Research Article Development of a Particle Interaction Kernel Function in MPS Method for Simulating Incompressible Free Surface Flow

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1 Journal of Applied Mathematics Volume 2, Article ID , 6 pages doi:.55/2/ Research Article Development of a Particle Interaction Kernel Function in MPS Method for Simulating Incompressible Free Surface Flow Ton W. H. Sheu,, 2, 3 Chenpeng Chiao, and Chinlong Huang Department of Engineering Science and Ocean Engineering, National Taiwan Universit, Taipei, 67, Taiwan 2 Taida Institute of Mathematical Science (TIMS), National Taiwan Universit, Taipei, 67, Taiwan 3 Center for Quantum Science and Engineering (CQSE), National Taiwan Universit, Taipei, 67, Taiwan Correspondence should be addressed to Ton W. H. Sheu, twhsheu@ntu.edu.tw Received 3 December 2; Accepted 2 Januar 2 Academic Editor: Zhangin Chen Copright q 2 Ton W. H. Sheu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. We aimed to derive a kernel function that accounts for the interaction among moving particles within the framework of particle method. To predict a computationall more accurate moving particle solution for the Navier-Stokes equations, kernel function is a ke to success in the development of interaction model. Since the smoothed quantit of a scalar or a vector at a spatial location is mathematicall identical to its collocated value provided that the kernel function is chosen to be the Dirac delta function, our guideline is to derive the kernel function that is closer to the delta function as much as possible. The proposed particle interaction model using the newl developed kernel function will be validated through the two investigated Navier-Stokes problems which have either the semianaltical or the benchmark solutions.. Introduction Numerical methods that are now available for performing flow simulations can be divided into the Eulerian and Lagrangian two classes. In the Eulerian method, flow equations are solved at a fied mesh sstem. On the contrar, in the Lagrangian method, which can be subdivided into the mesh-based and meshfree two subgroups, either meshes or particles, on the other hand, are advected with the fluid flow. The consequence is that in the transport equations we no longer have a need to approimate the convection terms. Avoidance of the approimation of convective terms is now known as one of the apparent advantages of emploing the Lagrangian approach. Numerical errors of the cross-wind diffusion tpe can, as a result, be well eliminated. There eists another mathematicall rigorous arbitrar Lagrangian-Eulerian ALE method which involves the use of Lagrangian and Eulerian solution steps in the course of moving mesh.

2 2 Journal of Applied Mathematics In the literature, several well-known grid-based methods, such as the marker-andcell MAC 2, volume-of-fluid VOF 3, and level-set 4 methods, have been developed to predict free surface flows. There eisted also a major class of meshfree methods, which are known as the particle methods. Particle methods deal with the prescribed particles moving in the Lagrangian sense. As a result, the convection terms can be directl calculated without incurring an numerical diffusion error. Particle methods can be also separated into the Eulerian particle method, such as the particle-in-cell PIC method 5 and the Lagrangian particle methods, which include the smoothed particle hdrodnamics SPHs and moving particle semi-implicit MPS two classes of meshless methods. SPH method, introduced firstl b Luc 6 and Gingold and Monaghan 7 in 977, was developed mainl for the simulation of compressible fluid flows based on the interpolation theor through the introduced kernel function or smoothing kernel. SPH method was later etended to simulate also the incompressible free surface flow 8. Another gridless particle method, called the MPS, was more recentl developed to simulate the incompressible Navier-Stokes fluid flows 9. The motion of each particle in this method is calculated through the interaction of its neighboring particles b means of the kernel or weight function. This means that all the spatial derivatives can be calculated b the deterministic particle interaction without the need of generating meshes in the flow. This eplains wh MPS method is graduall becoming effective for use in simulating man practical problems involving the complicated geometr and comple phsics. For the flow problems with an inflow or an outflow boundar, this method was however found to have difficult of tracing a particle easil. In addition, MPS method requires an etra CPU time to search for all its neighboring points. The rest of this paper will be organized as follows. In Section 2 we present the Navier- Stokes and the passive scalar equations, which involve convection and diffusion flu terms. This is followed b presenting the particle interaction models used for the approimation of the gradient and Laplacian differential operators. In Section 4 emphasis will be placed on subject to free surfaces the kernel function used in the particle method. In Section 5, we will validate the proposed particle model b solving two Navier-Stokes problems. In Section 6, conclusions will be drawn based on the predicted results. 2. Working Equations In this stud the following Navier-Stokes equations for the motion of an incompressible fluid flow will be considered in a simpl connected domain Ω: Du Dt ρ p ν 2 u f, Dρ u, ρ Dt where Du/Dt u/ t u u.in 2., which is the momentum equation, ρ stands for the densit of the investigated flow, ν the kinematic viscosit, and f the gravitational acceleration vector.

3 Journal of Applied Mathematics 3 The following model equation will be considered in the development of kernel function since it is the ke equation in the simulation of momentum equations for the incompressible fluid flow: u φ ( ) φ 2 v μ φ 2 φ Both of the velocit components u and v and the fluid viscosit μ in 2.3 are assumed to be the constant values to facilitate the development of the proposed kernel function in Section Particle Interaction Models for Two Differential Operators Due to the difficulties of appling the grid-based methods to simulate comple flow phsics, particle-based methods become graduall popular in the simulation of incompressible Navier-Stokes equations subject either to a free surface or an interface. In particle methods, the differential operators for the mass and momentum conservation equations need to be replaced b their corresponding particle interaction operators. In other words, partial differential equations written in the continuous contet will be transformed to their corresponding discrete particle interaction equations so that the transport equations under investigation can be approimated b the chosen moving particles and their interaction. The degree of success depends on the kernel or weighting function for the particles that are distanced from each other b the user s prescribed finite length for the approimation of velocit vectors and pressure in the incompressible fluid flows around these particles. Consider a particle at which a phsical quantit f i is defined at a position r i.one can represent f i approimatel at each location as follows b virtue of the following kernel function w r ( ) f r w ( ri r ) i i f i w ( ri r ). 3. It is important to note that the smoothed quantit f i for f at r i turns out to be eactl identical to the local value of f i if the kernel function in 3. is the Dirac delta function. This implies that the chosen kernel function w r, which should be constrained b the integral constraint equation w r dv shown in 3., determines the prediction qualit of the V particle methods. One can find different kernel functions in. For a scalar φ at a location r j, its value can be epanded in Talor series with respect to the value of φ at r i as follows: ) φ j φ i φ ij (r j H.O.T. 3.2 B dropping the higher-order terms shown above, we are led to get the first-order approimated equation φ j φ i φ ij r j. B multipling the term r j on both hand

4 4 Journal of Applied Mathematics sides of the resulting simplified equation, we can get φ ij ( φj φ i ) ( r j ) r j rj. 3.3 Let φ ij be f. We then substitute it into 3. to get the following smoothed representation of φ, which is denoted b φ at the node ij, in a domain with d dimensions φ ij d φ j φ i ) ) (r n j w (r j. 3.4 j / i r j In the above, n denotes the particle number densit and it is defined as j / i w r j. Note that n i n is true onl under the incompressible flow condition. One can similarl derive the Laplacian operator for a scalar φ, which has been derived in 9 as follows 2 φ i 2d ( ) ( ) rj φj φ λn i w, 3.5 j / i where rj 2 ( ) rj r V λ i w dv ( ). 3.6 rj dv V w Note that V shown above denotes the volume ecluding of a small interval that contains the point at i. One can also emplo the Laplacian operator given below 2 φ ij 2d φ j φ i ( ) rj w r n i. 3.7 j / i r j The advantage of emploing particle methods becomes clear in that all the spatial derivatives can be calculated from the chosen kernel function without the need of generating meshes in the phsical domain. Much of the computational effort in the generation of a good qualit mesh for performing an accurate numerical simulation in a domain with moving boundaries is therefore avoided. Unlike the SPH particle method, calculations of the values for φ and 2 φ in 3.4, 3.5, and 3.7 involve onl the kernel function w r. Since the derivative of kernel function needs not to be calculated in the approimation of differential operators and 2, numerical oscillations, which ma be generated in the fied grid Eulerian approach for the cases with high solution gradients, can be completel avoided. As a result, one has the fleibilit to choose the kernel function that has a slope as steep as the Dirac delta function.

5 Journal of Applied Mathematics 5 4. Development of the Kernel Functions In view of 3.4 and , we know that the qualit of approimating the operators φ ij and 2 φ i depends entirel on the chosen kernel function w r and the number of the prescribed particles. Moreover, the chosen particle locations and the emploed kernel function will determine the subsequent particle location in moving particle methods. For this reason, the chosen kernel function will be derived below in detail. In the light of 3., we know that the Dirac delta function δ r, which is constrained b δ r dr, is an ideal kernel function. Such a function is, unfortunatel, not implementable in computational practice. We have therefore to resort to its corresponding smoothed delta function when carring out the simulations based on the particle method. The smoothed function is sometimes called as the nascent delta function δ ε r, which is defined as lim ε δ ε r δ r. In the literature, one can find other nascent delta functions such as the Gaussian function, Lorentz line function, impulse function, and sinc function. There eists also the other class of kernel functions which were developed irrelevantl to the nascent delta functions. Tpical eamples include the eponential, cubic spline, and quadratic spline functions proposed b Beltschko et al. 2 and the kernel functions proposed b Koshizuka and Oka in and Koshizuka et al. in Development of Kernel Function for the Pure Diffusion Equation In this stud a new kernel function will be rigorousl developed and it will be used to simulate the incompressible Navier-Stokes equations using the particle methods. The guidelines of developing the proposed kernel function will be given below. The kernel function w r under current development falls into the categor of nascent delta function. This implies lim re w r,r e δ r,, where r e is the radius of a small circle. The weight between two particles that have a distance r apart will be diminished as r r e. For the sake of accurac, the kernel function will be developed to retain the following constraint condition in the Dirac delta function w ( r,r e ) dr. 4. Development of the current kernel function starts with representing it in terms of r/r e as a b ( ) r c ( ) r 2 d ( ) r 3 e ( ) r 4 ; r r e, r e r e r e r e r e r e r e r e r e w r ; r e <r. 4.2 Derivation of w r is followed b imposing the four constraint conditions given b w r r e w/ r r re w/ r r and r/r e w r dr /2. These imposed conditions

6 6 Journal of Applied Mathematics.2.8 w(r/re) a r/r e b Figure : Plot of the developed kernel function. a cast in r/r e coordinate; b plotted in plane. 6d H S.W.L. Channel bottom d Figure 2: Schematic of the solitar wave propagation problem. enable us to derive the algebraic equations given below a b c d e, b 2c 3d 4e, a b 2 c 3 d 4 e 5 2, 4.3 b. Given the above four equations, one can then easil epress a, b, c, d in terms of e as a /5 e, b, c 3 6/5 e, d 2 32/5 e. B substituting the resulting kernel function into 3.7 for 2 φ, one can get the corresponding discrete equation for the Laplace equation 2 φ/ 2 2 φ/ 2 based on

7 Journal of Applied Mathematics Y 2 5 Sheu et al. [6] Present Figure 3: Comparison of the predicted run-up heights of the solitar wave on the right wall. Time the particle method [ ( )] w h 2w 2h h 2 { (φi,j φ i,j φ i,j φ i,j 4φ i ) w h 4.4 ( φ i,j φ i,j φ i,j φ i,j 4φ i ) w ( 2h )}. B performing the modified equation analsis on 4.4, we are led to get the following modified equation: ( ) 2 φ 2 φ w 2h 2 2 ( ) 4 φ w h 2w 2h 2 4 φ φ h ( ) w 2h ( 2 w h 2w ( 2h )) ( 6 φ φ 2 4 ) 6 φ φ h Let w 2h /4 w h the first term on the right hand side of 4.5 turns out to be zero due to 2 φ/ 2 2 φ/ 2.

8 8 Journal of Applied Mathematics 2 p: a 2 p: b 2 p: c Figure 4: Continued.

9 Journal of Applied Mathematics 9 2 p: d 2 p: e 2 p: f Figure 4: The predicted free surfaces and pressures at different times. a t.5; b t 2.; c t 4.; d t 6.; e t 8.; f t..

10 Journal of Applied Mathematics L L 5L Figure 5: Schematic of the dam break problem. Provided that r e 2h, where h denotes the grid size, we can easil know from the following nine-point equation that the resulting approimated equation for 2.3 investigated at the condition of μ has accurac of the fourth order. φ i,j φ i,j φ i,j φ i,j 4φ i,j 4φ i,j 4φ i,j 4φ i,j 2φ i,j 6h 2 ( O h 4). 4.6 Note that 4.6 is derived under the condition of ( 3a 2 2 ) ( b 2 ) c 2 4 ( 2 ) ( d ) e The resulting five free parameters can then be uniquel determined from and 4.7 as a / , b, c / , d 2/ , e / In summar, the kernel function developed for simulating the pure diffusion equation in this stud is given as follows: ( ) r e 52 r 2 ( ) r e r e 52 r r e r e w r 48 ( ) r 4 ; r r e, r e r e ; r e <r 4.8 under the condition given below ( ) w 2h 4 w h. 4.9

11 Journal of Applied Mathematics a b Sheu et al. [6] Present c d Figure 6: Continued.

12 2 Journal of Applied Mathematics Sheu et al. [6] Present e f Figure 6: Comparison of the predicted free surfaces b the MPS and the level set methods at different times for the case with Re 5. a t.; b t.5; c t.; d t.5; e t 2.; f t 2.5. Note that the kernel function w r r e /r given in 3 has an infinitel large magnitude as the value of r approaches zero. The newl developed kernel function shown in Figure, on the other hand, has its local maimum value at r. It is also worth to mention that the currentl developed kernel function applied to predict the pure diffusion equation satisfies the following constraint condition, which is also embedded in the delta function r/re r/r e w r dr Numerical Results 5.. Solitar Wave Propagation Stud of the solitar wave propagation in a rectangular channel has received a considerable attention since the resulting analsis can provide engineers with useful information about the wave loading on the structures 4. The problem under consideration is schematic in

13 Journal of Applied Mathematics a b Sheu et al. [6] Present c d Figure 7: Continued.

14 4 Journal of Applied Mathematics Sheu et al. [6] Present e f Figure 7: Comparison of the predicted free surfaces b the MPS and the level set methods at different times for the case with Re 5. a t.; b t.5; c t.; d t.5; e t 2.; lf t 2.5. Figure 2 where the undisturbed water height is d and the channel width is 6d. Above the undisturbed water level, the crest of the investigated solitar wave is given b 5 d H sech 2 3H 4d Here, H 2 and d denote the initial wave height and the still water depth, respectivel. Initiall, the velocities are given b u gd H d sech2 3H 4d, 3 v ( H 3gd d ) 3/2( ) sech 2 3H d 4d tanh 3H 3 4d, 3 5.2

15 Journal of Applied Mathematics 5 where g 9.8 is the gravitational acceleration. In the current stud, the initial distance between the neighbor particles is.8, and 3482 particles were used in the computation. For the wave propagating towards the right wall, the predicted run-up height on the right wall in Figure 3 is compared with the result given in 5. The computed wave pressures at different times are shown in Figure Dam Break Flow The investigated dambreak problem is schematic Figure 5, where L denotes the height and the width of the stillwater in the tank. The computed results were compared with the results in 6. Figures 6 a 6 f show the results for the fluid flow with Re 5. Figures 7 a 7 f are the results predicted in the case with Re. The red lines stand for the simulated free surfaces, and the are compared with those shown in Concluding Remarks The idea of developing the proposed particle interaction model is to develop the kernel function which accommodates the propert r/r e r/r e w r dr embedded in the smoothed Dirac delta function. To justif the proposed kernel functions for simulating the realistic flow, we investigate two well-known benchmark problems for the validation sake. The predicted results of the dam break problem and the solitar wave problem clearl show that the solution predicted b the particle interaction model using the proposed kernel function can capture well the motion of the fluid flow subject to free surface. Acknowledgments The financial support b the National Science Council under Grants NSC E MY2 and NSC M-2-22 and CQSE project 97R66-69 are gratefull acknowledged. References C. W. Hirt, A. A. Amsden, and J. L. Cook, An arbitrar Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Phsics, vol. 4, no. 3, pp , F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phsics of Fluids, vol. 8, no. 2, pp , C. W. Hirt and B. D. Nichols, Volume of fluid VOF method for the dnamics of free boundaries, Journal of Computational Phsics, vol. 39, no., pp , S. Osher and R. P. Fedkiw, Level set methods: an overview and some recent results, Journal of Computational Phsics, vol. 69, no. 2, pp , 2. 5 F. H. Harlow, A machine calculation method for hdrodnamic problems, Los Alamos Scientific Laborator Report LAMS-956, L. B. Luc, A numerical approach to the testing of the fission hpothesis, The Astrophsical Journal, vol. 82, no. 2, pp. 3 24, R. R. Gingold and J. J. Monagham, Smoothed particle hdrodnamics: theor and application to non-spherical stars, The Astrophsical Journal, vol. 8, pp , J. J. Monaghan, Simulating free surface flows with SPH, Journal of Computational Phsics, vol., no. 2, pp , 994.

16 6 Journal of Applied Mathematics 9 S. Koshizuka, H. Tamako, and Y. Oka, A particle method for incompressible viscous flow with fluid fragmentation, Computational Fluid Dnamics Journal, vol. 4, no., pp , 995. B. Ataie-Ashtiani and L. Farhadi, A stable moving-particle semi-implicit method for free surface flows, Fluid Dnamics Research, vol. 38, no. 4, pp , 26. S. Koshizuka, A. Nobe, and Y. Oka, Numerical analsis of breaking waves using the moving particle semi-implicit method, International Journal for Numerical Methods in Fluids, vol. 26, no. 7, pp , T. Beltschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krsl, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, vol. 39, no. -4, pp. 3 47, S. Koshizuka and Y. Oka, Moving-particle semi-implicit method for fragmentation of incompressible fluid, Nuclear Science and Engineering, vol. 23, no. 3, pp , B. Ramaswam, Numerical simulation of unstead viscous free surface flow, Journal of Computational Phsics, vol. 9, no. 2, pp , T. W. H. Sheu and S. M. Lee, Large-amplitude sloshing in an oil tanker with baffle-plate/drilled holes, International Journal of Computational Fluid Dnamics, vol., no., pp. 45 6, T. W. H. Sheu, C. H. Yu, and P. H. Chiu, Development of a dispersivel accurate conservative level set scheme for capturing interface in two-phase flows, Journal of Computational Phsics, vol. 228, no. 3, pp , 29.

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