A Cartesian-to-curvilinear coordinate transformation in modified ghost fluid method for compressible multi-material flows. (English) Zbl 1491.76058
Summary: Modified ghost fluid method (MGFM) provides us an effective manner to simulate compressible multi-material flows. In most cases, the applications are limited in relatively simple geometries described by Cartesian grids. In this paper, the MGFM treatment with the level set (LS) technique is extended to curvilinear coordinate systems. The chain rule of differentiation (applicable to general curvilinear coordinates) and the orthogonal transformation (applicable to orthogonal curvilinear coordinates) are utilized to deduce the Cartesian-to-curvilinear coordinate transformation, respectively. The relationship between these two transformations for the extension of the LS/MGFM algorithm is analyzed in theory. It is shown that these two transformations are equivalent for orthogonal curvilinear grids. The extension of the LS/MGFM algorithm using the chain rule has a wider range of applications, as there is essentially no requirement for the orthogonality of the grids. Several challenging problems in two- or three-dimensions are utilized to validate the developed algorithm in curvilinear coordinates. The results indicate that this algorithm enables a simple and effective implementation for simulating interface evolutions, as in Cartesian coordinate systems. It has the potential to be applied in more complex computational domains.
MSC:
| 76M99 | Basic methods in fluid mechanics |
| 76N15 | Gas dynamics (general theory) |
| 76T30 | Three or more component flows |
Keywords:
modified ghost fluid method; level set method; multi-material Riemann problem; coordinate transformation; general curvilinear coordinate system; orthogonal curvilinear coordinatesReferences:
| [1] | E. Klaseboer, K. C. Hung, C. Wang, et al., Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure, J. Fluid Mech. 537 (2005) 387-413. http://dx.doi.org/10.1017/S0022112005005306 · Zbl 1138.76303 · doi:10.1017/S0022112005005306 |
| [2] | N. V. Petrov, A. A. Schmidt, Multiphase phenomena in underwater explosion, Exp. Therm. Fluid Sci. 60 (2015) 367-373. http://dx.doi.org/10.1016/j.expthermflusci.2014.05. 008 · doi:10.1016/j.expthermflusci.2014.05.008 |
| [3] | K. K. Haller, Y. Ventikos, D. Poulikakos, et al., Computational study of high-speed liquid droplet impact, J. Appl. Phys. 92 (5) (2002) 2821-2828. http://dx.doi.org/10.1063/1. 1495533 · doi:10.1063/1.1495533 |
| [4] | J. C. Hermanson, Dynamics of supersonic droplets of volatile liquids, AIAA J. 45 (3) (2007) 730-733. http://dx.doi.org/10.2514/1.26962 · doi:10.2514/1.26962 |
| [5] | J. J. Quirk, S. Karni, On the dynamics of a shock-bubble interaction, J. Fluid Mech. 318 (1996) 129-163. http://dx.doi.org/10.1017/S0022112096007069 · Zbl 0877.76046 · doi:10.1017/S0022112096007069 |
| [6] | N. Kyriazis, P. Koukouvinis, M. Gavaises, Numerical investigation of bubble dynamics us-ing tabulated data, Int. J. Multiphas. Flow 93 (2017) 158-177. http://dx.doi.org/10.1016/ j.ijmultiphaseflow.2017.04.004 · doi:10.1016/j.ijmultiphaseflow.2017.04.004 |
| [7] | J. Yang, T. Kubota, E. E. Zukoski, Applications of shock-induced mixing to supersonic com-bustion, AIAA J. 31 (5) (1993) 854-862. http://dx.doi.org/10.2514/3.11696 · doi:10.2514/3.11696 |
| [8] | N. Haehn, D. Ranjan, C. Weber, et al., Reacting shock bubble interaction, Combust. Flame 159 (3) (2012) 1339-1350. http://dx.doi.org/10.1016/j.combustflame.2011.10.015 · doi:10.1016/j.combustflame.2011.10.015 |
| [9] | H. Yong, C. L. Zhai, S. Jiang, et al., Numerical simulations of instabilities in the implosion process of inertial confined fusion in 2d cylindrical coordinates, Sci. China-Phys. Mech. As-tron. 59 (1) (2016) 614704. http://dx.doi.org/10.1007/s11433-015-5711-6 · doi:10.1007/s11433-015-5711-6 |
| [10] | G. Z. Voyiadjis, R. K. A. Al-Rub, A. N. Palazotto, Constitutive modeling and simulation of perforation of targets by projectiles, AIAA J. 46 (2) (2008) 304-316. http://dx.doi.org/10. 2514/1.26011 · doi:10.2514/1.26011 |
| [11] | S. Y. Wu, K. X. Liu, Q. Y. Chen, A three-dimensional Eulerian method for the numerical simulation of high-velocity impact problems, Chin. Phys. B 23 (3) (2014) 034601. http: //dx.doi.org/10.1088/1674-1056/23/3/034601 · doi:10.1088/1674-1056/23/3/034601 |
| [12] | R. Abgrall, How to prevent oscillations in multicomponent flow calculations: a quasi con-servative approach, J. Comput. Phys. 125 (1) (1996) 150-160. http://dx.doi.org/10.1006/ jcph.1996.0085 · Zbl 0847.76060 · doi:10.1006/jcph.1996.0085 |
| [13] | K.-M. Shyue, A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Grüneisen equation of state, J. Comput. Phys. 171 (2) (2001) 678-707. http://dx.doi. org/10.1006/jcph.2001.6801 · Zbl 1047.76573 · doi:10.1006/jcph.2001.6801 |
| [14] | E. Johnsen, T. Colonius, Implementation of WENO schemes in compressible multicompo-nent flow problems, J. Comput. Phys. 219 (2) (2006) 715-732. http://dx.doi.org/10.1016/ j.jcp.2006.04.018 · Zbl 1189.76351 · doi:10.1016/j.jcp.2006.04.018 |
| [15] | M. Capuano, C. Bogey, P. D. M. Spelt, Simulations of viscous and compressible gas-gas flows using high-order finite difference schemes, J. Comput. Phys. 361 (2018) 56-81. http://dx. doi.org/10.1016/j.jcp.2018.01.047 · Zbl 1391.76773 · doi:10.1016/j.jcp.2018.01.047 |
| [16] | M. Rudman, Volume-tracking methods for interfacial flow calculations, Int. J. Numer. Meth. Fluids 24 (7) (1997) 671-691. http://dx.doi.org/10.1002/(SICI)1097-0363(19970415) 24:7<671::AID-FLD508>3.0.CO;2-9 · Zbl 0889.76069 · doi:10.1002/(SICI)1097-0363(19970415)24:7<671 |
| [17] | W. J. Rider, D. B. Kothe, Reconstructing volume tracking, J. Comput. Phys. 141 (2) (1998) 112-152. http://dx.doi.org/10.1006/jcph.1998.5906 · Zbl 0933.76069 · doi:10.1006/jcph.1998.5906 |
| [18] | K. M. Shyue, F. Xiao, An Eulerian interface sharpening algorithm for compressible two-phase flow: The algebraic THINC approach, J. Comput. Phys. 268 (2014) 326-354. http: //dx.doi.org/10.1016/j.jcp.2014.03.010 · Zbl 1349.76388 · doi:10.1016/j.jcp.2014.03.010 |
| [19] | T. Nonomura, K. Kitamura, K. Fujii, A simple interface sharpening technique with a hyper-bolic tangent function applied to compressible two-fluid modeling, J. Comput. Phys. 258 (2014) 95-117. http://dx.doi.org/10.1016/j.jcp.2013.10.021 · Zbl 1349.76514 · doi:10.1016/j.jcp.2013.10.021 |
| [20] | S. Kokh, F. Lagoutière, An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model, J. Comput. Phys. 229 (8) (2010) 2773-2809. http://dx.doi.org/10.1016/j.jcp.2009.12.003 · Zbl 1302.76129 · doi:10.1016/j.jcp.2009.12.003 |
| [21] | K. K. So, X. Y. Hu, N. A. Adams, Anti-diffusion interface sharpening technique for two-phase compressible flow simulations, J. Comput. Phys. 231 (2012) 4304-4323. http://dx. doi.org/10.1016/j.jcp.2012.02.013 · Zbl 1426.76428 · doi:10.1016/j.jcp.2012.02.013 |
| [22] | V. Faucher, M. Bulik, P. Galon, Updated VOFIRE algorithm for fast fluid-structure tran-sient dynamics with multi-component stiffened gas flows implementing anti-dissipation on unstructured grids, J. Fluid. Struct. 74 (2017) 64-89. http://dx.doi.org/10.1016/j. jfluidstructs.2017.07.001 · doi:10.1016/j.jfluidstructs.2017.07.001 |
| [23] | R. K. Shukla, Nonlinear preconditioning for efficient and accurate interface capturing in simulation of multicomponent compressible flows, J. Comput. Phys. 276 (2014) 508-540. http://dx.doi.org/10.1016/j.jcp.2014.07.034 · Zbl 1349.65110 · doi:10.1016/j.jcp.2014.07.034 |
| [24] | D. P. Garrick, M. Owkes, J. D. Regele, A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension, J. Comput. Phys. 339 (2017) 46-67. http://dx.doi. org/10.1016/j.jcp.2017.03.007 · Zbl 1375.76099 · doi:10.1016/j.jcp.2017.03.007 |
| [25] | C. W. Hirt, A. A. Amsden, J. L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys. 135 (2) (1997) 203-216. http://dx.doi.org/10.1006/ jcph.1997.5702 · Zbl 0938.76068 · doi:10.1006/jcph.1997.5702 |
| [26] | A. J. Barlow, P.-H. Maire, W. J. Rider, et al., Arbitrary Lagrangian-Eulerian methods for mod-eling high-speed compressible multimaterial flows, J. Comput. Phys. 322 (2016) 603-665. http://dx.doi.org/10.1016/j.jcp.2016.07.001 · Zbl 1351.76093 · doi:10.1016/j.jcp.2016.07.001 |
| [27] | H. Terashima, G. Tryggvason, A front-tracking/ghost-fluid method for fluid interfaces in compressible flows, J. Comput. Phys. 228 (11) (2009) 4012-4037. http://dx.doi.org/10. 1016/j.jcp.2009.02.023 · Zbl 1171.76046 · doi:10.1016/j.jcp.2009.02.023 |
| [28] | M. A. Ullah, W. B. Gao, D. K. Mao, Towards front-tracking based on conservation in two space dimensions III, tracking interfaces, J. Comput. Phys. 242 (2013) 268-303. http://dx. doi.org/10.1016/j.jcp.2013.02.026 · Zbl 1314.65113 · doi:10.1016/j.jcp.2013.02.026 |
| [29] | S. Osher, R. P. Fedkiw, Level set methods: an overview and some recent results, J. Comput. Phys. 169 (2) (2001) 463-502. http://dx.doi.org/10.1006/jcph.2000.6636 · Zbl 0988.65093 · doi:10.1006/jcph.2000.6636 |
| [30] | F. Gibou, R. P. Fedkiw, S. Osher, A review of level-set methods and some recent applications, J. Comput. Phys. 353 (2018) 82-109. http://dx.doi.org/10.1016/j.jcp.2017.10.006 · Zbl 1380.65196 · doi:10.1016/j.jcp.2017.10.006 |
| [31] | R. P. Fedkiw, T. Aslam, B. Merriman, et al., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys. 152 (2) (1999) 457-492. http://dx.doi.org/10.1006/jcph.1999.6236 · Zbl 0957.76052 · doi:10.1006/jcph.1999.6236 |
| [32] | T. G. Liu, B. C. Khoo, K. S. Yeo, Ghost fluid method for strong shock impacting on ma-terial interface, J. Comput. Phys. 190 (2) (2003) 651-681. http://dx.doi.org/10.1016/ S0021-9991(03)00301-2 · Zbl 1076.76592 · doi:10.1016/S0021-9991(03 |
| [33] | C. W. Wang, T. G. Liu, B. C. Khoo, A real ghost fluid method for the simulation of multi-medium compressible flow, SIAM J. Sci. Comput. 28 (1) (2006) 278-302. http://dx.doi. org/10.1137/030601363 · Zbl 1114.35119 · doi:10.1137/030601363 |
| [34] | R. W. Houim, K. K. Kuo, A ghost fluid method for compressible reacting flows with phase change, J. Comput. Phys. 235 (2013) 865-900. http://dx.doi.org/10.1016/j.jcp.2012. 09.022 · doi:10.1016/j.jcp.2012.09.022 |
| [35] | Z. W. Feng, J. L. Rong, A. Kaboudian, et al., The modified ghost method for compressible multi-medium interaction with elastic-plastic solid, Commun. Comput. Phys. 22 (5) (2017) 1258-1285. http://dx.doi.org/10.4208/cicp.OA-2016-0212 · Zbl 1488.35434 · doi:10.4208/cicp.OA-2016-0212 |
| [36] | L. Xu, T.-G. Liu, Explicit interface treatments for compressible gas-liquid simulations, Com-put. Fluids 153 (2017) 34-48. http://dx.doi.org/10.1016/j.compfluid.2017.03.032 · Zbl 1390.76884 · doi:10.1016/j.compfluid.2017.03.032 |
| [37] | S. Fechter, C.-D. Munz, C. Rohde, et al., A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys. 336 (2017) 347-374. http://dx.doi.org/10.1016/j.jcp.2017.02.001 · Zbl 1375.76194 · doi:10.1016/j.jcp.2017.02.001 |
| [38] | Z. Y. Ge, J.-C. Loiseau1, O. Tammisola, et al., An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces, J. Comput. Phys. 353 (2018) 435-459. http://dx.doi.org/10.1016/j.jcp.2017.10.046 · Zbl 1380.76101 · doi:10.1016/j.jcp.2017.10.046 |
| [39] | R. C. Shi, Y. G. Qu, R. C. Batra, Numerical simulation of underwater explosion wave prop-agation in water-solid-air/water system using ghost fluid/solid method, J. Fluid. Struct. 90 (2019) 354-378. http://dx.doi.org/10.1016/j.jfluidstructs.2019.07.002 · doi:10.1016/j.jfluidstructs.2019.07.002 |
| [40] | Y. Hao, A. Prosperetti, A numerical method for three-dimensional gas-liquid flow computa-tions, J. Comput. Phys. 196 (2004) 126-144. http://dx.doi.org/10.1016/j.jcp.2003.10. 032 · Zbl 1109.76383 · doi:10.1016/j.jcp.2003.10.032 |
| [41] | W. Bo, X. Liu, J. Glimm, et al., A robust front tracking method: verification and application to simulation of the primary breakup of a liquid jet, SIAM J. Sci. Comput. 33 (4) (2011) 1505-1524. http://dx.doi.org/10.1137/10079135X · Zbl 1465.76084 · doi:10.1137/10079135X |
| [42] | H. T. Lu, J. Zhu, C. W. Wang, et al., A Riemann problem based method for solving compress-ible and incompressible flows, J. Comput. Phys. 330 (2017) 1-20. http://dx.doi.org/10. 1016/j.jcp.2016.10.047 · Zbl 1386.76174 · doi:10.1016/j.jcp.2016.10.047 |
| [43] | W. Bo, J. W. Grove, A volume of fluid method based ghost fluid method for compress-ible multi-fluid flows, Comput. Fluids 90 (2014) 113-122. http://dx.doi.org/10.1016/ j.compfluid.2013.11.013 · Zbl 1391.76524 · doi:10.1016/j.compfluid.2013.11.013 |
| [44] | T. Michael, J. M. Yang, F. Stern, A sharp interface approach for cavitation modeling using volume-of-fluid and ghost-fluid methods, J. Hydrodyn. 29 (6) (2017) 917-925. http://dx. doi.org/10.1016/S1001-6058(16)60806-5 doi:10.1016/S1001-6058(16)60806-5. · doi:10.1016/S1001-6058(16)60806-5 |
| [45] | A. Sarthou, S. Vincent, J. P. Caltagirone, A second-order curvilinear to Cartesian trans-formation of immersed interfaces and boundaries. Application to fictitious domains and multiphase flows, Comput. Fluids 46 (2011) 422-428. http://dx.doi.org/10.1016/j. compfluid.2010.11.008 · Zbl 1431.76105 · doi:10.1016/j.compfluid.2010.11.008 |
| [46] | J. F. Thompson, General curvilinear coordinate systems, Appl. Math. Comput. 10-11 (1982) 1-30. http://dx.doi.org/10.1016/0096-3003(82)90185-0 · Zbl 0493.65050 |
| [47] | J. Huang, P. M. Carrica, F. Stern, Coupled ghost fluid/two-phase level set method for curvi-linear body-fitted grids, Int. J. Numer. Meth. Fluids 55 (2007) 867-897. http://dx.doi.org/ 10.1002/fld.1499 · Zbl 1388.76253 · doi:10.1002/fld.1499 |
| [48] | J. Huang, P. M. Carrica, F. Stern, A geometry-based level set method for curvilinear overset grids with application to ship hydrodynamics, Int. J. Numer. Meth. Fluids 68 (2012) 494-521. http://dx.doi.org/10.1002/fld.2517 · Zbl 1427.76172 · doi:10.1002/fld.2517 |
| [49] | L. Xu, T. G. Liu, Accuracies and conservation errors of various ghost fluid methods for multi-medium Riemann problem, J. Comput. Phys. 230 (12) (2011) 4975-4990. http://dx.doi. org/10.1016/j.jcp.2011.03.021 · Zbl 1416.76233 · doi:10.1016/j.jcp.2011.03.021 |
| [50] | T. G. Liu, B. C. Khoo, C. W. Wang, The ghost fluid method for compressible gas-water simu-lation, J. Comput. Phys. 204 (1) (2005) 193-221. http://dx.doi.org/10.1016/j.jcp.2004. 10.012 · Zbl 1190.76160 · doi:10.1016/j.jcp.2004.10.012 |
| [51] | L. Xu, C. L. Feng, T. G. Liu, Practical techniques in ghost fluid method for compressible multi-medium flows, Commun. Comput. Phys. 20 (3) (2016) 619-659. http://dx.doi.org/ 10.4208/cicp.190315.290316a · Zbl 1373.76223 · doi:10.4208/cicp.190315.290316a |
| [52] | E. L. Lee, H. C. Hornig, J. W. Kury, Adiabatic expansion of high explosive detonation prod-ucts, Tech. Rep. UCRL-50422 (1968). http://dx.doi.org/10.2172/4783904 · doi:10.2172/4783904 |
| [53] | T. G. Liu, B. C. Khoo, K. S. Yeo, The simulation of compressible multi-medium flow I: A new methodology with test applications to 1D gas-gas and gas-water cases, Comput. Fluids 30 (3) (2001) 291-314. http://dx.doi.org/10.1016/S0045-7930(00)00022-0 · Zbl 1052.76046 |
| [54] | S. Sambasivan, H. S. UdayKumar, Ghost fluid method for strong shock interactions. Part 1: Fluid-fluid interfaces, AIAA J. 47 (12) (2009) 2907-2922. http://dx.doi.org/10.2514/1. 43148 · doi:10.2514/1.43148 |
| [55] | H. Terashima, G. Tryggvason, A front-tracking method with projected interface conditions for compressible multi-fluid flows, Comput. Fluids 39 (10) (2010) 1804-1814. http://dx. doi.org/10.1016/j.compfluid.2010.06.012 · Zbl 1245.76113 · doi:10.1016/j.compfluid.2010.06.012 |
| [56] | X. Y. Hu, B. C. Khoo, An interface interaction method for compressible multifluids, J. Com-put. Phys. 198 (1) (2004) 35-64. http://dx.doi.org/10.1016/j.jcp.2003.12.018 · Zbl 1107.76378 · doi:10.1016/j.jcp.2003.12.018 |
| [57] | Y. Y. Niu, Computations of two-fluid models based on a simple and robust hybrid primitive variable Riemann solver with AUSMD, J. Comput. Phys. 308 (2016) 389-410. http://dx. doi.org/10.1016/j.jcp.2015.12.045 · Zbl 1351.76119 · doi:10.1016/j.jcp.2015.12.045 |
| [58] | X. Bai, X. L. Deng, A sharp interface method for compressible multi-phase flows based on the Cut Cell and Ghost Fluid methods, Adv. Appl. Math. Mech. 9 (5) (2017) 1052-1075. http: //dx.doi.org/10.4208/aamm.2015.m1283 · Zbl 1488.65326 · doi:10.4208/aamm.2015.m1283 |
| [59] | C. Liu, C. H. Hu, Adaptive THINC-GFM for compressible multi-medium flows, J. Comput. Phys. 342 (2017) 43-65. http://dx.doi.org/10.1016/j.jcp.2017.04.032 · Zbl 1376.76056 · doi:10.1016/j.jcp.2017.04.032 |
| [60] | J. F. Haas, B. Sturtevant, Interaction of weak shock waves with cylindrical and spheri-cal gas inhomogeneities, J. Fluid Mech. 181 (1987) 41-76. http://dx.doi.org/10.1017/ S0022112087002003 · doi:10.1017/S0022112087002003 |
| [61] | W. Liu, L. Yuan, C.-W. Shu, A conservative modification to the ghost fluid method for compressible multiphase flows, Commun. Comput. Phys. 10 (4) (2011) 785-806. http: //dx.doi.org/10.4208/cicp.201209.161010a · Zbl 1373.76179 · doi:10.4208/cicp.201209.161010a |
| [62] | Y. Gorsse, A. Iollo, T. Milcent, H. Telib, A simple Cartesian scheme for compressible multi-materials, J. Comput. Phys. 272 (2014) 772-798. http://dx.doi.org/10.1016/j.jcp.2014. 04.057 · Zbl 1349.76333 · doi:10.1016/j.jcp.2014.04.057 |
| [63] | A. de Brauer, A. Iollo, T. Milcent, A Cartesian scheme for compressible multimaterial models in 3d, J. Comput. Phys. 313 (2016) 121-143. http://dx.doi.org/10.1016/j.jcp.2016.02. 032 · Zbl 1349.74370 · doi:10.1016/j.jcp.2016.02.032 |
| [64] | K. Shahbazi, Robust second-order scheme for multi-phase flow computations, J. Comput. Phys. 339 (2017) 163-178. http://dx.doi.org/10.1016/j.jcp.2017.03.025 · Zbl 1375.76101 · doi:10.1016/j.jcp.2017.03.025 |
| [65] | X. Y. Hu, N. A. Adams, G. Iaccarino, On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow, J. Comput. Phys. 228 (17) (2009) 6572-6589. http://dx. doi.org/10.1016/j.jcp.2009.06.002 · Zbl 1261.76023 · doi:10.1016/j.jcp.2009.06.002 |
| [66] | X. Deng, S. Inaba, B. Xie, et al., High fidelity discontinuity-resolving reconstruction for com-pressible multiphase flows with moving interfaces, J. Comput. Phys. 371 (2018) 945-966. http://dx.doi.org/10.1016/j.jcp.2018.03.036 · Zbl 1415.76465 · doi:10.1016/j.jcp.2018.03.036 |
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