A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: second-order, energy-stable numerical methods on adaptive octree based meshes. (English) Zbl 1512.76113
Summary: We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in [M. A. Khanwale et al., J. Comput. Phys. 419, Article ID 109674, 40 p. (2020; Zbl 07507235)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.
MSC:
| 76T06 | Liquid-liquid two component flows |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76M30 | Variational methods applied to problems in fluid mechanics |
Keywords:
two-phase flow; adaptive finite element method; fully-implicit, Crank-Nicolson time-marching scheme; variational multiscale method; cubic driven cavity flowCitations:
Zbl 07507235References:
| [1] | Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Annu. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051 |
| [2] | Teh, S.-y.; Lin, R.; Hung, L.-h.; Lee, A. P., Lab Chip, 8, 198 (2008) |
| [3] | Dangla, R.; Kayi, S. C.; Baroud, C. N., Proc. Natl. Acad. Sci., 110, 853-858 (2013) |
| [4] | Stoecklein, D.; Wu, C.-Y.; Owsley, K.; Xie, Y.; Di Carlo, D.; Ganapathysubramanian, B., Lab Chip, 14, 4197-4204 (2014) |
| [5] | Stoecklein, D.; Wu, C. Y.; Kim, D.; Di Carlo, D.; Ganapathysubramanian, B., Phys. Fluids, 28 (2016) |
| [6] | Khanwale, M. A.; Lofquist, A. D.; Sundar, H.; Rossmanith, J. A.; Ganapathysubramanian, B., J. Comput. Phys., 419, Article 109674 pp. (2020) · Zbl 07507235 |
| [7] | Osher, S.; Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 153 (2003), Springer · Zbl 1026.76001 |
| [8] | Yan, J.; Lin, S.; Bazilevs, Y.; Wagner, G. J., Comput. Fluids (2018) |
| [9] | Jacqmin, D., (34th AIAA Aerospace Sciences Meeting & Exhibit, vol. AIAA96 (1996), American Institute of Aeronautics and Astronautics: American Institute of Aeronautics and Astronautics Reno, Nevada, USA), 0858 |
| [10] | Jacqmin, D., J. Fluid Mech., 402, 57-88 (2000) · Zbl 0984.76084 |
| [11] | Guo, Z.; Lin, P.; Lowengrub, J.; Wise, S. M., Comput. Methods Appl. Mech. Eng., 326, 144-174 (2017) · Zbl 1439.76121 |
| [12] | Shokrpour Roudbari, M.; Şimşek, G.; van Brummelen, E. H.; van der Zee, K. G., Math. Models Methods Appl. Sci., 28, 733-770 (2018) · Zbl 1390.76047 |
| [13] | Xu, S.; Gao, B.; Lofquist, A.; Fernando, M.; Hsu, M.-C.; Sundar, H.; Ganapathysubramanian, B., Comput. Fluids, 214, Article 104764 pp. (2021) · Zbl 1521.76876 |
| [14] | Saurabh, K.; Gao, B.; Fernando, M.; Xu, S.; Khara, B.; Khanwale, M. A.; Hsu, M.-C.; Krishnamurthy, A.; Sundar, H.; Ganapathysubramanian, B. (2020), arXiv preprint |
| [15] | Volker, J., Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics Book, vol. 51 (2016), Springer · Zbl 1358.76003 |
| [16] | Burstedde, C.; Wilcox, L. C.; Ghattas, O., SIAM J. Sci. Comput., 33, 1103-1133 (2011) · Zbl 1230.65106 |
| [17] | Sundar, H.; Sampath, R. S.; Adavani, S. S.; Davatzikos, C.; Biros, G., (Proceedings of the 2007 ACM/IEEE Conference on Supercomputing (2007), ACM), 25 |
| [18] | Sundar, H.; Sampath, R. S.; Biros, G., SIAM J. Sci. Comput., 30, 2675-2708 (2008) · Zbl 1186.68554 |
| [19] | Fernando, M.; Sundar, H., paralab/Dendro-5.01: parallel adaptive octree library for finite different, finite volume and finite element computations (2020) |
| [20] | Kim, J.; Kang, K.; Lowengrub, J., J. Comput. Phys., 193, 511-543 (2004) · Zbl 1109.76348 |
| [21] | Feng, X., SIAM J. Numer. Anal., 44, 1049-1072 (2006) · Zbl 1344.76052 |
| [22] | Shen, J.; Yang, X., SIAM J. Sci. Comput., 32, 1159-1179 (2010) · Zbl 1410.76464 |
| [23] | Shen, J.; Yang, X., Chin. Ann. Math., Ser. B, 31, 743-758 (2010) · Zbl 1400.65049 |
| [24] | Dong, S., Comput. Methods Appl. Mech. Eng., 247-248, 179-200 (2012) · Zbl 1352.76119 |
| [25] | Chen, Y.; Shen, J., J. Comput. Phys., 308, 40-56 (2016) · Zbl 1352.65229 |
| [26] | Dong, S., Wall-bounded multiphase flows of N immiscible incompressible fluids: consistency and contact-angle boundary condition (2016) · Zbl 1415.76450 |
| [27] | Zhu, G.; Chen, H.; Yao, J.; Sun, S., Appl. Math. Model., 70, 82-108 (2019) · Zbl 1462.76133 |
| [28] | Chen, C.; Yang, X., ESAIM: Math. Model. Numer. Anal., 55, 2323-2347 (2021) · Zbl 1491.65091 |
| [29] | Han, D.; Wang, X., J. Comput. Phys., 290, 139-156 (2015) · Zbl 1349.76213 |
| [30] | Fu, G.; Han, D., Comput. Methods Appl. Mech. Eng., 387, Article 114186 pp. (2021) · Zbl 1507.76102 |
| [31] | Hughes, T. J.; Scovazzi, G.; Franca, L. P., Encyclopedia of Computational Mechanics, 1-64 (2018) |
| [32] | Hughes, T. J., Comput. Methods Appl. Mech. Eng., 127, 387-401 (1995) · Zbl 0866.76044 |
| [33] | Bazilevs, Y.; Calo, V.; Cottrell, J.; Hughes, T.; Reali, A.; Scovazzi, G., Comput. Methods Appl. Mech. Eng., 197, 173-201 (2007) · Zbl 1169.76352 |
| [34] | Mutlu, B. R.; Edd, J. F.; Toner, M., Proc. Natl. Acad. Sci., 115, 7682-7687 (2018) |
| [35] | Stoecklein, D.; Di Carlo, D., Anal. Chem., 91, 296-314 (2018) |
| [36] | Shen, J.; Yang, X., SIAM J. Numer. Anal., 53, 279-296 (2015) · Zbl 1327.65178 |
| [37] | Hughes, T. J.; Mazzei, L.; Jansen, K. E., Comput. Vis. Sci., 3, 47-59 (2000) · Zbl 0998.76040 |
| [38] | Ahmed, N.; Chacón Rebollo, T.; John, V.; Rubino, S., Arch. Comput. Methods Eng., 24, 115-164 (2017) · Zbl 1360.76105 |
| [39] | Coupez, T.; Hachem, E., Comput. Methods Appl. Mech. Eng., 267, 65-85 (2013) · Zbl 1286.76029 |
| [40] | Hachem, E.; Feghali, S.; Codina, R.; Coupez, T., Comput. Struct., 122, 88-100 (2013) |
| [41] | Hachem, E.; Khalloufi, M.; Bruchon, J.; Valette, R.; Mesri, Y., Comput. Methods Appl. Mech. Eng., 308, 238-255 (2016) · Zbl 1439.76070 |
| [42] | Ishii, M.; Fernando, M.; Saurabh, K.; Khara, B.; Ganapathysubramanian, B.; Sundar, H., (SC’19: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’19 (2019), Association for Computing Machinery: Association for Computing Machinery New York, NY, USA), 1-61 |
| [43] | Bao, H.; Bielak, J.; Ghattas, O.; Kallivokas, L. F.; O’Hallaron, D. R.; Shewchuk, J. R.; Xu, J., Comput. Methods Appl. Mech. Eng., 152, 85-102 (1998) · Zbl 0961.74056 |
| [44] | Fernando, M.; Neilsen, D.; Lim, H.; Hirschmann, E.; Sundar, H., SIAM J. Sci. Comput., 41, C97-C138 (2019) · Zbl 1416.83050 |
| [45] | Sundar, H.; Sampath, R. S.; Biros, G., SIAM J. Sci. Comput., 30, 2675-2708 (2008) · Zbl 1186.68554 |
| [46] | Burstedde, C.; Wilcox, L. C.; Ghattas, O., SIAM J. Sci. Comput., 33, 1103-1133 (2011) · Zbl 1230.65106 |
| [47] | Fernando, M.; Duplyakin, D.; Sundar, H., (Proceedings of the 26th International Symposium on High-Performance Parallel and Distributed Computing, HPDC ’17 (2017), ACM: ACM New York, NY, USA), 231-242 |
| [48] | Magaletti, F.; Picano, F.; Chinappi, M.; Marino, L.; Casciola, C. M., J. Fluid Mech., 714, 95-126 (2013) · Zbl 1284.76116 |
| [49] | Conti, M.; Giorgini, A., The three-dimensional Cahn-Hilliard-Brinkman system with unmatched viscosities (2018) |
| [50] | Giorgini, A.; Miranville, A.; Temam, R., SIAM J. Math. Anal., 51, 2535-2574 (2019) · Zbl 1419.35160 |
| [51] | Zeidler, E., Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators (1985), Springer: Springer Berlin · Zbl 0583.47051 |
| [52] | Tezduyar, T.; Mittal, S.; Ray, S.; Shih, R., Comput. Methods Appl. Mech. Eng., 95, 221-242 (1992) · Zbl 0756.76048 |
| [53] | Brooks, A.; Hughes, T., Comput. Methods Appl. Mech. Eng., 32, 199-259 (1982) · Zbl 0497.76041 |
| [54] | Hughes, T. J.; Sangalli, G., SIAM J. Numer. Anal., 45, 539-557 (2007) · Zbl 1152.65111 |
| [55] | Balay, S.; Gropp, W. D.; McInnes, L. C.; Smith, B. F., (Arge, E.; Bruaset, A. M.; Langtangen, H. P., Modern Software Tools in Scientific Computing (1997), Birkhäuser Press), 163-202 |
| [56] | Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Karpeyev, D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H. (2019), PETSc Web page |
| [57] | Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Karpeyev, D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H., PETSc Users Manual (2019), Argonne National Laboratory, Technical Report ANL-95/11 - Revision 3.11 |
| [58] | Sundar, H.; Sampath, R.; Biros, G., SIAM J. Sci. Comput., 30, 2675-2708 (2008) · Zbl 1186.68554 |
| [59] | Bern, M.; Eppstein, D.; Teng, S.-H., Int. J. Comput. Geom. Appl., 9, 517-532 (1999) · Zbl 1074.68630 |
| [60] | Sundar, H.; Sampath, R. S.; Adavani, S. S.; Davatzikos, C.; Biros, G., (SC’07: Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis (2007), ACM/IEEE), 1-12 |
| [61] | Hysing, S.-R.; Turek, S.; Kuzmin, D.; Parolini, N.; Burman, E.; Ganesan, S.; Tobiska, L., Int. J. Numer. Methods Fluids, 60, 1259-1288 (2009) · Zbl 1273.76276 |
| [62] | Aland, S.; Voigt, A., Int. J. Numer. Methods Fluids, 69, 747-761 (2012) |
| [63] | Yuan, H. Z.; Chen, Z.; Shu, C.; Wang, Y.; Niu, X. D.; Shu, S., J. Comput. Phys., 345, 404-426 (2017) · Zbl 1378.76120 |
| [64] | Xie, Y.; Wodo, O.; Ganapathysubramanian, B., Comput. Fluids, 141, 223-234 (2015) · Zbl 1390.76370 |
| [65] | Tryggvason, G.; Unverdi, S. O., Phys. Fluids A, Fluid Dyn., 2, 656-659 (1990) |
| [66] | Li, X.; Jin, B.; Glimm, J., J. Comput. Phys., 126, 343-355 (1996) · Zbl 0858.76055 |
| [67] | Guermond, J.-L.; Quartapelle, L., J. Comput. Phys., 165, 167-188 (2000) · Zbl 0994.76051 |
| [68] | Tryggvason, G., J. Comput. Phys., 75, 253-282 (1988) · Zbl 0638.76056 |
| [69] | Ding, H.; Spelt, P. D.; Shu, C., J. Comput. Phys., 226, 2078-2095 (2007) · Zbl 1388.76403 |
| [70] | Waddell, J. T.; Niederhaus, C. E.; Jacobs, J. W., Phys. Fluids, 13, 1263-1273 (2001) · Zbl 1184.76579 |
| [71] | Hunt, J.; Wray, A.; Moin, P., 193-208 (1988), Center for Turbulence Research Report CTR-S88 |
| [72] | Liang, H.; Li, Q. X.; Shi, B. C.; Chai, Z. H., Phys. Rev. E, 93, Article 1 pp. (2016) |
| [73] | Jain, S. S.; Mani, A.; Moin, P., J. Comput. Phys., 418, Article 109606 pp. (2020) · Zbl 07506170 |
| [74] | Chakravarthy, V.; Ottino, J., Chem. Eng. Sci., 51, 3613-3622 (1996) |
| [75] | Chella, R.; Viñals, J., Phys. Rev. E, 53, 3832-3840 (1996) |
| [76] | Park, K.; Dorao, C. A.; Fernandino, M., (ASME 2016 Fluids Engineering Division Summer Meeting, Volume 1B, Symposia: Fluid Mechanics (Fundamental Issues and Perspectives; Industrial and Environmental Applications); Multiphase Flow and Systems (Multiscale Methods; Noninvasive Measurements; Numerical Methods; Heat Transfer; Performance); Transport Phenomena (2016), American Society of Mechanical Engineers), 1-9 |
| [77] | Ishii, M.; Fernando, M.; Saurabh, K.; Khara, B.; Ganapathysubramanian, B.; Sundar, H., (Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2019)), 1-61 |
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