|Table of Contents|

Citation:
 Lehua Xiao,Ting Long.An Axisymmetric Adaptive Multiresolution SPH for Modeling Strongly Compressible Multiphase Flows[J].Journal of Marine Science and Application,2025,(4):682-707.[doi:10.1007/s11804-024-00511-5]
Click and Copy

An Axisymmetric Adaptive Multiresolution SPH for Modeling Strongly Compressible Multiphase Flows

Info

Title:
An Axisymmetric Adaptive Multiresolution SPH for Modeling Strongly Compressible Multiphase Flows
Author(s):
Lehua Xiao12 Ting Long12
Affilations:
Author(s):
Lehua Xiao12 Ting Long12
1. State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, Guangxi University, Nanning 530004, China;
2. School of Mechanical Engineering, Guangxi University, Nanning 530004, China
Keywords:
Axisymmetric smooth particle hydrodynamicsAdaptive multiresolution schemeStrongly compressible multiphase flowsShock waveUnderwater explosion
分类号:
-
DOI:
10.1007/s11804-024-00511-5
Abstract:
Multiphase flows widely exist in various scientific and engineering fields, and strongly compressible multiphase flows commonly occur in practical applications, which makes them an important part of computational fluid dynamics. In this study, an axisymmetric adaptive multiresolution smooth particle hydrodynamics (SPH) model is proposed to solve various strongly compressible multiphase flow problems. In the present model, the governing equations are discretized in cylindrical polar coordinates, and an improved volume adaptive scheme is developed to better solve the problem of excessive volume change in strongly compressible multiphase flows. On this basis, combined with the adaptive particle refinement technique, an adaptive multiresolution scheme is proposed in this study. In addition, the high-order differential operator and diffusion correction term are utilized to improve the accuracy and stability. The effectiveness of the model is verified by testing four typical strongly compressible multiphase flow problems. By comparing the results of adaptive multiresolution SPH with other numerical results or experimental data, we can conclude that the present SPH method effectively models strongly compressible multiphase flows.

References:

[1] Adami S, Hu XY, Adams NA (2012) A generalized wall boundary condition for smoothed particle hydrodynamics. Journal of Computational Physics 231(21): 7057-7075. DOI: 10.1016/j.jcp.2012.05.005
[2] Alimi JM, Serna A, Pastor C, Bernabeu G (2003) Smooth particle hydrodynamics: importance of correction terms in adaptive resolution algorithms. Journal of Computational Physics 192(1): 157-174. DOI: 10.1016/S0021-9991(03)00351-6
[3] Antuono M, Colagrossi A, Marrone S, Molteni D (2010) Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Computer Physics Communications 181(3): 532-549. DOI: 10.1016/j.cpc.2009.11.002
[4] Avesani D, Dumbser M, Bellin A (2014) A new class of Moving-Least-Squares WENO-SPH schemes. Journal of Computational Physics 270: 278-299. DOI: 10.1016/j.jcp.2014.03.041
[5] Balsara DS (1995) Von Neumann stability analysis of smoothed particle hydrodynamics—Suggestions for optimal algorithms. Journal of Computational Physics 121(2): 357-372. DOI: 10.1016/S0021-9991(95)90221-X
[6] Barcarolo DA, Le Touzé D, Oger G, De Vuyst F (2014) Adaptive particle refinement and derefinement applied to the smoothed particle hydrodynamics method. Journal of Computational Physics 273: 640-657. DOI: 10.1016/j.jcp.2014.05.040
[7] Benz W (1990) Smooth particle hydrodynamics: A review. In: Buchler JB. (Eds.) The Numerical Modelling of Nonlinear Stellar Pulsations: Problems and Prospects. Kluwer Academi C, Doredrecht, 269-288. DOI: 10.1007/978-94-009-0519-1
[8] Berger MJ, Oliger J (1984) Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics 53(3): 484-512. DOI: 10.1016/0021-9991(84)90073-1
[9] Brookshaw L (2002) Smooth particle hydrodynamics in cylindrical coordinates. ANZIAM Journal 44: C114-C139. http://anziamj.austms.org.au/V44/CTAC2001/Broo
[10] Bui HH, Fukagawa R, Sako K, Ohno S (2008) Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive model. International Journal for Numerical and Analytical Methods in Geomechanics 32(12): 1537-1570. DOI: 10.1002/nag.688
[11] Chen X, Wan D (2019) GPU accelerated MPS method for large-scale 3-D violent free surface flows. Ocean Engineering 171: 677-694. DOI: 10.1016/j.oceaneng.2018.11.009
[12] Colagrossi A, Landrini M (2003) Numerical simulation of interfacial flows by smoothed particle hydrodynamics. Journal of Computational Physics 191(2): 448-475. DOI: 10.1016/S0021-9991(03)00324-3
[13] Crespo AJ, Domínguez JM, Rogers BD, Gómez-Gesteira M, Longshaw S, Canelas RJ, García-Feal O (2015) DualSPHysics: Open-source parallel CFD solver based on Smoothed Particle Hydrodynamics (SPH). Computer Physics Communications 187: 204-216. DOI: 10.1016/j.cpc.2014.10.004
[14] Cui P, Zhang AM, Wang SP (2016) Small-charge underwater explosion bubble experiments under various boundary conditions. Physics of Fluids 28(11): 117103. DOI: 10.1063/1.4967700
[15] Dobratz BM (1981) LLNL explosives handbook: properties of chemical explosives and explosives and explosive simulants (No. UCRL-52997). Lawrence Livermore National Lab. (LLNL), Livermore, USA
[16] Fang XL, Colagrossi A, Wang PP, Zhang AM (2022) An accurate and robust axisymmetric SPH method based on Riemann solver with applications in ocean engineering. Ocean Engineering 244: 110369. DOI: 10.1016/j.oceaneng.2021.110369
[17] Feldman J, Bonet J (2007) Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems. International Journal for Numerical Methods in Engineering 72(3): 295-324. DOI: 10.1002/nme.2010
[18] Ferrari A, Dumbser M, Toro EF, Armanini A (2009) A new 3D parallel SPH scheme for free surface flows. Computers & Fluids 38(6): 1203-1217. DOI: 10.1016/j.compfluid.2008.11.012
[19] Freret L, Williamschen M, Groth CP (2022) Enhanced anisotropic block-based adaptive mesh refinement for three-dimensional inviscid and viscous compressible flows. Journal of Computational Physics 458: 111092. DOI: 10.1016/j.jcp.2022.111092
[20] Fu L, Ji Z (2019) An optimal particle setup method with Centroidal Voronoi Particle dynamics. Computer Physics Communications 234: 72-92. DOI: 10.1016/j.cpc.2018.08.002
[21] García-Senz D, Relano A, Cabezón RM, Bravo E (2009) Axisymmetric smoothed particle hydrodynamics with self-gravity. Monthly Notices of the Royal Astronomical Society 392(1): 346-360. DOI: 10.1111/j.1365-2966.2008.14058.x
[22] Gibou F, Fedkiw R, Osher S (2018) A review of level-set methods and some recent applications. Journal of Computational Physics 353: 82-109. DOI: 10.1016/j.jcp.2017.10.006
[23] Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society 181(3): 375-389. DOI: 10.1093/mnras/181.3.375
[24] Gong K, Shao S, Liu H, Wang B, Tan SK (2016) Two-phase SPH simulation of fluid-structure interactions. Journal of Fluids and Structures 65: 155-179. DOI: 10.1016/j.jfluidstructs.2016.05.012
[25] Gotoh H, Khayyer A (2018) On the state-of-the-art of particle methods for coastal and ocean engineering. Coastal Engineering Journal 60(1): 79-103. DOI: 10.1080/21664250.2018.1436243
[26] Hammani I, Marrone S, Colagrossi A, Oger G, Le Tou?e D (2020) Detailed study on the extension of the δ -SPH model to multiphase flow. Computer Methods in Applied Mechanics and Engineering 368: 113189. DOI: 10.1016/j.cma.2020.113189
[27] Hu XY, Adams NA (2006) A multi-phase SPH method for macroscopic and mesoscopic flows. Journal of Computational Physics 213(2): 844-861. DOI: 10.1016/j.jcp.2005.09.001
[28] Hu XY, Adams NA (2007) An incompressible multi-phase SPH method. Journal of Computational Physics 227(1): 264-278. DOI: 10.1016/j.jcp.2007.07.013
[29] Hu XY, Adams NA, Iaccarino G (2009) On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow. Journal of Computational Physics 228(17): 6572-6589. DOI: 10.1016/j.jcp.2009.06.002
[30] Huang X, Sun P, Lyu H, Zhang AM (2022) Water entry problems simulated by an axisymmetric SPH model with vas scheme. Journal of Marine Science and Application 21(2): 1-15. DOI: 10.1007/s11804-022-00265-y
[31] Joshi S, Franc JP, Ghigliotti G, Fivel M (2021) An axisymmetric solid SPH solver with consistent treatment of particles close to the symmetry axis. Computational Particle Mechanics 8: 35-49. DOI: 10.1007/s40571-019-00310-8
[32] Kazemi E, Koll K, Tait S, Shao S (2020) SPH modelling of turbulent open channel flow over and within natural gravel beds with rough interfacial boundaries. Advances in Water Resources 140: 103557. DOI: 10.1016/j.advwatres.2020.103557
[33] Khayyer A, Gotoh H (2013) Enhancement of performance and stability of MPS mesh-free particle method for multiphase flows characterized by high density ratios. Journal of Computational Physics 242: 211-233. DOI: 10.1016/j.jcp.2013.02.002
[34] Khayyer A, Gotoh H, Shimizu Y (2019) A projection-based particle method with optimized particle shifting for multiphase flows with large density ratios and discontinuous density fields. Computers & Fluids 179: 356-371. DOI: 10.1016/j.compfluid.2018.10.018
[35] Khayyer A, Shimizu Y, Gotoh H, Nagashima K (2021a) A coupled incompressible SPH-Hamiltonian SPH solver for hydroelastic FSI corresponding to composite structures. Applied Mathematical Modelling 94: 242-271. DOI: 10.1016/j.apm.2021.01.011
[36] Khayyer A, Shimizu Y, Gotoh H, Hattori S (2021b) Multi-resolution ISPH-SPH for accurate and efficient simulation of hydroelastic fluid-structure interactions in ocean engineering. Ocean Engineering 226: 108652. DOI: 10.1016/j.oceaneng.2021.108652
[37] Khayyer A, Shimizu Y, Gotoh T, Gotoh H (2023) Enhanced resolution of the continuity equation in explicit weakly compressible SPH simulations of incompressible free-surface fluid flows. Applied Mathematical Modelling 116: 84-121. DOI: 10.1016/j.apm.2022.10.037
[38] Kitsionas S, Whitworth AP (2002) Smoothed particle hydrodynamics with particle splitting, applied to self-gravitating collapse. Monthly Notices of the Royal Astronomical Society 330(1): 129-136. DOI: 10.1046/j.1365-8711.2002.05115.x
[39] Kitsionas S, Whitworth AP (2007) High-resolution simulations of clump-clump collisions using SPH with particle splitting. Monthly Notices of the Royal Astronomical Society 378(2): 507-524. DOI: 10.1111/j.1365-2966.2007.11707.x
[40] Lauer E, Hu XY, Hickel S, Adams N.A. (2012) Numerical modelling and investigation of symmetric and asymmetric cavitation bubble dynamics. Computers & Fluids 69: 1-19. DOI: 10.1016/j.compfluid.2012.07.020
[41] Li MK, Zhang AM, Ming FR, Sun PN, Peng YX (2020a) An axisymmetric multiphase SPH model for the simulation of rising bubble. Computer Methods in Applied Mechanics and Engineering 366: 113039. DOI: 10.1016/j.cma.2020.113039
[42] Li S, van der Meer D, Zhang AM, Prosperetti A, Lohse D (2020b) Modelling large scale airgun-bubble dynamics with highly non-spherical features. International Journal of Multiphase Flow 122: 103143. DOI: 10.1016/j.ijmultiphaseflow.2019.103143
[43] Li S, Zhang AM, Han R, Ma Q (2019a) 3D full coupling model for strong interaction between a pulsating bubble and a movable sphere. Journal of Computational Physics 392: 713-731. DOI: 10.1016/j.jcp.2019.05.001
[44] Li T, Zhang AM, Wang SP, Li S, Liu WT (2019b) Bubble interactions and bursting behaviors near a free surface. Physics of Fluids 31(4): 042104. DOI: 10.1063/1.5088528
[45] Liang C, Huang W, Chen D (2023) A pressure-dependent adaptive resolution scheme for smoothed particle hydrodynamics simulation of underwater explosion. Ocean Engineering 270: 113695. DOI: 10.1016/j.oceaneng.2023.113695
[46] Lind SJ, Xu R, Stansby PK, Rogers BD (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. Journal of Computational Physics 231(4): 1499-1523. DOI: 10.1016/j.jcp.2011.10.027
[47] Liu M, Zhang Z (2019) Smoothed particle hydrodynamics (SPH) for modeling fluid-structure interactions. Science China Physics, Mechanics & Astronomy 62: 1-38. DOI: 10.1007/s11433-018-9357-0
[48] Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Archives of Computational Methods in Engineering 17: 25-76. DOI: 10.1007/s11831-010-9040-7
[49] Long T, Hu D, Wan D, Zhang C, Yang G (2017) An arbitrary boundary with ghost particles incorporated in coupled FEM-SPH model for FSI problems. Journal of Computational Physics 350: 166-183. DOI: 10.1016/j.jcp.2017.08.044
[50] Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. The Astronomical Journal 8(12): 1013-1024. DOI: 10.1086/112/64
[51] Luo M, Koh CG, Bai W, Gao M (2016) A particle method for two - phase flows with compressible air pocket. International Journal for Numerical Methods in Engineering 108(7): 695-721. DOI: 10.1002/nme.5230
[52] Lyu HG, Sun PN (2022) Further enhancement of the particle shifting technique: Towards better volume conservation and particle distribution in SPH simulations of violent free-surface flows. Applied Mathematical Modelling 101: 214-238. DOI: 10.1016/j.apm.2021.08.014
[53] Lyu HG, Sun PN, Miao JM, Zhang AM (2022) 3D multi-resolution SPH modeling of the water entry dynamics of free-fall lifeboats. Ocean Engineering 257: 111648. DOI: 10.1016/j.oceaneng.2022.111648
[54] Marrone S, Antuono M, Colagrossi A, Colicchio G, Le Touzé D, Graziani G (2011a) δ -SPH model for simulating violent impact flows. Computer Methods in Applied Mechanics and Engineering 200(13-16): 1526-1542. DOI: 10.1016/j.cma.2010.12.016
[55] Marrone S, Colagrossi A, Antuono M, Lugni C, Tulin MP (2011b) A 2D+t SPH model to study the breaking wave pattern generated by fast ships. Journal of Fluids and Structures 27(8): 1199-1215. DOI: 10.1016/j.jfluidstructs.2011.08.003
[56] Marsh A, Oger G, Le Touzé D, Guibert D (2011) Validation of a conservative variable-resolution SPH scheme including ▽h terms. In 6th Int. SPHERIC Workshop (SPHERIC 2011)
[57] Ming F, Sun P, Zhang A (2014) Investigation on charge parameters of underwater contact explosion based on axisymmetric SPH method. Applied Mathematics and Mechanics 35(4): 453-468. DOI: 10.1007/s10483-014-1804-6
[58] Mokos A, Rogers BD, Stansby PK, Domínguez JM (2015) Multiphase SPH modelling of violent hydrodynamics on GPUs. Computer Physics Communications 196: 304-316. DOI: 10.1016/j.cpc.2015.06.020
[59] Monaghan JJ (1989) On the problem of penetration in particle methods. Journal of Computational Physics 82(1): 1-15. DOI: 10.1016/0021-9991(89)90032-6
[60] Monaghan JJ (1992) Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics 30: 543-574. https://doi.org/10.1146/annurev.aa.30.090192.002551
[61] Monaghan JJ (1994) Simulating free surface flows with SPH. Journal of Computational Physics 110(2): 399-406. DOI: 10.1006/jcph.1994.1034
[62] Monaghan JJ (2005) Smoothed particle hydrodynamics. Reports on Progress in Physics 68(8): 1703. DOI: 10.1088/0034-4885/68/8/R01
[63] Monaghan JJ (2012) Smoothed particle hydrodynamics and its diverse applications. Annual Review of Fluid Mechanics 44: 323-346. DOI: 10.1146/annurev-fluid-120710-101220
[64] Monaghan JJ, Lattanzio JC (1985) A refined particle method for astrophysical problems. Astronomy and Astrophysics 149: 135-143. DOI: 10.1002/asna.2113060608
[65] Monaghan JJ, Rafiee A (2013) A simple SPH algorithm for multi-fluid flow with high density ratios. International Journal for Numerical Methods in Fluids 71(5): 537-561. DOI: 10.1002/fld.3671
[66] Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible flows using SPH. Journal of Computational Physics 136(1): 214-226. DOI: 10.1006/jcph.1997.5776
[67] Nazeer M, Hussain F, Hameed MK, Khan MI, Ahmad F, Malik MY, Shi QH (2021) Development of mathematical modeling of multiphase flow of Casson rheological fluid: Theoretical approach. Chaos Solitons & Fractals 150(11): 111198. DOI: 10.1016/j.chaos.2021.111198
[68] Nonoyama H, Moriguchi S, Sawada K, Yashima A (2015) Slope stability analysis using smoothed particle hydrodynamics (SPH) method. Soils and Foundations 55(2): 458-470. DOI: 10.1016/j.sandf.2015.02.019
[69] Oger G, Le Touzé D, Guibert D, De Leffe M, Biddiscombe J, Soumagne J, Piccinal JG (2016) On distributed memory MPI-based parallelization of SPH codes in massive HPC context. Computer Physics Communications 200: 1-14. DOI: 10.1016/j.cpc.2015.08.021
[70] Omang M, Børve S, Trulsen J (2006) SPH in spherical and cylindrical coordinates. Journal of Computational Physics 213(1): 391-412. DOI: 10.1016/j.jcp.2005.08.023
[71] Omidvar P, Stansby PK, Rogers BD (2012) Wave body interaction in 2D using smoothed particle hydrodynamics (SPH) with variable particle mass. International Journal for Numerical Methods in Fluids 68(6): 686-705. DOI: 10.1002/fld.2528
[72] Petalas N, Aziz KA (2000) Mechanistic model for multiphase flow in pipes. Journal of Canadian Petroleum Technology 39(6): 00-06-04. DOI: 10.2118/98-39
[73] Plesset MS, Prosperetti A (1977) Bubble dynamics and cavitation. Annual Review of Fluid Mechanics 9(1): 145-185. DOI: 10.1146/annurev.fl.09.010177.001045
[74] Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Computer Methods in Applied Mechanics and Engineering 139(1-4): 375-408. DOI: 10.1016/S0045-7825(96)01090-0
[75] Reyes López Y, Roose D, Recarey Morfa C (2013) Dynamic particle refinement in SPH: application to free surface flow and non-cohesive soil simulations. Computational Mechanics 51: 731-741. DOI: 10.1007/s00466-012-0748-0
[76] Sedov LI (2018) Similarity and dimensional methods in mechanics. CRC Press
[77] Shadloo MS, Oger G, Le Touzé D (2016) Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges. Computers & Fluids 136: 11-34. DOI: 10.1016/j.compfluid.2016.05.029
[78] Shi H, Huang Y (2022) A GPU-based δ -Plus-SPH model for non-newtonian multiphase flows. Water 14(11): 1734. DOI: 10.3390/w14111734
[79] Sigalotti LD, López H, Donoso A, Sira E, Klapp J (2006) A shock-capturing SPH scheme based on adaptive kernel estimation. Journal of Computational Physics 212(1): 124-149. DOI: 10.1016/j.jcp.2005.06.016
[80] Sod GA (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics 27(1): 1-31. DOI: 10.1016/0021-9991(78)90023-2
[81] Steinberg DJ (1987) Spherical explosions and the equation of state of water (No. UCID-20974). Lawrence Livermore National Lab. (LLNL), Livermore, USA
[82] Sun PN, Colagrossi A, Marrone S, Antuono M, Zhang AM (2019a) A consistent approach to particle shifting in the δ-Plus-SPH model. Computer Methods in Applied Mechanics and Engineering 348: 912-934. DOI: 10.1016/j.cma.2019.01.045
[83] Sun PN, Colagrossi A, Zhang AM (2018) Numerical simulation of the self-propulsive motion of a fishlike swimming foil using the δ+-SPH model. Theoretical and Applied Mechanics Letters 8(2): 115-125. DOI: 10.1016/j.taml.2018.02.007
[84] Sun PN, Le Touzé D, Oger G, Zhang AM (2021a) An accurate SPH Volume Adaptive Scheme for modeling strongly-compressible multiphase flows. Part 1: Numerical scheme and validations with basic 1D and 2D benchmarks. Journal of Computational Physics 426: 109937. DOI: 10.1016/j.jcp.2020.109937
[85] Sun PN, Le Touzé D, Oger G, Zhang AM (2021b) An accurate SPH Volume Adaptive Scheme for modeling strongly-compressible multiphase flows. Part 2: Extension of the scheme to cylindrical coordinates and simulations of 3D axisymmetric problems with experimental validations. Journal of Computational Physics 426: 109936. DOI: 10.1016/j.jcp.2020.109936
[86] Sun PN, Luo M, Le Touzé D, Zhang AM (2019b) The suction effect during freak wave slamming on a fixed platform deck: Smoothed particle hydrodynamics simulation and experimental study. Physics of Fluids 31(11): 117108. DOI: 10.1063/1.5124613
[87] Toro EF (2013) Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Science & Business Media
[88] Vacondio R, Rogers BD, Stansby PK (2012) Accurate particle splitting for smoothed particle hydrodynamics in shallow water with shock capturing. International Journal for Numerical Methods in Fluids 69(8): 1377-1410. DOI: 10.1002/fld.2646
[89] Wang PP, Zhang AM, Fang XL, Khayyer A, Meng ZF (2022) Axisymmetric Riemann-smoothed particle hydrodynamics modeling of high-pressure bubble dynamics with a simple shifting scheme. Physics of Fluids 34(11): 112122. DOI: 10.1063/5.0123106
[90] Xie F, Zhao W, Wan D (2021) Numerical simulations of liquid-solid flows with free surface by coupling IMPS and DEM. Applied Ocean Research 114: 102771. DOI: 10.1016/j.apor.2021.102771
[91] Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. Journal of computational Physics 228(18): 6703-6725. DOI: 10.1016/j.jcp.2009.05.032
[92] Yang Q, Xu F, Yang Y, Dai Z, Wang J (2023a) A GPU-accelerated adaptive particle refinement for multi-phase flow and fluid-structure coupling SPH. Ocean Engineering 279: 114514. DOI: 10.1016/j.oceaneng.2023.114514
[93] Yang X, Feng S, Wu J, Zhang G, Liang G, Zhang Z (2023b) Study of the water entry and exit problems by coupling the APR and PST within SPH. Applied Ocean Research 139: 103712. DOI: 10.1016/j.apor.2023.103712
[94] Yilmaz A, Kocaman S, Demirci M (2021) Numerical modeling of the dam-break wave impact on elastic sluice gate: A new benchmark case for hydroelasticity problems. Ocean Engineering 231: 108870. DOI: 10.1016/j.oceaneng.2021.108870
[95] Zamyshlyaev BV, Yakovlev YS (1973) Dynamic loads in underwater explosion. Naval Intelligence Support Center, Washington, DC, USA
[96] Zhang AM, Sun PN, Ming FR, Colagrossi A (2017) Smoothed particle hydrodynamics and its applications in fluid-structure interactions. Journal of Hydrodynamics 29(2): 187-216. DOI: 10.1016/S1001-6058(16)60730-8
[97] Zhang AM, Cui P, Cui J, Wang QX (2015a) Experimental study on bubble dynamics subject to buoyancy. Journal of Fluid Mechanics 776: 137-160. DOI: 10.1017/jfm.2015.323
[98] Zhang AM, Li SM, Cui P, Li S, Liu YL (2023) A unified theory for bubble dynamics. Physics of Fluids 35(3): 033323. DOI: 10.1063/5.0145415
[99] Zhang AM, Sun PN, Ming FR (2015b) An SPH modeling of bubble rising and coalescing in three dimensions. Computer Methods in Applied Mechanics and Engineering 294: 189-209. DOI: 10.1016/j.cma.2015.05.014
[100] Zhang S, Wang SP, Liu YL, Zhang AM, Cui P (2019) Interaction of clustered air gun bubbles in marine prospecting. Ocean Engineering 191: 106523. DOI: 10.1016/j.oceaneng.2019.106523
[101] Zhang ZL, Liu MB (2018) A decoupled finite particle method for modeling incompressible flows with free surfaces. Applied Mathematical Modelling 60: 606-633. DOI: 10.1016/j.apm.2018.03.043

Memo

Memo:
Received date:2024-3-10;Accepted date:2024-5-10。<br>Foundation item:Supported by the Guangxi Natural Science Foundation (Grant No. 2021GXNSFBA196008) and the Guangxi Science and Technology Development Program (Grant No. GuikeAD22035189).<br>Corresponding author:Ting Long,E-mail:longting@gxu.edu.cn
Last Update: 2025-08-27