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New Numerical Algorithms in SUPER CE/SE and Their Applications in Explosion Mechanics

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New Numerical Algorithms in SUPER CE/SE and Their Applications in Explosion Mechanics SCIENCE CHINA Physics, Mechanics & Astronomy • Research Paper • February 2010 Vol. 53 No.2: 237−243 Special Topics: Computational Explosion Mechanics doi: 10.1007/s11433-009-0266-z New numerical algorithms in SUPER CE/SE and their applications in explosion mechanics WANG Gang1,2, WANG JingTao1 & LIU KaiXin1,3* 1 LTCS and College of Engineering, Peking University, Beijing 100871, China; 2 LHD, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China; 3 Center for Applied Physics and Technology, Peking University, Beijing 100871, China Received June 28, 2009; accepted November 11, 2009 The new numerical algorithms in SUPER/CESE and their applications in explosion mechanics are studied. The researched al- gorithms and models include an improved CE/SE (space-time Conservation Element and Solution Element) method, a local hybrid particle level set method, three chemical reaction models and a two-fluid model. Problems of shock wave reflection over wedges, explosive welding, cellular structure of gaseous detonations and two-phase detonations in the gas-droplet system are simulated by using the above-mentioned algorithms and models. The numerical results reveal that the adopted algorithms have many advantages such as high numerical accuracy, wide application field and good compatibility. The numerical algo- rithms presented in this paper may be applied to the numerical research of explosion mechanics. new algorithm, SUPER CE/SE, numerical simulation, explosion mechanics The research of explosion mechanics, usually involving jing Institute of Applied Physics and Computational many disciplines such as compressible flows, incompressi- Mathematics) and MMIC [2] by BIT (Beijing Institute of ble flows, elastic-plastic flows, chemically reacting flows Technology). The present software packages of computa- and two-phase flows, contributes significantly to the field of tional explosion mechanics mainly adopt well-developed industry, aeronautics, and military engineering. Numerical numerical algorithms and models, for example, VOF [3] simulation of explosion mechanics needs not only accurate (Volume of Fluid) method for interface capturing and tra- numerical schemes and interface capturing methods, but jectory method [4] for particle tracing. also a perfect combination of the selected numerical algo- This paper has studied the new numerical algorithms in rithms and the physical and chemical models. SUPER CE/SE and their applications in explosion mechan- The imported commercial software, such as LS-DYNA ics. SUPER CE/SE uses an improved CE/SE (space-time and AUTODYN, currently plays a dominant role in the ap- Conservation Element and Solution Element) method [5,6] plication of explosion mechanics in China. Further devel- to discrete governing equations, a local hybrid particle level opment based on the imported commercial software is quite set method to capture the interface, and a two-fluid model to difficult, because the kernel modules of the software are treat two-phase flows. SUPER CE/SE is also able to adopt always held as commercial secrets. The software of compu- common chemical reaction models, such as two-step, de- tational explosion mechanics in China has undergone a tailed and Sichel’s two-step chemical reaction models. With step-by-step development. The well-established software the above algorithms and models, problems of shock wave packages include METH [1] developed by BIAPCM (Bei- reflection over wedges, explosive welding, cellular structure of gaseous detonations and two-phase detonations in gas-droplet system are investigated. Comparisons and dis- *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2010 phys.scichina.com www.springerlink.com 238 WANG Gang, et al. Sci China Phys Mech Astron February (2010) Vol. 53 No. 2 cussions on the numerical results reveal the applicability of advantages such as simple mesh structure, clear physics the above algorithms in computational explosion mechan- concept, easy to extend to three-dimensional cases and ics. convenient high-accuracy Taylor expansions. The physical variables Q, E and F at every mesh point (t, x, y)∈SE(j, k, n) are approximated by their second-order Taylor expansions 1 Numerical algorithms and models Q*, E* and F* as * Computational explosion mechanics refers to many areas of M (,δδδxyt ,)P′ study including computational mechanics, fluid mechanics, =+MMPP′′()xytδδδxyt + () M P ′ + () M P ′ solid mechanics, physics and chemistry. In this section, we 111222 mainly introduce the basic algorithms in SUPER CE/SE +++()()()()()()MMMxx PPP′′′δ xytyy δδtt including the improved CE/SE scheme, the local hybrid 222 2 (1) particle level set method, the common chemical reaction +++()MMMxy PPP′′′δδxy ()yt δδ yt ()xt δδ xt , models and the two-fluid model. The models of compressi- ble flow, incompressible flow and elastic-plastic flow can where M=Q, E, F, δ xxx= − P′ , δ yyy=−P′ , δ ttt= − P′ . be found in ref. [7,8]. A second-order accuracy CE/SE scheme can be constructed by eq. (1). 1.1 Improved CE/SE method 1.2 Local hybrid particle level set method The CE/SE method [9] originally proposed by Chang in 1995 is a new numerical framework for solving hyperbolic How to accurately trace the interface of materials is a chal- conservation laws. The CE/SE method substantially differs lenge confronted in the Eulerian method for multi-material from the existing well-established CFD methods in that it simulations. The Level Set method [12], originally proposed contains many excellent features such as a unified treatment by Osher and Sethian in 1988, is an effective Eulerian algo- of space and time, satisfying both local and global flux rithm for interface capturing. The level set method essen- conservation in space and time and simple treatments for tially refers to a level set function whose zero isoline ex- boundary conditions. As a result of its simplicity, generality actly determines the material interface is introduced to de- and accuracy, the CE/SE method has been successfully ap- scribe the development of the material interface. The level plied to scientific exploration and engineering. However set method can deal with very complicated topologic researchers have found defects in the original CE/SE changes of interfaces, because it is not necessary to recon- schemes, such as the complication of mesh structure, the struct interface fronts just like some “wave front tracing” absence of high-order accuracy schemes and the difficulty methods. to extend to three-dimensional situation. To solve these Adalsteinsson [13] proposed a fast local algorithm for the problems, Zhang et al. [10], Zhang et al. [11] and Liu et al. level set method which confines the solving region to a [5,6] have made some progress in this area. narrow band surrounding the interface. This algorithm has The improved CE/SE method used in SUPER CE/SE the time complexity of O(Nw), where N is the number of constructs SEs and CEs shown in Figure 1 on rectangular grid points and w is the width of the narrow band. Applying meshes and is extended to second-order accuracy. This con- the idea of fast local algorithm, Peng [14] proposed a fast struction of SE and CE on rectangular meshes has many level set method based on PDE (Partial Difference Equa- Figure 1 Mesh construction of the improved CE/SE method. (a) Solution element SE(P′); (b) conservation element CE(P′). WANG Gang, et al. Sci China Phys Mech Astron February (2010) Vol. 53 No. 2 239 tions), which has merits such as easy to be implemented and 1.4 Two-fluid model time complexity of O(N). In this paper, we assume the gas-droplet system has the fol- Enright [15] introduced non-mass Lagrangian particles to lowing characteristics: the flow is unsteady; the initial tem- the level set method to solve the problem of mass losing. perature of the gas and the droplets is the same; the radii of Based on this idea, he proposed a hybrid particle Level Set droplets are uniform; droplets are homogenously distributed method in 2002. initially; the phase of droplets is considered a continuous medium; the shape of droplets always remains spherical 1.3 Common chemical reaction models even in the process of separation, and evaporation; the tem- perature distribution in every droplet is uniform; the total The two-step reaction model simplifies a complicated volume of droplets can be ignored compared with the vol- chemical reaction into an induction reaction and an exo- ume of gas; the interactions between droplets are ignored; thermic reaction. The rates of a (progress parameter for in- when the fuel reaches the gaseous state, chemical reactions duction reaction) and β (progress parameter for exothermic occur and accomplish immediately; the chemical energy is reaction), ω and ω , are given as below [16]: α β absorbed only by gas, which is considered ideal. With these assumptions, the governing equations for gas ⎧ ⎛⎞Eα ⎪ωραα=−k exp⎜⎟ − , phase can be expressed as ⎪ ⎝⎠RT ⎪ 0 (α > 0), ∂∂∂QEF ⎧ S ⎪ ⎪ + +=, (4) ⎪ ⎡ E ∂∂∂txy ⎨ ⎪ +− 22 ⎛⎞β (2) ωωωββββ=+=−⎪ kp⎢ βexp⎜⎟ − ⎪ ⎢ RT ⎪ ⎨⎣⎝⎠ where ⎪ ⎪ EQ+ ⎤ Q = (ρ, ρu, ρv, E), ⎪ 2 ⎛⎞β 2 ⎪ −−(1βα ) exp⎜⎟ −⎥ (0), ≤ E = ( ρu, ρu +p, ρuv, (E+p)u), ⎪ RT 2 ⎩⎪ ⎩ ⎝⎠⎦⎥ F = (ρv, ρuv, ρv +p, (E+p)v), S = (Ip, −Fx+upIp, −Fy+vpIp, −(upFx+ vpFy) where ρ is the mass density, p the pressure, T the temperature, 2 2 +((up +up )/2+qr)Ip). R the gas constant, Q the heat release parameter, kα and kβ The governing equations for droplet phase can be expressed the constants of reaction rates, and Eα and Eβ the activation as energies. The two-step reaction model does not need large ∂∂∂QEF computing resource because it only takes the heat release pp++=p S , (5) parameter Q into account. However, the two-step reaction ∂∂∂txyp model can describe the essential characteristics of detona- tions. where The detailed chemical reaction model is extensively used Qp = (ρp, ρpup, ρpvp, N), E = (ρ u , ρ u 2, ρ u v Nu ), to describe the transformation of reactants into products at p p p p p p p p, p F = (ρ v , ρ u v ρ v 2, Nv ), the molecular level through a large number of elementary p p p p p p, p p p S = −(I , −F +u I , −F +v I , 0).
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