent nodes for interpolation. The template for the fifth order WENO scheme

requires seven grid points, which is obviously wider than the traditional lower order schemes.

For such grid points close to the border of any processor, the corresponding information

required for interpolation may be located in other adjacent processors, as shown in Fig.

2. It can be seen from Fig. 2 that the computation of several columns of grid points

located at the right side of the yellow middle processor would need to use information in

9

Fig. 1: Computational domain divided equally into subdomains for parallel implementation.

the right adjacent processor. Therefore, the information of the adjacent processors must be

transmitted into the current processor for parallel computation.

Fig. 2: Data communication required by a finite difference WENO scheme.

For an ordinary finite difference scheme, it generally only requires to communicate data

of the current processor with its adjacent processors in the orthogonal directions. That is

to say, information in the processors in the diagonal directions are not needed. For the

numerical simulations of the air explosion based on the RGFM combined with the WENO

scheme, however, the situation may be different. If the interface formed by the explosion

products and the air is very close to a corner of any subdomain belonging to a computational

processor, information such as pressure and density at some grid points of the adjacent

diagonal processor to that corner may be needed in the construction of the local Riemann

solver, as shown in Fig. 3. Considering such special requirements for the RGFM method to

10

Fig. 3: Possible data communication required by RGFM.

data communication, a fully enclosed type data communication mode is presented, in which

the enclosed communication ghost layers with a width of several grid points are created on

the periphery of the computational subdomain for any processor. Through the real-time data

communication based on the presented parallel method, it can ensure that, similar to the

serial run environment, any subdomain can be computed efficiently. In a three-dimensional

flow field, for a processor not located at the border of the computational domain, there are

26 adjacent processors. Therefore, 26 “send” buffers corresponding to 26 “receive” buffers

are established. Fig. 4 shows the computational subdomain of an individual processor.

The cubic area surrounded by the light color lines in the middle is an actual computational

subdomain using the WENO scheme for the current processor. Others are not only the ghost

layers but also the 26 “receive” buffers. It should be noted that, without loss of generality,

the corresponding 26 “send” buffers are not illustrated in Fig. 4.

Fig. 4: An enclosed type data communication style for an individual processor.

11

Table 3 shows the number of all the “send” buffers and their processors corresponding to

all the “receive” buffers for the n-th subdomain, and the number of all the “receive” buffers

and their processors corresponding to all the “send” buffers for the n-th subdomain. It is

noted that notations x and y refer to the respective numbers of processors in the horizontal

and vertical directions.

Tab. 3: Communication mode for the n-th receive (send) processor.

Processor number Receive (send) Send (receive) Processor numberof receive (send) buffer buffer number buffer number of send (receive) buffer

n 1 8 n-xy-x-1n 2 7 n-xy-x+1n 3 6 n-xy+x-1n 4 5 n-xy+x+1n 5 4 n+xy-x-1n 6 3 n+xy-x+1n 7 2 n+xy+x-1n 8 1 n+xy+x+1n 9 12 n-xy-xn 10 11 n-xy+xn 11 10 n+xy-xn 12 9 n+xy+xn 13 16 n-xy-1n 14 15 n-xy+1n 15 14 n+xy-1n 16 13 n+xy+1n 17 20 n-x-1n 18 19 n-x+1n 19 18 n+x-1n 20 17 n+x+1n 21 22 n-1n 22 21 n+1n 23 24 n-xn 24 23 n+xn 25 26 n-xyn 26 25 n+xy

The classical fifth order WENO scheme used in this paper needs three layers of ghost

points for the current computational processor. As for the RGFM method, the grid points

beyond the three ghost layers are definitely not needed when constructing the local Riemann

12

Problem. Therefore, the enclosed type MPI-based parallel procedure containing three ghost

layers of grid points are employed in all the following numerical examples.

5. Numerical investigation on the three-dimensional air explosion

5.1. Numerical investigation on the air explosion for the TNT charge

In this subsection, simulations of the air explosion for the TNT charge are performed uti-

lizing a parallel high order code described previously. The explosion products are described

by the JWL EOS, and its parameters can be found in Table 1. The initial pressure of the

explosion products achieved in this subsection is 8.6 GPa. The computational model can be

seen in Fig. 5 for the air explosion.

Fig. 5: The parallel computational model for the air explosion.

Before performing a series of numerical simulations, the convergence of the computational

code with increasing resolution for the air explosion is validated for this study. The com-

putational domain with the TNT charge radius of 0.3m is 7.8m×7.8m×7.8m. To accelerate

the speed of computations, 64 processors are employed to compute this problem. When the

difference of pressure between two mesh sizes monitored at the location (5.6m, 3.9m, 3.9m)

is small enough, grid convergence is declared. Fig. 6 illustrates the evolution of the explosion

overpressure for three different mesh sizes (0.03m, 0.02m and 0.01m). It is observed clearly

from Fig. 6 that grid convergence can be attained with a mesh size of 0.02m.

In order to demonstrate and validate the stability and reliability of the numerical results,

the explosion wave overpressure values obtained at different locations are compared with the

13

Fig. 6: The convergence of mesh refinements.

empirical formula in Henrych [26]. 64 processors are employed to compute this problem with

up to 59.78 million grid points. The size of a uniform cubic mesh is 0.02m×0.02m×0.02m,

while the total physical domain with the TNT charge radius of 0.21m is also 7.8m×7.8m×7.8m.

The comparison is shown in Fig. 7. It is clear that, except for the region of the near field (in

which the proportional distance is less than 0.348 m/kg1/3), the explosion wave overpressure

obtained by the numerical simulation is in good agreement with the empirical formula, with

a relative error less than 8.3 percent. As a result, the material interface treatment combining

level-set technique with the RGFM method based on a shock-shock Riemann solver can be

applied successfully to the numerical simulations of the air explosion. At the same time, the

comparison also shows that neglecting the details of the explosion process has little effect

on the results of the numerical calculation by simplifying the explosive products into a high

temperature and high pressure gas.

The numerical results reveal that the essential physics of the expansion of the interface

is the movement of the contact discontinuity for a local Riemann problem at the interface,

and the pressure, normal speed and density close to the interface determine the evolution of

the interface. Thus, solving the local Riemann problem is an effective way to quantitatively

reveal the evolution mechanism of the air explosion. Solving the local Riemann problem can

obtain accurately flow characteristics near the interface and inside the flow field. Therefore,

the unique advantages based on the RGFM procedure and the local Riemann solver make

14

Fig. 7: The explosion wave overpressure peaks obtained by both the numerical simulationand the empirical formula.

the procedure ideal for simulating the air explosion.

The computational results at several temporal moments are shown in Figures 8 to 10.

In order to observe directly the distribution of the physical variables such as pressure and

density inside the three-dimensional internal explosion field, the results in only part of the

computational domain are shown. It can be seen from Figures 8 to 10 that the contours of

density and pressure are very symmetrical and nonphysical oscillations do not appear. The

distribution of density and pressure in the air explosion field can also be revealed clearly.

Fig. 8: The numerical results at 0.50ms: (a) Pressure contour; (b) Density contour.

Fig. 11 shows the distribution of pressure and density for the spherical explosion field

along the central axis at 0.772 ms. It is obvious that there are two density peaks in the

typical explosion field. The first peak is inside the explosion products close to the interface

15

Fig. 9: The numerical results at 1.01ms: (a) Pressure contour; (b) Density contour.

Fig. 10: The numerical results at 1.71ms: (a) Pressure contour; (b) Density contour.

16

of the material, while the other one appears in the explosion wave front in the air. The curve

describing the evolution of the pressure has only one peak, and the transition of pressure

is continuous at the interface between the explosion products and the air. It can also be

found that, in the rapid expansion process of the air caused by the explosion products with

an extremely high temperature and pressure, the rarefaction wave propagates back into the

explosion products and converges to the center of the spherical explosion products, which

dramatically attenuates the pressure in the zones swept by the rarefaction wave. At 0.772

ms, the density distributes from 0.5 kg/m3 to 1.0 kg/m3, which is slightly lower than that

under the ambient temperature and pressure; the pressure is less than 0.1 MPa, which is also

below the normal atmospheric pressure. However, a narrow banded zone with relatively high

pressure and density is located inside the explosion products and close to the propagating

interface. At 0.772 ms, the maximum pressure and density can reach up to 0.86 MPa and 3.3

kg/m3, respectively, which are much higher than the atmospheric ones under the ambient

temperature and pressure.

Fig. 11: The distributions of pressure and density along the center axis at 0.772ms: (a)Pressure curve; (b) Density curve.

5.2. Numerical investigation on the air explosion of aluminized explosives

In this subsection simulations of the air explosion for aluminized explosives with the

radius of 0.21m are performed. The JWL-Miller EOS is used to describe the explosion

products, and Table 2 gives all the parameters used. For simplicity, the ignition and reaction

process of the aluminized explosives are again omitted in the air explosion. The initial

pressure used in this subsection of explosion products is 6.26 GPa. The numerical results

17

are again obtained with 64 processors and the total number of grid points is 59.78 million.

The mesh size is 0.02m×0.02m×0.02m. The computational domain is 7.8m×7.8m×7.8m.

Fig. 12: The numerical results for the aluminized explosive: (a) Pressure contour; (b) Densitycontour; (c) Level-set contour; (d) Aluminum powder reaction fraction contour.

Fig. 12 shows the contours of the computational results for part of the computational

domain for the aluminized explosives. Table 4 shows the evolution of the explosion wave

overpressure peak for the aluminized explosive with the change of the proportional distance.

Fig. 13 compares explosion wave overpressure peaks of the TNT charge with that of the

aluminized explosives with the variation of proportional distance. It can be seen from Fig. 13

that the explosion wave overpressure peaks caused by the aluminized explosive are obviously

lower than that of TNT in the near field. However, because the secondary energy release

process caused by the aluminum powder afterburning can continuously supplement energy,

the energy loss in the propagating process of the explosion wave is replenished to some

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Tab. 4: The explosion wave overpressure peaks for the aluminized explosive at differentproportional distances.

Proportional 0.182 0.249 0.332 0.415 0.497 0.580 0.663 0.746 0.829distance(m/kg1/3)

Overpressure 8.09 6.85 5.31 4.10 3.22 2.52 1.97 1.55 1.25peak(MPa)

extent. Therefore, the explosion wave attenuation of the aluminized explosive is relatively

slow, and, its overpressure will gradually catch up with and then surpass that formed by the

TNT as the explosion wave spreads forward to the far field.

Fig. 13: The comparison of the explosion wave overpressure peaks between TNT and alu-minized explosives in the air explosion.

6. Concluding remarks

Combining the RGFM multi-medium interface treatment method with a fifth order fi-

nite difference WENO scheme, and based on a Riemann solver presented for the nonlinear

equations of state and the MPI-based enclose type parallel mode, the large-scale air explo-

sion parallel code is developed to successfully simulate the three-dimensional air explosion

processes for the TNT charge and aluminized explosives. First of all, the air explosions of

the three-dimensional spherical TNT are carried out numerically, and then the distribution

19

of the physical quantities such as density and pressure are obtained in the explosive flow

field. The validation of the code based on the RGFM for simulating air explosion is given

through comparing the explosion wave overpressures of the numerical results with those of

the Henrych empirical formula [26]. The errors are less than 8.3 percent except for the region

of the near field close to the interface in the air. At the same time, it is also demonstrated

that, neglecting the reaction process in the air explosion, as is done in this paper to simply

the computation, has little impact on the numerical results of the far field explosion over-

pressure. Next the air explosion of the spherical aluminized explosive is simulated to obtain

the pressure and density distribution in the explosion field. The overpressure peaks of the

TNT charge and the aluminized explosive are compared with the variation of proportional

distance. This comparison shows that the subsequent aluminum powder afterburning gives

rise to secondary energy release and forces the relatively slow attenuation of the explosion

wave overpressure peaks of the aluminized explosive.

Acknowledgments

The research of Wang, Ding and Li is supported by the National Natural Science Founda-

tion of China under grants 11325209 and 11221202, the Specialized Research Foundation for

the Doctoral Program of Higher Education of China under grant 20121101110004, and the

Beijing Natural Science Foundation under grant 8132034. The research of Shu is supported

by ARO grant W911NF-15-1-0226 and NSF grant DMS-1418750.

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