Authors

Pankaj Jagad, Abdullah Abukhwejah, Mamdouh Mohamed, Ravi Samtaney

Abstract

A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier–Stokes equations is performed. An existing DEC method [M. S. Mohamed, A. N. Hirani, and R. Samtaney, “Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes,” J. Comput. Phys. 312, 175–191 (2016)] is modified to this end and is extended to include the energy-preserving time integration and the Coriolis force to enhance its applicability to investigate the late-time behavior of flows on rotating surfaces, i.e., that of the planetary flows. The simulation experiments show second order accuracy of the scheme for the structured-triangular meshes and first order accuracy for the otherwise unstructured meshes. The method exhibits a second order kinetic energy relative error convergence rate with mesh size for inviscid flows. The test case of flow on a rotating sphere demonstrates that the method preserves the stationary state and conserves the inviscid invariants over an extended period of time.

Citation

  • Journal: Physics of Fluids
  • Year: 2021
  • Volume: 33
  • Issue: 1
  • Pages:
  • Publisher: AIP Publishing
  • DOI: 10.1063/5.0035981

BibTeX

@article{Jagad_2021,
  title={{A primitive variable discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes}},
  volume={33},
  ISSN={1089-7666},
  DOI={10.1063/5.0035981},
  number={1},
  journal={Physics of Fluids},
  publisher={AIP Publishing},
  author={Jagad, Pankaj and Abukhwejah, Abdullah and Mohamed, Mamdouh and Samtaney, Ravi},
  year={2021}
}

Download the bib file

References

  • Marsden, J. E. & West, M. Discrete mechanics and variational integrators. Acta Numerica vol. 10 357–514 (2001) – 10.1017/s096249290100006x
  • Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2006)
  • Compatible Spatial Discretizations (2007)
  • Perot, J. B. & Subramanian, V. Discrete calculus methods for diffusion. Journal of Computational Physics vol. 224 59–81 (2007) – 10.1016/j.jcp.2006.12.022
  • Poincaré, H. Sur les résidus des intégrales doubles. Acta Mathematica vol. 9 321–380 (1887) – 10.1007/bf02406742
  • Cartan, É. Sur certaines expressions différentielles et le problème de Pfaff. Annales scientifiques de l’École normale supérieure vol. 16 239–332 (1899) – 10.24033/asens.467
  • Goursat, E. Sur certains systèmes d’équations aux différentiels totales et sur une généralisation du problème de Pfaff. Annales de la faculté des sciences de Toulouse Mathématiques vol. 7 1–58 (1915) – 10.5802/afst.298
  • The classical theory of electricity and magnetism. (1932)
  • Leçons sur la géométrie des espaces de riemann (1946)
  • Variétés Différentiables: Formes, Courants, Formes Harmoniques (1955)
  • Discrete exterior calculus. Int. J. Comput. Methods Eng. Sci. Mech. (2015)
  • Discrete differential forms for computational modeling. Discrete Differential Geometry (2008)
  • Discrete exterior calculus for variational problems in computer vision and graphics.
  • Discrete differential geometry: An applied introduction. (2006)
  • Elcott, S., Tong, Y., Kanso, E., Schröder, P. & Desbrun, M. Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics vol. 26 4 (2007) – 10.1145/1189762.1189766
  • Mullen, P., Crane, K., Pavlov, D., Tong, Y. & Desbrun, M. Energy-preserving integrators for fluid animation. ACM Transactions on Graphics vol. 28 1–8 (2009) – 10.1145/1531326.1531344
  • Digital geometry processing with discrete exterior calculus. (2013)
  • de Goes, F., Desbrun, M., Meyer, M. & DeRose, T. Subdivision exterior calculus for geometry processing. ACM Transactions on Graphics vol. 35 1–11 (2016) – 10.1145/2897824.2925880
  • Mohamed, M. S., Hirani, A. N. & Samtaney, R. Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes. Journal of Computational Physics vol. 312 175–191 (2016) – 10.1016/j.jcp.2016.02.028
  • Nitschke, I., Reuther, S. & Voigt, A. Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation. Advances in Mathematical Fluid Mechanics 177–197 (2017) doi:10.1007/978-3-319-56602-3_7 – 10.1007/978-3-319-56602-3_7
  • Jagad, P., Mohamed, M. S. & Samtaney, R. Investigation of flow past a cylinder embedded on curved and flat surfaces. Physical Review Fluids vol. 5 (2020) – 10.1103/physrevfluids.5.044701
  • Zhang, X., Schmidt, D. & Perot, B. Accuracy and Conservation Properties of a Three-Dimensional Unstructured Staggered Mesh Scheme for Fluid Dynamics. Journal of Computational Physics vol. 175 764–791 (2002) – 10.1006/jcph.2001.6973
  • Frank, J. BIT Numerical Mathematics vol. 43 41–55 (2003) – 10.1023/a:1023620100065
  • Mittal, R. & Moin, P. Suitability of Upwind-Biased Finite Difference Schemes for Large-Eddy Simulation of Turbulent Flows. AIAA Journal vol. 35 1415–1417 (1997) – 10.2514/2.253
  • Simulating Jupiter’s great red spot with discrete exterior calculus. (2020)
  • Perot, J. B. Discrete Conservation Properties of Unstructured Mesh Schemes. Annual Review of Fluid Mechanics vol. 43 299–318 (2011) – 10.1146/annurev-fluid-122109-160645
  • Flow discretization by complementary volume techniques. (1989)
  • Hall, C. A., Cavendish, J. C. & Frey, W. H. The dual variable method for solving fluid flow difference equations on Delaunay triangulations. Computers & Fluids vol. 20 145–164 (1991) – 10.1016/0045-7930(91)90017-c
  • Nicolaides, R. A. Direct Discretization of Planar Div-Curl Problems. SIAM Journal on Numerical Analysis vol. 29 32–56 (1992) – 10.1137/0729003
  • Perot, B. Conservation Properties of Unstructured Staggered Mesh Schemes. Journal of Computational Physics vol. 159 58–89 (2000) – 10.1006/jcph.2000.6424
  • Lindborg, E. & Mohanan, A. V. A two-dimensional toy model for geophysical turbulence. Physics of Fluids vol. 29 (2017) – 10.1063/1.4985990
  • Rahman, Sk. M., Ahmed, S. E. & San, O. A dynamic closure modeling framework for model order reduction of geophysical flows. Physics of Fluids vol. 31 (2019) – 10.1063/1.5093355
  • Espa, S., Cabanes, S., King, G. P., Di Nitto, G. & Galperin, B. Eddy–wave duality in a rotating flow. Physics of Fluids vol. 32 (2020) – 10.1063/5.0006206
  • Hamed, A. M., Kamdar, A., Castillo, L. & Chamorro, L. P. Turbulent boundary layer over 2D and 3D large-scale wavy walls. Physics of Fluids vol. 27 (2015) – 10.1063/1.4933098
  • Vortices on closed surfaces. Geometry, Mechanics, and Dynamics (2015)
  • Hu, D., Zhang, P. & E, W. Continuum theory of a moving membrane. Physical Review E vol. 75 (2007) – 10.1103/physreve.75.041605
  • Rivera, M. K., Aluie, H. & Ecke, R. E. The direct enstrophy cascade of two-dimensional soap film flows. Physics of Fluids vol. 26 (2014) – 10.1063/1.4873579
  • Kellay, H. Hydrodynamics experiments with soap films and soap bubbles: A short review of recent experiments. Physics of Fluids vol. 29 (2017) – 10.1063/1.4986003
  • Liao, Y. & Ouellette, N. T. Geometry of scale-to-scale energy and enstrophy transport in two-dimensional flow. Physics of Fluids vol. 26 (2014) – 10.1063/1.4871107
  • Mohamed, M. S., Hirani, A. N. & Samtaney, R. Numerical convergence of discrete exterior calculus on arbitrary surface meshes. International Journal for Computational Methods in Engineering Science and Mechanics vol. 19 194–206 (2018) – 10.1080/15502287.2018.1446196
  • Shi, J.-M., Gerlach, D., Breuer, M., Biswas, G. & Durst, F. Heating effect on steady and unsteady horizontal laminar flow of air past a circular cylinder. Physics of Fluids vol. 16 4331–4345 (2004) – 10.1063/1.1804547
  • Russell, D. & Jane Wang, Z. A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow. Journal of Computational Physics vol. 191 177–205 (2003) – 10.1016/s0021-9991(03)00310-3
  • Kawaguti, M. Numerical Solution of the Navier-Stokes Equations for the Flow around a Circular Cylinder at Reynolds Number 40. Journal of the Physical Society of Japan vol. 8 747–757 (1953) – 10.1143/jpsj.8.747
  • Le, D. V., Khoo, B. C. & Peraire, J. An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries. Journal of Computational Physics vol. 220 109–138 (2006) – 10.1016/j.jcp.2006.05.004
  • Henderson, R. D. & Karniadakis, G. E. Unstructured Spectral Element Methods for Simulation of Turbulent Flows. Journal of Computational Physics vol. 122 191–217 (1995) – 10.1006/jcph.1995.1208
  • SHEARD, G. J., HOURIGAN, K. & THOMPSON, M. C. Computations of the drag coefficients for low-Reynolds-number flow past rings. Journal of Fluid Mechanics vol. 526 257–275 (2005) – 10.1017/s0022112004002836
  • Stålberg, E., Brüger, A., Lötstedt, P., Johansson, A. V. & Henningson, D. S. High Order Accurate Solution of Flow Past a Circular Cylinder. Journal of Scientific Computing vol. 27 431–441 (2006) – 10.1007/s10915-005-9043-y
  • Laminar flow past a circular cylinder at Reynolds number varying from 50 to 5000. (2005)
  • Dennis, S. C. R. & Chang, G.-Z. Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. Journal of Fluid Mechanics vol. 42 471–489 (1970) – 10.1017/s0022112070001428
  • Kawaguti, M. & Jain, P. Numerical Study of a Viscous Fluid Flow past a Circular Cylinder. Journal of the Physical Society of Japan vol. 21 2055–2062 (1966) – 10.1143/jpsj.21.2055
  • Taneda, S. Experimental Investigation of the Wakes behind Cylinders and Plates at Low Reynolds Numbers. Journal of the Physical Society of Japan vol. 11 302–307 (1956) – 10.1143/jpsj.11.302
  • The Steady Flow of a Viscous Fluid Past a Circular Cylinder at Reynolds Numbers 40 and 44 (1961)
  • MITTAL, S. & KUMAR, B. Flow past a rotating cylinder. Journal of Fluid Mechanics vol. 476 303–334 (2003) – 10.1017/s0022112002002938
  • Kurtulus, D. F. On the Unsteady Behavior of the Flow around NACA 0012 Airfoil with Steady External Conditions at Re=1000. International Journal of Micro Air Vehicles vol. 7 301–326 (2015) – 10.1260/1756-8293.7.3.301
  • Connors, J. Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model. Numerical Methods for Partial Differential Equations vol. 26 1328–1350 (2009) – 10.1002/num.20493
  • Bell, J. B., Colella, P. & Glaz, H. M. A second-order projection method for the incompressible navier-stokes equations. Journal of Computational Physics vol. 85 257–283 (1989) – 10.1016/0021-9991(89)90151-4
  • Shi, Q., Chen, Y. & Xie, X. Interplay of surface geometry and vorticity dynamics in incompressible flows on curved surfaces. Applied Mathematics and Mechanics vol. 38 1191–1212 (2017) – 10.1007/s10483-017-2238-8
  • Turner, A. M., Vitelli, V. & Nelson, D. R. Vortices on curved surfaces. Reviews of Modern Physics vol. 82 1301–1348 (2010) – 10.1103/revmodphys.82.1301
  • Reuther, S. & Voigt, A. The Interplay of Curvature and Vortices in Flow on Curved Surfaces. Multiscale Modeling & Simulation vol. 13 632–643 (2015) – 10.1137/140971798
  • DOI not foun – 10.1175/1520-0469(1946)003<0053:tmohwi>2.0.co;2
  • Integration of the Nondivergent Barotropic Vorticity Equation with an Icosahedral-Hexagonal Grid for the Sphere (1968)
  • Blair Perot, J. & Zusi, C. J. Differential forms for scientists and engineers. Journal of Computational Physics vol. 257 1373–1393 (2014) – 10.1016/j.jcp.2013.08.007
  • Differential Forms with Applications to the Physical Sciences by Harley Flanders (1963)
  • Manifolds, tensor Analysis, and Applications (2012)