“A clebsch method for free-surface vortical flow simulation” by Xiong, Wang, Wang and Zhu

We propose a novel Clebsch method to simulate the free-surface vortical flow. At the center of our approach lies a level-set method enhanced by a wave-function correction scheme and a wave-function extrapolation algorithm to tackle the Clebsch method’s numerical instabilities near a dynamic interface. By combining the Clebsch wave function’s expressiveness in representing vortical structures and the level-set function’s ability on tracking interfacial dynamics, we can model complex vortex-interface interaction problems that exhibit rich free-surface flow details on a Cartesian grid. We showcase the efficacy of our approach by simulating a wide range of new free-surface flow phenomena that were impractical for previous methods, including horseshoe vortex, sink vortex, bubble rings, and free-surface wake vortices.

1. M. Aanjaneya, S. Patkar, and R. Fedkiw. 2013. A monolithic mass tracking formulation for bubbles in incompressible flow. J. Comput. Phys. 247 (2013), 17–61.Google ScholarCross Ref
2. H. Bateman. 1929. Notes on a differential equation that occurs in the two-dimensional motion of a compressible fluid and the associated variational problems. Proc. R. Soc. London Ser. A 125 (1929), 598–618.Google ScholarCross Ref
3. N. Chentanez and M. Müller. 2011. Real-time Eulerian water simulation using a restricted tall cell grid. ACM Trans. Graph. 30 (2011), 10.Google ScholarDigital Library
4. A. Chern. 2017. Fluid dynamics with incompressible Schrödinger flow. Ph.D. Dissertation. California Institute of Technology.Google Scholar
5. A. Chern, F. Knöppel, U. Pinkall, and P. Schröder. 2017. Inside fluids: Clebsch maps for visualization and processing. ACM Trans. Graph. 36 (2017), 142.Google ScholarDigital Library
6. A. Chern, F. Knöppel, U. Pinkall, P. Schröder, and S. Weißmann. 2016. Schrödinger’s smoke. ACM Trans. Graph. 35 (2016), 77.Google ScholarDigital Library
7. A. Clebsch. 1859. Ueber die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56 (1859), 1–10.Google ScholarCross Ref
8. M. Coquerelle and G. H. Cottet. 2008. A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys. 227 (2008), 9121–9137.Google ScholarDigital Library
9. C. Eckart. 1938. The electrodynamics of material media. Phys. Rev. 54 (1938), 920–923.Google ScholarCross Ref
10. D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. 2002. A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183 (2002), 83–116.Google ScholarDigital Library
11. H. Ertel. 1942. Ein neuer hydrodynamischer Wirbelsatz. Wirbelsatz. Meteorol. Z. 59 (1942), 271–281.Google Scholar
12. R. Fedkiw and S. Osher. 2002. Level set methods and dynamic implicit surfaces. Surfaces 44 (2002), 77.Google Scholar
13. R. Fedkiw, J. Stam, and H. W. Jensen. 2001. Visual simulation of smoke. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques. 15–22.Google Scholar
14. R. P. Fedkiw, T. Aslam, B. Merriman, and S. Osher. 1999. A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999), 457–492.Google ScholarDigital Library
15. F. Gibou, R. Fedkiw, and S. Osher. 2018. A review of level-set methods and some recent applications. J. Comput. Phys. 353 (2018), 82–109.Google ScholarCross Ref
16. J. Hao, S. Xiong, and Y. Yang. 2019. Tracking vortex surfaces frozen in the virtual velocity in non-ideal flows. J. Fluid Mech. 863 (2019), 513–544.Google ScholarCross Ref
17. P. He and Y. Yang. 2016. Construction of initial vortex-surface fields and Clebsch potentials for flows with high-symmetry using first integrals. Phys. Fluids 28 (2016), 037101.Google ScholarCross Ref
18. H. Helmholtz. 1858. Uber integrale der hydrodynamischen Gleichungen welche den Wirbel-bewegungen ensprechen. J. Reine Angew. Math 55 (1858), 25–55.Google ScholarCross Ref
19. H. Hopf. 1931. Über die Abbildungen der Dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104 (1931), 637–665.Google ScholarCross Ref
20. B. Houston, M. B. Nielsen, C. Batty, O. Nilsson, and K. Museth. 2006. Hierarchical RLE level set: A compact and versatile deformable surface representation. ACM Trans. Graph. 25 (2006), 151–175.Google ScholarDigital Library
21. D. L. Hu, B. Chan, and J. W. M. Bush. 2003. The hydrodynamics of water strider locomotion. Nature 424 (2003), 663–666.Google ScholarCross Ref
22. L. Huang and D. L. Michels. 2020. Surface-only ferrofluids. ACM Trans. Graph. 39 (2020), 6.Google ScholarDigital Library
23. L. Huang, Z. Qu, X. Tan, X. Zhang, D. L. Michels, and C. Jiang. 2021. Ships, splashes, and waves on a vast ocean. ACM Trans. Graph. 40 (2021), 203.Google ScholarDigital Library
24. C. J. Hughes, R. Grzeszczuk, E. Sifakis, D. Kim, S. Kumar, A. P. Selle, J. Chhugani, M. Holliman, and Y. Chen. 2007. Physical simulation for animation and visual effects: parallelization and characterization for chip multiprocessors. SIGARCH Comput. Archit. News 35 (2007), 220–231.Google ScholarDigital Library
25. C. Jiang, C. Schroeder, A. Selle, J. Teran, and A. Stomakhin. 2015. The affine particle-in-cell method. ACM Trans. Graph. 34 (2015), 51.Google ScholarDigital Library
26. H. Kedia, D. Foster, M. R. Dennis, and W. T. M. Irvine. 2016. Weaving knotted vector fields with tunable helicity. Phys. Rev. Lett. 117 (2016), 274501.Google ScholarCross Ref
27. E. A. Kuznetsov and A. V. Mikhailov. 1980. On the topological meaning of canonical Clebsch variables. Phys. Lett. A 77 (1980), 37–38.Google ScholarCross Ref
28. F. Losasso, R. Fedkiw, and S. Osher. 2006. Spatially adaptive techniques for level set methods and incompressible flow. Comput. Fluids 35 (2006), 995–1010.Google ScholarCross Ref
29. F. Losasso, F. Gibou, and R. Fedkiw. 2004. Simulating water and smoke with an octree data structure. In ACM Trans. Graph. 457–462.Google ScholarDigital Library
30. F. Losasso, J. Talton, N. Kwatra, and R. Fedkiw. 2008. Two-way coupled SPH and particle level set fluid simulation. IEEE Trans. Vis. Comput. Graph. 14 (2008), 797–804.Google ScholarDigital Library
31. R. W. MacCormack. 2003. The effect of viscosity in hypervelocity impact cratering. J. Spacecr. Rockets. 40 (2003), 757–763.Google ScholarCross Ref
32. E. Madelung. 1926. Eine anschauliche Deutung der Gleichung von Schrödinger. Naturwissenschaften 14 (1926), 1004–1004.Google ScholarCross Ref
33. E. Madelung. 1927. Quantentheorie in hydrodynamischer Form. Z. Phys. 40 (1927), 322–326.Google ScholarCross Ref
34. H. Mazhar, T. Heyn, A. Pazouki, D. Melanz, A. Seidl, A. Bartholomew, A. Tasora, and D. Negrut. 2013. Chrono: a parallel multi-physics library for rigid-body, flexible-body, and fluid dynamics. Mech. Sci. 4 (2013), 49–64.Google ScholarCross Ref
35. R. I. McLachlan, C. Offen, and B. K. Tapley. 2019. Symplectic integration of PDEs using Clebsch variables. J. Comput. Dyn. 6 (2019), 111–130.Google Scholar
36. S. Mochizuki, K. Suzukawa, K. Saga, and H. Osaka. 2008. Vortex Structures around a Flat Paddle Impeller in a Stirred Vessel. J. Fluid Sci. 3 (2008), 241–249.Google ScholarCross Ref
37. H. K. Moffatt. 1969. The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35 (1969), 117–129.Google ScholarCross Ref
38. J. J. Moreau. 1961. Constantes d’un îlot tourbillonnaire en fluide parfait barotrope. C. R. Acad. Sci. Paris 252 (1961), 2810–2812.Google Scholar
39. M. Müller, D. Charypar, and M. H. Gross. 2003. Particle-based fluid simulation for interactive applications.. In Symposium on Computer Animation. 154–159.Google Scholar
40. S. Osher and R. P. Fedkiw. 2001. Level set methods: an overview and some recent results. J. Comput. Phys. 169 (2001), 463–502.Google ScholarDigital Library
41. S. Osher and J. A. Sethian. 1988. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 160 (1988), 151–178.Google Scholar
42. M. Padilla, A. Chern, F. Knöppel, U. Pinkall, and P. Schröder. 2019. On bubble rings and ink chandeliers. ACM Trans. Graph. 38 (2019), 1–14.Google ScholarDigital Library
43. L. M. Pismen and L. M Pismen. 1999. Vortices in nonlinear fields: from liquid crystals to superfluids, from non-equilibrium patterns to cosmic strings. Vol. 100. Oxford University Press.Google Scholar
44. Z. Qu, X. Zhang, M. Gao, C. Jiang, and B. Chen. 2019. Efficient and conservative fluids using bidirectional mapping. ACM Trans. Graph. 38 (2019), 4.Google ScholarDigital Library
45. N. Rasmussen, D. Q. Nguyen, W. Geiger, and R. Fedkiw. 2003. Smoke simulation for large scale phenomena. ACM Trans. Graph. 22 (2003), 703–707.Google ScholarDigital Library
46. R. Salmon. 1988. Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. 20 (1988), 225–256.Google ScholarCross Ref
47. R. Saye. 2016. Interfacial gauge methods for incompressible fluid dynamics. Sci. Adv. 2 (2016), e1501869.Google ScholarCross Ref
48. A. Selle, N. Rasmussen, and R. Fedkiw. 2005. A vortex particle method for smoke, water and explosions. ACM Trans. Graph. 24 (2005), 910–914.Google ScholarDigital Library
49. A. L. Sorokin. 2001. Madelung transformation for vortex flows of a perfect liquid. Dokl. Phys. 46 (2001), 576–578.Google ScholarCross Ref
50. M. Sussman. 2003. A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles. J. Comput. Phys. 187 (2003), 110–136.Google ScholarDigital Library
51. R. Tao, H. Ren, Y. Tong, and S. Xiong. 2021. Construction and evolution of knotted vortex tubes in incompressible Schrödinger flow. Phys. Fluids 33 (2021), 077112.Google ScholarCross Ref
52. G. I. Taylor and A. E. Green. 1937. Mechanism of the production of small eddies from large ones. Proc. Roy. Soc. Lond. A 158 (1937), 499–521.Google ScholarCross Ref
53. W. Thomson. 1869. On vortex motion. Trans. R. Soc. Edinburgh 25 (1869), 217–260.Google ScholarCross Ref
54. B. Tings and D. Velotto. 2018. Comparison of ship wake detectability on C-band and X-band SAR. Int. J. Remote Sens 39 (2018), 4451–4468.Google ScholarCross Ref
55. K. Marten; K. Shariff; S. Psarakos; D. J. White. 1996. Ring bubbles of dolphins. Sci. Am. 275 (1996), 82.Google Scholar
56. J. Z. Wu, H. Y. Ma, and M. D. Zhou. 2006. Vorticity and Vortex Dynamics. Springer.Google Scholar
57. J. Z. Wu, H. Y. Ma, and M. D. Zhou. 2015. Vortical Flows. Springer.Google Scholar
58. S. Xiong, R. TAO, Y. Zhang, F. Feng, and B. ZHU. 2021. Incompressible Flow Simulation on Vortex Segment Clouds. ACM Trans. Graph. 40 (2021), 98.Google ScholarDigital Library
59. S. Xiong and Y. Yang. 2017. The boundary-constraint method for constructing vortex-surface fields. J. Comput. Phys. 339 (2017), 31–45.Google ScholarDigital Library
60. S. Xiong and Y. Yang. 2019a. Construction of knotted vortex tubes with the writhe-dependent helicity. Phys. Fluids 31 (2019), 047101.Google ScholarCross Ref
61. S. Xiong and Y. Yang. 2019b. Identifying the tangle of vortex tubes in homogeneous isotropic turbulence. J. Fluid Mech. 874 (2019), 952–978.Google ScholarCross Ref
62. S. Xiong and Y. Yang. 2020. Effects of twist on the evolution of knotted magnetic flux tubes. J. Fluid Mech. 895 (2020), A28.Google ScholarCross Ref
63. S. Yang, S. Xiong, Y. Zhang, F. Feng, J. Liu, and B. Zhu. 2021. Clebsch gauge fluid. ACM Trans. Graph. 40 (2021), 99.Google ScholarDigital Library
64. Y. Yang and D. I. Pullin. 2010. On Lagrangian and vortex-surface fields for flows with Taylor-Green and Kida-Pelz initial conditions. J. Fluid Mech. 661 (2010), 446–481.Google ScholarCross Ref
65. Y. Yang and D. I. Pullin. 2011. Evolution of vortex-surface fields in viscous Taylor-Green and Kida-Pelz flows. J. Fluid Mech. 685 (2011), 146–164.Google ScholarCross Ref
66. X. Zhang and R. Bridson. 2014. A PPPM fast summation method for fluids and beyond. ACM Trans. Graph. 33 (2014), 6.Google ScholarDigital Library
67. Y. Zhu and R. Bridson. 2005. Animating sand as a fluid. ACM Trans. Graph. 24 (2005), 965–972.Google ScholarDigital Library