Clebsch gauge fluid

Abstract

We propose a novel gauge fluid solver based on Clebsch wave functions to solve incompressible fluid equations. Our method combines the expressive power of Clebsch wave functions to represent coherent vortical structures and the generality of gauge methods to accommodate a broad array of fluid phenomena. By evolving a transformed wave function as the system's gauge variable enhanced by an additional projection step to enforce pressure jumps on the free boundaries, our method can significantly improve the vorticity generation and preservation ability for a broad range of gaseous and liquid phenomena. Our approach can be easily implemented by modifying a standard grid-based fluid simulator. It can be used to solve various fluid dynamics, including complex vortex filament dynamics, fluids with different obstacles, and surface-tension flow.

References

1. Alexis Angelidis and Fabrice Neyret. 2005. Simulation of smoke based on vortex filament primitives. In Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation. 87--96.
2. Landon Boyd and Robert Bridson. 2012. MultiFLIP for energetic two-phase fluid simulation. ACM Transactions on Graphics (TOG) 31, 2 (2012), 1--12.
3. Jeremiah U Brackbill and Hans M Ruppel. 1986. FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. J. Comput. Phys. 65, 2 (1986), 314--343.
4. Axel Brandenburg. 2010. Magnetic field evolution in simulations with euler potentials. MON. NOT. R. ASTRON. SOC. 401 (2010), 347--354.
5. Robert Bridson, Jim Houriham, and Marcus Nordenstam. 2007. Curl-noise for procedural fluid flow. ACM Transactions on Graphics (ToG) 26, 3 (2007).
6. Tyson Brochu, Todd Keeler, and Robert Bridson. 2012. Linear-time smoke animation with vortex sheet meshes. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Citeseer, 87--95.
7. Tomas F. Buttke. 1993. Velicity Methods: Lagrangian Numerical Methods which Preserve the Hamiltonian Structure of Incompressible Fluid Flow. Springer Netherlands, Dordrecht, 39--57.
8. Thomas F. Buttke and Alexandre J. Chorin. 1993. Turbulence calculations in magnetization variables. Applied Numerical Mathematics 12, 1 (1993), 47 -- 54. SPECIAL ISSUE.
9. C. Cartes, M. D. Bustamante, and M. E. Brachet. 2007. Generalized Eulerian-Lagrangian description of Navier-Stokes dynamics. Phys. Fluids 19 (2007), 077101.
10. A. Chern. 2017. Fluid Dynamics with Incompressible Schrödinger Flow. Ph.D. Dissertation. California institute of technology.
11. A. Chern, F. Knöppel, U. Pinkall, and P. Schröder. 2017. Inside fluids: Clebsch maps for visualization and processing. ACM Trans. Graph. 36 (2017), 142.
12. A. Chern, F. Knöppel, U. Pinkall, P. Schröder, and S. Weißmann. 2016. Schrödinger's smoke. ACM Trans. Graph. 35 (2016), 77.
13. A. Clebsch. 1859. Ueber die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56 (1859), 1--10.
14. Georges-Henri Cottet, Petros D Koumoutsakos, et al. 2000. Vortex methods: theory and practice. Vol. 8. Cambridge university press Cambridge.
15. Ounan Ding, Tamar Shinar, and Craig Schroeder. 2020. Affine particle in cell method for MAC grids and fluid simulation. J. Comput. Phys. 408 (2020), 109311.
16. Aleksandar Donev, Andy Nonaka, Yifei Sun, Thomas Fai, Alejandro Garcia, and John Bell. 2014. Low Mach number fluctuating hydrodynamics of diffusively mixing fluids. Communications in Applied Mathematics and Computational Science 9, 1 (May 2014), 47--105.
17. Weinan E and Jian-Guo Liu. 1997. Finite Difference Schemes for Incompressible Flows in the Velocity-Impulse Density Formulation. J. Comput. Phys. 130, 1 (1997), 67 -- 76.
18. Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schröder, and Mathieu Desbrun. 2007. Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics (TOG) 26, 1 (2007).
19. Ronald Fedkiw, J. Stam, and H. Jensen. 2001. Visual simulation of smoke. Proceedings of the 28th annual conference on Computer graphics and interactive techniques (2001).
20. N. Foster and Ronald Fedkiw. 2001. Practical animation of liquids. Proceedings of the 28th annual conference on Computer graphics and interactive techniques (2001).
21. C. Fu, Q. Guo, Theodore F. Gast, Chenfanfu Jiang, and J. Teran. 2017. A polynomial particle-in-cell method. ACM Transactions on Graphics (TOG) 36 (2017), 1--12.
22. S. Gagniere, David Hyde, A. Marquez-Razon, C. Jiang, Z. Ge, X. Han, Q. Guo, and J. Teran. 2020. A Hybrid Lagrangian/Eulerian Collocated Advection and Projection Method for Fluid Simulation. ArXiv abs/2003.12227 (2020).
23. C. R. Graham and F. S. Henyey. 2000. Clebsch representation near points where the vorticity vanishes. Phys. Fluids 12 (2000), 744--746.
24. 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.
25. H. Hopf. 1931. Über die Abbildungen der Dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104 (1931), 637--665.
26. J. Jeong and F. Hussain. 1995. On the identification of a vortex. J. Fluid. Mech. 285 (1995), 69--94.
27. Chenfanfu Jiang, Craig Schroeder, Andrew Selle, Joseph Teran, and Alexey Stomakhin. 2015. The affine particle-in-cell method. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1--10.
28. Myungjoo Kang, Ronald P Fedkiw, and Xu-Dong Liu. 2000. A boundary condition capturing method for multiphase incompressible flow. Journal of Scientific Computing 15, 3 (2000), 323--360.
29. 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.
30. ByungMoon Kim, Y. Liu, I. Llamas, and J. Rossignac. 2005. FlowFixer: Using BFECC for Fluid Simulation. In NPH.
31. Doyub Kim, Oh-Young Song, and Hyeongseok Ko. 2009. Stretching and wiggling liquids. ACM SIGGRAPH Asia 2009 papers (2009).
32. Theodore Kim, Nils Thürey, Doug James, and Markus Gross. 2008. Wavelet turbulence for fluid simulation. ACM Transactions on Graphics (TOG) 27, 3 (2008), 1--6.
33. Dustin Kleckner and William TM Irvine. 2013. Creation and dynamics of knotted vortices. Nat. Phys. 9, 4 (2013), 253--258.
34. P. Robert Kotiuga. 1991. Clebsch potentials and the visualization of three-dimensional solenoidal vector fields. IEEE T. MAGN 27 (1991), 3986--3989.
35. Petros Koumoutsakos, Georges-Henri Cottet, and Diego Rossinelli. 2008. Flow simulations using particles-Bridging Computer Graphics and CFD. In SIGGRAPH 2008-35th International Conference on Computer Graphics and Interactive Techniques. ACM, 1--73.
36. G. A. Kuz'min. 1983. Ideal incompressible hydrodynamics in terms of the vortex momentum density. Phys. Lett. A 96 (1983), 88--90.
37. H. Lamb. 1932. Hydrodynamics (6 ed.). Cambridge University Press.
38. A Leonard. 1980. Vortex methods for flow simulation. J. Comput. Phys. 37, 3 (1980), 289--335.
39. TT Lim. 1989. An experimental study of a vortex ring interacting with an inclined wall. Exp. Fluids 7, 7 (1989), 453--463.
40. TT Lim. 1997. A note on the leapfrogging between two coaxial vortex rings at low Reynolds numbers. Phys. Fluids 9, 1 (1997), 239--241.
41. Miao'er Liu, Yu-Xin Ren, and Hanxin Zhang. 2004. A class of fully second order accurate projection methods for solving the incompressible Navier-Stokes equations. J. Comput. Phys. 200, 1 (2004), 325 -- 346.
42. John H. Maddocks and Robert L. Pego. 1995. An unconstrained Hamiltonian formulation for incompressible fluid flow. Comm. Math. Phys. 170, 1 (1995), 207--217.
43. Patrick Mullen, Keenan Crane, Dmitry Pavlov, Yiying Tong, and Mathieu Desbrun. 2009. Energy-preserving integrators for fluid animation. ACM Transactions on Graphics (TOG) 28, 3 (2009), 1--8.
44. V I Oseledets. 1989. On a new way of writing the Navier-Stokes equation. The Hamiltonian formalism. Russian Mathematical Surveys 44, 3 (jun 1989), 210--211.
45. S. Park and M. Kim. 2005. Vortex fluid for gaseous phenomena. In SCA '05.
46. Tobias Pfaff, Nils Thuerey, and Markus Gross. 2012. Lagrangian vortex sheets for animating fluids. ACM Transactions on Graphics (TOG) 31, 4 (2012), 1--8.
47. Ziyin Qu, Xinxin Zhang, Ming Gao, Chenfanfu Jiang, and Baoquan Chen. 2019. Efficient and conservative fluids using bidirectional mapping. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--12.
48. R. Saye. 2016. Interfacial gauge methods for incompressible fluid dynamics. Sci. Adv. 2 (2016), e1501869.
49. Robert Saye. 2017a. Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: Part I. J. Comput. Phys. 344 (2017), 647 -- 682.
50. Robert Saye. 2017b. Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: Part II. J. Comput. Phys. 344 (2017), 683 -- 723.
51. A. Selle, Ronald Fedkiw, ByungMoon Kim, Y. Liu, and J. Rossignac. 2008. An Unconditionally Stable MacCormack Method. Journal of Scientific Computing 35 (2008), 350--371.
52. Andrew Selle, Nick Rasmussen, and Ronald Fedkiw. 2005. A vortex particle method for smoke, water and explosions. In ACM SIGGRAPH 2005 Papers. 910--914.
53. C. B. Smiet, S. Candelaresi, and D. Bouwmeester. 2017. Ideal relaxation of the Hopf fibration. Phys. Plasmas 24 (2017), 072110.
54. C. B. Smiet, S. Candelaresi, A. Thompson, J. Swearngin, J.W. Dalhuisen, and D. Bouwmeester. 2015. Self-organizing knotted magnetic structures in plasma. Phys. Rev. Lett. 115 (2015), 095001.
55. J. Stam. 1999. Stable fluids. In SIGGRAPH '99.
56. Mark J Stock, Werner JA Dahm, and Grétar Tryggvason. 2008. Impact of a vortex ring on a density interface using a regularized inviscid vortex sheet method. J. Comput. Phys. 227, 21 (2008), 9021--9043.
57. D.M. Summers. 2000. A Representation of Bounded Viscous Flow Based on Hodge Decomposition of Wall Impulse. J. Comput. Phys. 158, 1 (2000), 28 -- 50.
58. D M Summers and A J Chorin. 1996. Numerical vorticity creation based on impulse conservation. Proceedings of the National Academy of Sciences 93, 5 (1996), 1881--1885.
59. S. Weißmann and U. Pinkall. 2010. Filament-based smoke with vortex shedding and variational reconnection. ACM Trans. Graph. 29 (2010), 115.
60. S. Xiong and Y. Yang. 2017. The boundary-constraint method for constructing vortex-surface fields. J. Comput. Phys. 339 (2017), 31--45.
61. S. Xiong and Y. Yang. 2019. Identifying the tangle of vortex tubes in homogeneous isotropic turbulence. J. Fluid Mech. 874 (2019), 952--978.
62. S. Xiong and Y. Yang. 2020. Evolution and helicity analysis of linked vortex tubes in viscous flows. Sci. Sin-Phys. Mech. Astron. 50 (2020), 040005.
63. 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.
64. V. E. Zakharov and E. A. Kuznetsov. 1997. Hamiltonian formalism for nonlinear waves. Phys.-Usp. 40 (1997), 1087--1116.
65. Xinxin Zhang, Robert Bridson, and Chen Greif. 2015. Restoring the missing vorticity in advection-projection fluid solvers. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1--8.
66. Y. Zhao, S. Xiong, Y. Yang, and S. Chen. 2018. Sinuous distortion of vortex surfaces in the lateral growth of turbulent spots. Phys. Rev. Fluids 3 (2018), 074701.
67. Y. Zhao, Y. Yang, and S. Chen. 2016. Vortex reconnection in the late transition in channel flow. J. Fluid Mech. 802 (2016), R4.
68. Yongning Zhu and Robert Bridson. 2005. Animating sand as a fluid. ACM Transactions on Graphics (TOG) 24, 3 (2005), 965--972.

Index Terms

• Computing methodologies

  • Modeling and simulation