THE LEVEL SET METHOD APPLIED TO THREE-DIMENSIONAL DETONATION WAVE PROPAGATION. Wen Shanggang, Sun Chengwei, Zhao Feng, Chen Jun

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1 THE LEVEL SET METHOD APPLIED TO THREE-DIMENSIONAL DETONATION WAVE PROPAGATION We Shaggag, Su Chegwei, Zhao Feg, Che Ju Laboratory for Shock Wave ad Detoatio Physics Research, Southwest Istitute of Fluid Physics, P.O.Box ,Miayag, 61900, Chia O the basis of previous works o D detoatio pr4opagatio, the code LS3D employig the level set method to calculate 3D detoatio wave propagatio has developed i this paper. With this code some propagatio processes of detoatio waves i HEs without boudaries ad i differet shaped IHE JB9014 charges, both iitiated i differet ways, have bee studied, icludig multi-poit iitiatio, rate stick, plae wave les ad two detoatio wave iteractio. The calculated results for JB9014 explosive agree well with the experimetal results. 1 INTRODUCTION The level set (LS) method developed by Osher ad Sethia et al [1~] describes the evolutio of iterface that propagate especially accordig to the depedece of their ow curvature, for example the detoatio waves i explosives. I LS method, a Hamilto-Jacobi called evolutio equatio here ad describig the iterface movemets should be itroduced from hydrodyamics or experimetal research. The umerical solutios of the evolutio equatio uder a set of boudary ad iitial coditios are performed with the LS codes to calculate the level-set fuctios. The the iterfaces are determied with a series of special values of the level-set fuctio. The kid of method hadles automatically ay frot shape ad ay topological chages i explosive charges. The physical foudatio of the LS method is the same for the DSD (detoatio Shock Dyamics) theor i.e., the detoatio frot velocity is curvature-depedet. The a geometrical model of detoatio propagatio ca be proposed with a form similar to the eikoal equatio i geometrical optics. However, the eikoal equatio is of high order ad high o-liear ad very difficult to be solved umerically. I the D case without iteractio betwee adjacet segmets of the detoatio frot, the geometrical approach is cosiderably effective, sice the frot at ext time step ca be described as the evelope of the wavelets origiatig from the frot at last time step [3]. Meawhile the mechaics

2 boudary coditios i D case are ease matched for the above geometrically determied detoatio frot. Nevertheless, it is very difficult to exted the geometrical approach to the 3D case. We have to ask for the LS method which is a effective treatmet for complicated iterfaces ad oted for its simple logic ad programmig. Aslam ad Bdzil [4] have applied this method to modelig detoatio shock dyamics (DSD). They have developed a umerical method to implemet arbitrarily complex D boudary coditios. Based o their works, we apply the LS method to 3D detoatio wave propagatio i explosive, ad compile the calculatio code LS3D. The code LS3D ca be used i calculatig the detoatio wave propagatio i explosive charges iitiated i differet way. It ca deal with differet shaped ad compoet multi-medium explosive charges. Employig this code, we simulate some detoatio propagatio problems umerically i this paper. THE LEVEL-SET EQUATION OF THREE-DIMENSIONAL DETONA- TION WAVE PROPAGATION Cosiderig the detoatio wave propagatig i 3D space of explosives, it is assumed that the detoatio frot is ifiitely thi. I the space a rectagular coordiate system ( x, z ) has bee defied, where this detoatio frot is described as the zero level set of a fuctio ϕ ( x, z; t). Let a level set of poits be defied by ϕ ( x, z, t = 0) = d (1) where d is the distace from poit ( x, z) to the detoatio frot. The set is correspodig to a cotour surface. Peculiarly the detoatio frot is assiged by the level set ϕ = 0, while the uburt explosive is deoted by ϕ > 0 ad the burt explosive ϕ < 0. Sice a level surface is give by ϕ ( x, z, t) = cost, it follows that its total derivatives with respect to t idetically equals zero, i.e., ϕ ϕ dx ϕ dy ϕ dz = 0 () t x dt y dt z dt where the time derivative d x / dt, dy / dt ad dz / dt are the compoets of the surface velocity D, i.e., the displacemet vector of this poit withi a ifiite small iterval. The Eq.() ca be deoted with a Hamilto-Jacobilike equatio as ϕ + D ( κ ) ϕ = 0 (3) t where κ is the curvature of the surface, D (κ ) is the curvature κ depedet ormal velocity of detoatio frot that ca be obtaied by theoretical calculatios or by experimets. I 3D Cartesia coordiates, the curvature ca be deduced as ϕ x ( ϕ κ= yy + ϕ ) + ϕ ( ϕ zz x y ( ϕ + ϕ + ϕ ) x y y + ϕ ) + ϕ ( ϕ xz y ) 3/ z 3 / z z ( ϕ xϕ yϕ xy + ϕ xϕ zϕ + ϕ ϕ zϕ yz ) ( ϕ + ϕ + ϕ xx zz xx + ϕ yy )

3 Usually the D (κ ) has the followig form of fuctio, D x, z) = D ( D, x, z; κ, R, T, K) (4) ( CJ 0 0 Where D CJ is the ideal steady detoatio velocity i a boudless explosive bulk. R 0 is the characteristic size coefficiet of the explosive charge, for example the diameter of a cylidrical explosive, T 0 is the ambiet temperature. I this paper, a simple liear relatio D = DCJ (1 ακ ) is utilized, where α is a costat of the explosive. The solutio of equatio (3) uder give iitial ad boudary coditios geerates the field ϕ ( x, z, t), ad the locatio of the detoatio frot at time t is the simply assiged by a search for the level surface ϕ = 0. The iitial coditio for the level-set equatio is the LS fuctio value of each poit at t = 0, ϕ ( x, z; t = 0) = 0 represets the iitial detoatio frot. The recipe for the boudary coditio is same as Aslam ad Bdzil s [4]:(1) Whe the flow i the explosive is supersoic, the cotiuatio boudary is applied. () Whe the flow i the explosive is soic or subsoic, the soic coditio is applied or the boudary agle must be adjusted util the pressure i the iert ad the explosive are equilibrated. See details i [4]. propagatio Assumed that the detoatio wave propagatig at a costat velocity D CJ i the explosive charge. I this paper, we let D =8.0 km/s ad calculate the CJ detoatio frot propagatig i explosives iitiated at seve poits or o the ceter lie at the bottom Seve-poit iitiated detoatio wave The explosive cube with side legth 16 mm is iitiated at seve poits at the bottom whose coordiates are (4,4,0), (4,8,0), (4,1,0), (1,4,0), (1,8,0), (1,1,0) ad (8,8,0) respectively. I the calculatio the space step ad the time step are 0.4 mm ad 0.01 µs respectively. The detoatio frots at differet time are show i Figure 1, the hemisphere like detoatio waves appear at first, the they merge with each other ad itersect the edges of the cube, ad evetually the frot burs out of the cube. 3 NUMERICAL CALCULATIONS 3.1 The calculatio of ideal detoatio (a) t=0.30 µs

the itersectio curves of detoatio frot with the sectio plae x=10 at differet time are show i Figure 3

These figures show that the detoatio frot shape evolves from cylidrical to plaar oe. (b) t=0.

1. Sigle lie iitiated detoatio wave A cube explosive with side legth 0 mm is iitiated o a ceter lie at

4 the itersectio curves of detoatio frot with the sectio plae x=10 at differet time are show i Figure 3 (the time iterval betwee adjacet curves is µs). These figures show that the detoatio frot shape evolves from cylidrical to plaar oe. (b) t=0.90 µs (a) t=0.078 µs (c) t= 1.90 µs FIG. 1. SEVEN-POINT INITIATED DETONATION FRONT AT DIFFERENT TIME 3.1. Sigle lie iitiated detoatio wave A cube explosive with side legth 0 mm is iitiated o a ceter lie at the bottom, the calculatio space step ad time step are 0. mm ad µs respectively. The detoatio frots at differet time are show i Figure, ad (b) t=1.865 µs

The D ( κ ) relatio D = 7.71 (1 0.875κ ) (km/s) is used i the calculatio. 3..1 Two-poit iitiated detoatio wave i the JB9014 slab (c) t=.486 µs FIG.

5 TATB-based isesitive explosive (IHE) with better thermal stability ad lower sesitivity to impact. Employig the code LS3D, the detoatio propagatio i JB9014 slab or cylider iitiated i differet way has bee umerically simulated; some calculatio results are compared with the experimetal data. The D ( κ ) relatio D = 7.71 ( κ ) (km/s) is used i the calculatio Two-poit iitiated detoatio wave i the JB9014 slab (c) t=.486 µs FIG.. SINGLE LINE INITIATED DETONATION AT DIFFERENT TIME A JB9014 explosive slab of 00 mm i side legth ad 5 mm i thickess is iitiated at two poits whose coordiates are (100,50,0) ad (100,150,0). The space step ad time step i calculatio are 0.5 mm ad 0.01 µs respectively. The calculatio results of detoatio wave at differet time are show i Figure 4. Figure 5 shows the detoatio frot breakig out o the top lie (100,5). The calculated waveform (dashed lie) agrees well with the experimetal observatio (solid lie) [5]. FIG.3. THE INTERSECTION CURVES OF DETONATION FRONTS WITH THE SECTION PLANE X=10 AT DIFFERENT TIME ( t = µs) 3. Numerical simulatios of detoatio waves i explosive JB9014 slab ad cylider The explosive JB9014 is a (a) t=0.88 µs

3.. Detoatio propagatio i a JB9014 cylider A JB9014 explosive cylider of 0 mm i diameter ad 70 mm i legth is iitiated at the bottom whe t = 0. The space step ad time step i calculatio are 0.5 mm ad 0.

6 3.. Detoatio propagatio i a JB9014 cylider A JB9014 explosive cylider of 0 mm i diameter ad 70 mm i legth is iitiated at the bottom whe t = 0. The space step ad time step i calculatio are 0.5 mm ad µs respectively. (b) t=4.03 µs The calculatio results of detoatio frot (dashed lies) at differet time are show i Figure 6. The time correspodig to the curves upwards are 1.05, 1.95,.015,.0, 4.315, 4.496, 6.535, 6.80, ad µs respectively. The solid lies are the experimetal results [5]. (c) t=7.19 µs FIG.4. TWO-POINT DETONATION, DETONATION FRONTS AT DIFFERENT TIME IN JB Experimetal LS3D calculatio y / mm t / µs FIG.5. THE BREAK OUT OF THE DETONATION FRONT ON TOP LINE OF THE EXPLOSIVE SLAB FIG.6. DETONATION FRONTS AT DIFFERENT TIME IN JB9014 CYLINDER

3.3 Other calculatio results 3.3.1 Detoatio wave propagatio i the plae wave les A explosive plae wave les of 100 mm i diameter ad 50 mm i height is combied by two parts of explosives with differet detoatio velocity.

7 3.3 Other calculatio results Detoatio wave propagatio i the plae wave les A explosive plae wave les of 100 mm i diameter ad 50 mm i height is combied by two parts of explosives with differet detoatio velocity. The upper oe is Comp. B with detoatio velocity 7.8 km/s, ad the lower oe is Baratol with detoatio velocity 4.7 km/s. The propagatig process of detoatio wave i the les has bee simulated umerically with the code LS3D, the space step ad time step are 0.4 mm ad 0.00 µs respectively. legth 16 mm each is iitiated at two poit whose coordiates are (4,8,0) ad (1,8,0) respectively. Assumed that the radius of both iitial spherical detoatio waves is 3.5 mm. The propagatig process of the two spherical detoatio waves i the explosive cube has bee simulated umerically. The space step ad time step i calculatio are 0. mm ad µs respectively. The D ( κ ) relatio D = 8.04 (1 0.16κ ) is used The detoatio frots are show i Figure 7, ad the evolutio from a divergig wave to a plaar oe ca be displayed distictly. z / mm x / mm FIG.8 THE INTERACTION OF TWO SPHERICAL DETONATION WAVES ( t = 0.56 µs) FIG.7. DETONATION WAVE PROPAGATING IN THE EXPLOSIVE PLANE WAVE LENS ( t = 0.56 µs) 3.3. Iteractio of two-poit iitiated detoatio waves A comp. B explosive cube with side Whe two diverget spherical detoatio waves collide, regular reflectio is first observed, ad subsequetl irregular reflectio develops. The mach stem is the bridge that develops betwee the spheres. It is very difficult to simulate the iteractig process of two spherical detoatio waves i three-dimesioal space

8 umerically. I code LS3D, the iteractig process of two spherical detoatio waves is simplified as followig: whe the agle formed by the ormal of the two spherical frot at the iteractio poit meet the mach reflectio coditio, the the propagatio velocity of detoatio frot at the iteractio poit is adjusted to higher oe. Else, o chage is made. The calculatio results are show i Figure 8. The two detoatio waves itersect, the merges, ad evetually evolves a flat wave gradually. 4 CONCLUSIONS All the calculated results show that: the code LS3D ca be used i calculatig the detoatio wave propagatio i explosive charges iitiated i differet way. It ca deal with differet shaped ad compoet multi-medium explosive charges. The validity of calculated results depeds o the accurate frot curvature depedet detoatio velocity ad the assigmet of edge agle betwee explosive with air or medium. The measuremet of these key parameters is a importat study aspect of detoatio wave propagatio. It is very difficult ad ieffective to simulate the three-dimesioal detoatio wave propagatio by meas of commo fluid mechaics code especially for that i IHE. But the Level Set method is a practicable approach. The combiatio of LS method ad the ghost fluid method is promisig i egieerig calculatios of IHE devices. Refereces [1] Osher, S. ad Sethia, J. A., Frot Propagatio with Curvature-Depedet Speed: Algorithm Based o Hamilto-Jacobi Formulatios, J. Comput. Phys., 79(1), 1988, pp [] Adalsteisso, D. ad Sethia, J. A., A Fast Level Set Method for Propagatig Iterfaces, J. Comp. Phys., 118(), pp [3] Su Chegwei, Gao We et al, The Geeralized Geometrical Optics Model for the Detoatio Shock Dyamics, Proc. of 4th Iteratioal Symposium o High Dyamic Pressure, [4] Aslam, T. D., Bdzil, J. B. ad Stewart, D. S., Level Set Methods Applied to Modelig Detoatio Shock Dyamics, J. Comput. Phys., 16,1996, pp [5] We Shaggag, Zhao Feg ad Su Chegwei, Experimetal Study o Multi-Poit Iitiated Detoatio Waves i Isesitive Explosive JB9014, to be published. [6] Fag Qig ad Wei Yuzhag, The Propagatio Behavior of Diverget Detoatio Wave i Plastic-Boded Explosive TATB, private commuicatio.

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