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 hydrodynamics (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 demonstrate 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 modifications 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 velocity 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 interpolation 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]

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

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.

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 1 , P < pK , 2 2 Œ1 P aK D r1 pK ˆ 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 . RcR C LcL Then SPH scheme is translated: XN i D mjWij,(9) jD1

du XN 2P i D m r W , (10) dt j i j jD1 i j

de XN P i D 2 m .u U/r W . (11) dt j i i j jD1 i j

3.2. Construction of the Riemann problem To construct Riemann problem, we start from the equations that are written in conservation form, @u @f.u/ C D 0. @t @x n n n In the grid methods, we use discrete function u Dfuj , j D 1, 2, ....g to construct an appropriate reconstructing function R.x, u /, which satisfying the following condition: Z xjC1=2 n 1 n n Rj D R.x, u /dx D uj . x xj1=2 Then we take R.x, un/ as an initial value and construct local Riemann problem’s exact solution in the middle of conjoint grid at t. Thus, all of exact solutions in local area construct a globe solution at tn t tnC1. Finally, the mean value of Riemann solution at n C 1is nC1 nC1 regarded as discrete function u Dfuj , j D 1, 2, ....g. This is a computational circle of Godunov scheme [24]. At the beginning, Godunov took subsection constant as reconstructing function:

n n n n n u .x/ D R.x, u / D uj x 2 .xj1=2, xjC1=2/, This scheme’s numerical solution has no oscillation, but resolving power in disconnection is low. To increase precision, Van Leer brought forward MUSCL scheme [19, 21] that replaces constant reconstructing function by linear interpolation and has two-order precision. Based on Godunov scheme and MUSCL scheme, in SPH of meshless, we introduce two modes to decide interaction between one particle and each of its neighbors by interparticle contact algorithms(suppose i < j):

1. Choosing two particles’ state to decide initial state of Riemann problem

UL D ui, UR D uj, (12) but it only has one-order precision. 2. By a piecewise linear interpolation to decide initial state of Riemann problem, 1 1 U D u C ıu , U D u ıu , (13) L i 2 i R j 2 j

where the slope ıui D minmod.uiC1 ui, ui ui1/. If the distribution of particles is dense, we can substitute i C m, i m for i C 1, i 1, m is an integer that is larger than 1. We call (9)–(11) and (13) MUSCL–SPH scheme.

The construction of the slope in (13) is different from [13–18]. In [13], author deﬁned the linear distribution of physical variable f.x/ in both the i-th region and j-th region, and the gradient of f.x/ in each region was assigned by the SPH summation, then the initial values of Riemann problem was the average values of each domain of dependence. Iwasaki extent it to magnetohydrodynamics problem in [16], for simplicity, computed the slope only with the information of particle i and j, moreover, the interface location s of Riemann problem was adopted 0. A technique to apply slope-limiting was presented in [18], and the equivalent of the local gradient at three adjacent grid cells is deﬁned by projecting the SPH approximation of the gradient along the vector of connectivity. In the expression of (13), the initial value in each particle’s region can be simply assigned by the neighbors particle, and j particle is not always the most nearest to i particle in the i–j line. Moreover, when some physical problems need massive particles to compute, the advantage of less calculation is shown in this method.

3.3. Choice of the neighbors in Monotone Upwind Scheme of Conservation Law –smoothed particles hydrodynamics scheme

The realization of MUSCL–SPH scheme in 2D is more difﬁcult than that in 1D, the reason is that for the term ıui in the expression of (13), i C 1andi 1 are not easy to decide. In a plane, the particles are out-of-order distributed and move ceaselessly with the time. According to the meaning of the interparticle contact algorithms, we present a way to choose the neighbors of particle in 2D.

Figure 1. Choice of the neighbors in Monotone Upwind Scheme of Conservation Law–smoothed particles hydrodynamics scheme.

First of all, we consider the interaction of i particle and j particle, which are in a line that connects with them, illustrated Figure 1. Secondly, i C 1particleandi 1 particle, which are the neighbors of i particle, will be searched near the i–j line. If there is no particle between i and j,thenj particle will be chosen as i C 1 particle. Lastly, the same process will be performed to every particle in the whole computational domain. Thus, UL and UR of the expression (13) can be determined on 2D.

4. Numerical tests

4.1. Sod’s problem(1D) At ﬁrst, we consider a shock tube problem for a perfect gas that considered by Sod [22]. In this test, a diaphragm is placed at x D 0, which separates two regions of constant density and pressure. The initial conditions are

L D 1, pL D 1, eL D 2.5 x < 0,

R D 0.125, pR D 0.1, eR D 2.0 x > 0. In the equation of state, D 1.4. We take 40 particles in the low-density region and 320 particles in the other region. At t D 0.2, the results for a typical calculation with the artificial viscosity(8) are compared with the exact solution in Figure 2, and the results for a calculation with scheme .12/ and .13/, respectively, are compared with the exact solution in Figures 3 and 4. The results show that the great improvement of the pressure profile is achieved at the contact discontinuity. The scheme with the artificial viscosity has oscillations in the contact surface, and when we take .12/, the oscillations decrease in evidence and the resolving power of shock wave is raised, but the rarefaction wave is rather stretched. The rarefaction wave and shock wave are improved greatly when .13/ is replaced by .12/. In [15], its rarefaction is stretched. Comparing Figure 3 with Figure 4, we know that MUSCL–SPH scheme has higher accuracy than the scheme in [15].

4.2. Noh’s problem(2D) The uniform implosion of an ideal gas was conceived by Noh [23] as a stringent test case for shock capturing codes. The problem consists of a D 5=3 gas of ﬁnite uniform density and zero pressure moving radially inward at constant speed. The initial conditions for cylindrical mass distribution of unit radius are as follows:

1 1

0.9 0.9

0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 Velocity

0.4 Pressure 0.4 0.3 0.2 0.3 0.1 0.2 0 0.1 -0.5 0 0.5 -0.5 0 0.5 X X

Figure 2. Velocity and pressure ﬁgures with smoothed particles hydrodynamics formulations (5)–(7) and artiﬁcial viscosity (8) at t D 0.2. Solid line: exact solution, Dot line: numerical solution.

1 1

0.9 0.9

0.8 0.8 0.7 0.7 0.6 0.6 0.5 Velocity

Pressure 0.5 0.4 0.4 0.3 0.2 0.3 0.1 0.2 0 0.1 -0.5 0 0.5 -0.5 0 0.5 X X

Figure 3. Velocity and pressure ﬁgures with smoothed particles hydrodynamics formulations (9)–(11) and (12) at t D 0.2. Solid line: exact solution, Dot line: numerical solution.

1 1

0.9 0.9

0.8 0.8 0.7 0.7 0.6 0.6 0.5

Pressure 0.5 Velocity 0.4 0.4 0.3 0.2 0.3 0.1 0.2 0 0.1 -0.5 0 0.5 -0.5 0 0.5 X X

Figure 4. Velocity and pressure ﬁgures with Monotone Upwind Scheme of Conservation Law–smoothed particles hydrodynamics formulations (9)–(11) and (13) at t D 0.2. Solid: exact solution, Dot line: numerical solution.

den 15.9991 0.25 14.2799 13.4203 12.5607 11.7011 10.8415 9.98184 9.12223 8.26262

Y 0 7.40301 6.5434 5.68378 4.82417 3.96456 2.2453

-0.25

-0.2 00.20.4 X

Figure 5. Contour maps of density with Monotone Upwind Scheme of Conservation Law–smoothed particles hydrodynamics formulations (9)–(11) and (13) at t D 0.6.

D 1, p D 0, u D1, e D 0.

For cylindrical geometry, Noh found the analytical solution to be a shock moving radially outward with speed 1/3 and postshock and preshock values given by D 16, p D 16=3, u D 0, e D 1=2and D .1Ct=r/, p D 0, u D1, e D 0., respectively. At the shock position, the value of the preshock density is exactly 4. We use 5400 particles to simulate this problem. At the beginning, the particles is uniform. In Figure 5, the contour maps of the density with MUSCL–SPH formulations (9)–(11) and (13) at t D 0.6 are showed, we can see that it is symmetrical and the boundary arrives at r D 0.4 exactly. For comparing the numerical solutions with the exact solutions well, we mapped the density and the pressure in a radius direction. In Figure 6, the numerical solutions and exact solutions of density and pressure at t D 0.6 are plotted. We can see that the density is lower than 16. This phenomenon was called ‘wall heating’ [23]. In [24], this problem was computed by SPH too.

4.3. Shock–bubble interaction problem(2D) In this section, we examine an air shock collapse of a helium bubble [25]. Consider a domain Œ0, 325 Œ44.5, 44.5, bubble’s center and shock wave are at .175, 0/ and x D 225, respectively, bubble’s radius is 25, shock wave spreads left, as of Figure 7(a). The upper and lower boundary conditions are reﬂections for rigid wall boundaries. The left and right boundary conditions are free surfaces. There are 7290 particles that are distributed uniformly in the computational domain at the initial time. The nondimensionalized initial conditions are D 1.3764, p D 1.5698, u D0.394, v D 0, D 1.4, x > 225,

D 1, p D 1, u D 0, v D 0, D 1.4, x

D 0.138, p D 1, u D 0, v D 0, D 5=3, .x 175/2 C y2 < 625. Figure 7(b) is the particles’ distributions using MUSCL–SPH formulations (9)–(11) and (13) at t D 150. From ﬁgures, we can see this method can simulate the shock wave and contact discontinuities well.

(a) 16 15 14 13 12 11 10 9

Density 8 7 6 5 4 3 2 00.10.20.3 0.4 r

(b) 6

3 Pressure

0 0 0.1 0.2 0.3 0.4 r

Figure 6. Density and pressure ﬁgures with Monotone Upwind Scheme of Conservation Law–smoothed particles hydrodynamics formulations (9)–(11) and (13) at t D 0.6. Solid line: exact solution; dashed line: numerical solution.

(a) (b) 150 150

100 100

50 50 Y 0 Y 0

-50 -50

-100 -100

-150 -150 0 100 200 300 0 100 200 300 X X

Figure 7. (a) Initial physical conditions ﬁgures and (b) particles’ distributions ﬁgures with Monotone Upwind Scheme of Conservation Law–smoothed particles hydrodynamics formulations (9)–(11) and (13) at t D 150.

5. Conclusions

In this paper, based on the MUSCL scheme in grid methods, we adopt interparticle contact algorithm and construct MUSCL–SPH meshless reformulation, which has two-order accuracy. The algorithm is employed to simulate shock waves. Finally, we compare its results with the analytical solutions of the 1D Sod shock tube problem and 2D Noh problem, which shows that the shock is much better resolved.

Acknowledgement

Supported by the National Natural Science Foundation of China (Grant Nos.11201033,91230205,11372051).

References 1. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996; 139(1):3–47. 2. Li SF, Liu WK. Meshfree Particle Methods. Springer: Berlin, 2004. 3. Liu MB, Liu GR. Smoothed particle hydrodynamics(SPH): an overview and recent developments. Archives of Computational Methods in Engineering 2010; 17:25–76. 4. Lucy L.B. A numerical approach to the testing of the fission hypothesis. Astron. J. 1977; 82(12):1013–1024. 5. Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and applications to non-spherical stars. Monthly Notices of the Royel Astronomical Society 1977; 18:375–389. 6. Monaghan JJ. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics 1992; 30:543–574. 7. Swegle JW, Hicks DL, Attaway SW. Smoothed particle hydrodynamics stability analysis. Journal of Computational Physics 1995; 116(1):123–134. 8. Morris JP. Analysis of smoothed particle hydrodynamics with applications, Ph.D. thesis, Monash University, 1996. 9. Liu GR, Liu MB. Smoothed Particle Hydrodynamics. Word Scientific Publishing Co.Pte.Ltd: Singapore, 2003. 10. Monaghan JJ. Why particle methods work. SIAM Journal on Scientific and Statistical Computing 1982; 3(4):422–433. 11. Liu WK, Chen Y. Wavelet and multiple-scale reproducing kernel methods. International Journal for Numerical Methods in Fluids 1995; 21:901–931. 12. Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids 2005; 20(8):1081–1106. 13. Inutsuka S. Reformulation of smoothed particle hydrodynamics with Riemann Solver. Journal of Computational Physics 2002; 179(1):238–267. 14. Monaghan JJ. SPH and Riemann solver. Journal of Computational Physics 1997; 136(1):298–307. 15. Parshikov AN, Medin SA, Loukashenko II, Milekhin VA. Improvements in SPH method by means of interparticle contact algorithms and analysis of perforation tests at moderate projectile velocities. International Journal of Impact Engineering 2000; 24:779–796. 16. Iwasaki K, Inutsuka S. Smoothed particle magnetohydrodynamics with a Riemann solver and the method of characteristics. MNRAS 2011; 418: 1668–1690. 17. Cha S, Inutsuka S, Nayakshin S. Kelvin–Helmholtz instabilities with Godunov smoothed particle hydrodynamics. MNRAS 2010; 403:1165–1174. 18. Murante G, Borgani S, Brunino R, Cha S. Hydrodynamic simulations with the Godunov smoothed particle hydrodynamics. MNRAS 2011; 417: 136–153. 19. Van LB. Towards the ultimate conservation difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics 1979; 32:101–135. 20. Monaghan JJ. Smoothed particle hydrodynamics. Reports on Progress in Physics 2005; 68:1703–1759. 21. Leveque RJ. Numerical Methods for Conservation Laws. Birkh äuser Verlag, 1992. 22. Sod GA. Review a survey of several finite difference method of system of nonlinear hyperbolic conservation laws. Journal of Computational Physics 1978; 27(1):1–31. 23. Noh WF. Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. Journal of Computational Physics 1978; 72(1): 78–120. 24. Libersky LD, Petschek AG, Carney TC, Hipp JR, Allahdadi FA. High strain Lagrangian hydrodynamics. Journal of Computational Physics 1993; 109(1): 67–75. 25. Fedkiw RP, Aslam T, Merriman B, Osher S. A non-oscillatory Eulerian approach to interfaces in multimaterial flows(the ghost fluid method). Journal of Computational Physics 1999; 152(2):457–492.