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A MUSCL Smoothed Particles Hydrodynamics for Compressible Multi-Material Flows

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A MUSCL Smoothed Particles Hydrodynamics for Compressible Multi-Material Flows Research Article Received 1 October 2014 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.3551 MOS subject classification: 74s30 A MUSCL smoothed particles hydrodynamics for compressible multi-material flows Xiaoyan Hu*†, Song Jiang and Ruili Wang Communicated by S. Wise By incorporating the Monotone Upwind Scheme of Conservation Law (MUSCL) scheme into the smoothed particles hydro- dynamics (SPH) method and making use of an interparticle contact algorithm, we present a MUSCL–SPH scheme of second order for multifluid computations, which extends the Riemann-solved-based SPH method. The numerical tests demon- strate high accuracy and resolution of the scheme for both shocks, contact discontinuities, and rarefaction waves in the one-dimensional shock tube problem. For the two-dimensional cylindrical Noh and shock-bubble interaction problems, the MUSCL–SPH scheme can resolve shocks well. Copyright © 2015 John Wiley & Sons, Ltd. Keywords: meshless methods; SPH method; MUSCL scheme; Riemann solver 1. Introduction In the last several decades, meshless methods have attracted a lot of attention both from engineers and mathematicians because of their gridless and flexibility, and much progress has been made in the development and applications of the meshless methods (e.g., [1–3]). One of the simplest and most well-known meshless methods is the so-called smoothed particles hydrodynamics (SPH) method, which was originally devised to simulate a wide variety of problems in astrophysics in 1977 [4, 5]. Because of its simplicity and robust- ness, it has rapidly become a useful tool for applications in numerous other areas, such as elasticity, fracture, and fluid dynamics [6]. The application of SPH to a wide range of problems has led to significant extensions and improvements of the original SPH method. However, some drawbacks of the SPH method have been realized. For example, Swegle, et al. in 1995 identified the tensile instabil- ity problem associated with the SPH method that is important for materials with strength [7], while in 1996, Morris [8] observed the particle inconsistency of the SPH method, which can lead to poor accuracy in simulations. Moreover, spurious oscillations of the SPH method across discontinuities in fluid dynamics simulations are also well-known. To circumvent such drawbacks, different modifica- tions or corrections have been tried to restore the consistency and to improve the accuracy of the SPH method over the past years [9]. These modifications lead to various versions of the SPH method and corresponding formulations. Monaghan in 1982 proposed a sym- metrization formulation that was reported to have better effects [10]. Liu and Chen introduced the reproducing kernel particle method that can result in better accuracy in the particle approximation [11, 12]. Another approach in extending the SPH method is to use Rie- mann solvers in the computation of discontinuities to improve accuracy and resolution and to eliminate spurious oscillations across the discontinuities, for example, [13–18]. Parshikov and Medin proposed a Riemann solver-based SPH method in 2000 [15], where they first solved the Riemann problem arising from the interaction between one particle and each of its neighbors, and then inserted the veloc- ity and pressure at the contact point determined by the Riemann solver into the SPH formulation. As pointed out in [15], this method is only of first order and resolves shock waves and the contact discontinuities quite well in the one-dimensional shock tube problem. However, it is easy to observe from the numerical tests in [15] that rarefaction waves are not so good resolved (rather stretched). In 2002, Inutsuka used the exact Riemann solver into the strictly conservative particle method and introduced a piecewise linear inter- polation of the physical variables to determine the initial state of Riemann problem [13]. Iwasaki and Inutsuka applied this method to magnetohydrodynamics in [16]. Recently, based on the method that was developed by Inutsuka, Cha et al provided a much improved description of KH instabilities through two-dimensional tests [17], and Murante also used this method to simulate the hydrodynamics in 3D [18]. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China * Correspondence to: Xiaoyan Hu, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China. † E-mail: [email protected] Copyright © 2015 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2015 X. HU, S. JIANG AND R. WANG The aim of the present paper is to construct a shock-capturing SPH scheme of second order by using an approximate Riemann solver, which works well for both strong, weak shocks, and rarefaction waves. Our scheme is based on appropriate incorporating the MUSCL(Monotone Upwind Scheme of Conservation Law) scheme [19] into the SPH method, resulting in a so-called second-order MUSCL–SPH scheme, which is different from [13–18]. The numerical tests in Section 6 demonstrate high accuracy and resolution of our MUSCL–SPH scheme for both shocks, contact discontinuities, and rarefaction waves. The paper is organized as follows: In Section 2, we introduce the SPH formulation and the choice of parameters in formulation. In Section 2.3, time integration and time step are presented. In Section 3, we give a description of interparticle contact algorithms and construction of the Riemann problem. In Section 4, several numerical tests are presented, which demonstrate the high accuracy of the present scheme. Finally we conclude in section 5. 2. Smoothed particles hydrodynamics equations 2.1. Governing equations We consider the following set of equations for non-radiating inviscid fluid: d Dru,(1) dt du 1 D rp,(2) dt d.e C 1 u2/ 1 2 D rpu,(3) dt with the equation of state is p D . 1/e,(4) where , u, p,ande are the material density, velocity, pressure, and specific internal energy, respectively; is the specific heat ratio. 2.2. Smoothed particles hydrodynamics formulation There is a variety of the SPH versions of fluid dynamics equations. For the conservation of the total energy, we select the following equations as the SPH approximations [13]: XN i D mjWij,(5) jD1 du XN p C p i D m i j r W ,(6) dt j i j jD1 i j de 1 XN p C p i D m i j .u u /r W .(7) dt 2 j i j i j jD1 i j where Wij is the cubic spline kernel function. In the kernel function, the smoothing length h is very important, which has direct influence on the efficiency of the computation and the accuracy of the solution. We choose  à 1 mi d hi D , A C i where , A are suitable constants, d is the number of dimensions [20]. Thus, an upper bound on hi when becomes very small is desirable to prevent strong interactions between a very low and a very high density region, and h varies in time and space. To simulate shocks, this Monghan type qij is the most widely used artificial viscosity in the pressure terms of SPH literatures so far [6]. The detailed formulation is as follows: ( 2 ˛cijijCˇij , uij rij < 0, qij D ij (8) 0, uij rij 0, hijuij.rij/ 2 2 where ij D 2 2 , uij D ui uj, rij D ri rj, D 0.01h , hij D .hi C hj/=2, cij D .ci C cj/=2, ij D .i C j/=2, ˛ and ˇ are constants. .rij/ C” ij ci is sound velocity in the ri. Copyright © 2015 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2015 X. HU, S. JIANG AND R. WANG 2.3. Time integration Just as other explicit hydrodynamic methods, the discrete SPH equations can be integrated with standard methods, such as the second- order accurate Leap-Frog, predictor-corrector and Runge–Kutta schemes, and so on. In this paper ,we use predictor-corrector schemes. The explicit time integration schemes are subject to the Courant–Friedrichs–Lewy condition for stability. In this paper, time step is t D CN min.t1, t2/ ,CN and t1 are the defined constants. hi t2 D min . i ci C 1.2.˛ci C ˇmaxjjjij/ 3. Riemann solver 3.1. Interparticle contact algorithms In the computational domain, each basic particle i exchanges momentum and energy with surrounding particle j within the interaction distance. Because in the contact point O, the interaction of particles is considered to be equivalent to that at the contact surface in continuous fluid. We can decide a state using Riemann solver, then its velocity and pressure take place of mean values between velocities and pressure of basic and surrounding particles. It should be noticed that the artificial viscosity is not necessary during computing. This is the interparticle contact algorithms. Assuming uL, pL, aL and uR, pR, aR are both sides’ states of the contact discontinuity. We can get velocity U and pressure P in contact point O by Riemann solver. The interaction of particles is substituted by U and P: 1 .p.i/ C p. j// ! P, 2 1 .u.i/ C u. j// ! U, 2 where U and P can be decided by iterativeness in following formula: a u C a u C p p U D L L R R L R , aR C aL a p C a p C .u u /a a P D L R R L L R L R . aR C aL This 8 ˆ 1 P ˆ pL < K cK Á 1 , P < pK , 2 2 Œ1 P aK D r1 pK ˆ Á ˆ C1 P : K cK 1 C 1, P > pK , 2 pK where K D L, R. If the ratio of initial pressure is not large, we can use Riemann approximation to improve the efficiency of the computation: c u C c u .p p / U D R R R L L L R L , RcR C LcL c p C c p c c .u u / P D R R L L L R R R L L R L .
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