A primitive variable discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes

A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier-Stokes equations is performed. An existing DEC method (Mohamed, M. S., Hirani, A. N., Samtaney, R. (2016). Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes. Journal of Computational Physics, 312, 175-191) 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 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.

[1]  I. Variétés différentiables , 2020, Intégrales singulières.

[2]  G. P. King,et al.  Eddy–wave duality in a rotating flow , 2020, Physics of Fluids.

[3]  R. Samtaney,et al.  Investigation of flow past a cylinder embedded on curved and flat surfaces , 2020, Physical Review Fluids.

[4]  R. Ashing.  Stable , 2020, Definitions.

[5]  Omer San,et al.  A dynamic closure modeling framework for model order reduction of geophysical flows , 2019, Physics of Fluids.

[6]  Mikio Nakahara.  Manifolds , 2018, Geometry, Topology and Physics.

[7]  Ravi Samtaney,et al.  Numerical convergence of discrete exterior calculus on arbitrary surface meshes , 2018, 1802.04506.

[8]  Ashwin Vishnu Mohanan,et al.  A two-dimensional toy model for geophysical turbulence , 2017 .

[9]  H. Kellay.  Hydrodynamics experiments with soap films and soap bubbles: A short review of recent experiments , 2017 .

[10]  Xilin Xie,et al.  Interplay of surface geometry and vorticity dynamics in incompressible flows on curved surfaces , 2017 .

[11]  Axel Voigt,et al.  Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation , 2016, 1611.04392.

[12]  Mark Meyer,et al.  Subdivision exterior calculus for geometry processing , 2016, ACM Trans. Graph..

[13]  L. Castillo,et al.  Turbulent boundary layer over 2D and 3D large-scale wavy walls , 2015 .

[14]  Dilek Funda Kurtulus,et al.  On the Unsteady Behavior of the Flow around NACA 0012 Airfoil with Steady External Conditions at Re=1000 , 2015 .

[15]  Anil N. Hirani,et al.  Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes , 2015, J. Comput. Phys..

[16]  Axel Voigt,et al.  The Interplay of Curvature and Vortices in Flow on Curved Surfaces , 2014, Multiscale Model. Simul..

[17]  N. Ouellette,et al.  Geometry of scale-to-scale energy and enstrophy transport in two-dimensional flow , 2014 .

[18]  H. Aluie,et al.  The direct enstrophy cascade of two-dimensional soap film flows , 2013, 1309.4894.

[19]  Keenan Crane,et al.  Digital geometry processing with discrete exterior calculus , 2013, SIGGRAPH '13.

[20]  J. Blair Perot,et al.  Discrete Conservation Properties of Unstructured Mesh Schemes , 2011 .

[21]  Jeffrey M. Connors,et al.  Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model , 2010 .

[22]  D. Nelson,et al.  Vortices on curved surfaces , 2010 .

[23]  Keenan Crane,et al.  Energy-preserving integrators for fluid animation , 2009, ACM Trans. Graph..

[24]  Anil N. Hirani,et al.  Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus , 2008, ArXiv.

[25]  J. Koiller,et al.  Vortices on Closed Surfaces , 2008, 0802.4313.

[26]  J. Blair Perot,et al.  Discrete calculus methods for diffusion , 2007, J. Comput. Phys..

[27]  Pingwen Zhang,et al.  Continuum theory of a moving membrane. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Yiying Tong,et al.  Stable, circulation-preserving, simplicial fluids , 2007, TOGS.

[29]  Boo Cheong Khoo,et al.  An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries , 2006, J. Comput. Phys..

[30]  Yiying Tong,et al.  Stable, circulation-preserving, simplicial fluids , 2006, SIGGRAPH Courses.

[31]  Dan S. Henningson,et al.  High Order Accurate Solution of Flow Past a Circular Cylinder , 2006, J. Sci. Comput..

[32]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[33]  Yiying Tong,et al.  Discrete differential forms for computational modeling , 2005, SIGGRAPH Courses.

[34]  Yiying Tong,et al.  Discrete differential forms for computational modeling , 2005, SIGGRAPH Courses.

[35]  Mark C. Thompson,et al.  Computations of the drag coefficients for low-Reynolds-number flow past rings , 2005, Journal of Fluid Mechanics.

[36]  F. Durst,et al.  Heating effect on steady and unsteady horizontal laminar flow of air past a circular cylinder , 2004 .

[37]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[38]  Anil N. Hirani,et al.  Discrete exterior calculus for variational problems in computer vision and graphics , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[39]  Z. J. Wang,et al.  A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow , 2003 .

[40]  S. Mittal,et al.  Flow past a rotating cylinder , 2003, Journal of Fluid Mechanics.

[41]  D. Schmidt,et al.  Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics , 2002 .

[42]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[43]  B. Perot.  Conservation Properties of Unstructured Staggered Mesh Schemes , 2000 .

[44]  George Em Karniadakis,et al.  Unstructured spectral element methods for simulation of turbulent flows , 1995 .

[45]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

[46]  J. Cavendish,et al.  The dual variable method for solving fluid flow difference equations on Delaunay triangulations , 1991 .

[47]  P. Colella,et al.  A second-order projection method for the incompressible navier-stokes equations , 1989 .

[48]  R. A. Nicolaides,et al.  Flow discretization by complementary volume techniques , 1989 .

[49]  S. Dennis,et al.  Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100 , 1970, Journal of Fluid Mechanics.

[50]  Akio Arakawa,et al.  Integration of the Nondivergent Barotropic Vorticity Equation with AN Icosahedral-Hexagonal Grid for the SPHERE1 , 1968 .

[51]  Mitutosi Kawaguti,et al.  Numerical Study of a Viscous Fluid Flow past a Circular Cylinder , 1966 .

[52]  Sadatoshi Taneda,et al.  Experimental Investigation of the Wakes behind Cylinders and Plates at Low Reynolds Numbers , 1956 .

[53]  Mitutosi Kawaguti,et al.  Numerical Solution of the Navier-Stokes Equations for the Flow around a Circular Cylinder at Reynolds Number 40 , 1953 .

[54]  S. Neamtan.  THE MOTION OF HARMONIC WAVES IN THE ATMOSPHERE , 1946 .

[55]  E. Cartan,et al.  Leçons sur la géométrie des espaces de Riemann , 1928 .

[56]  Henri Poincaré,et al.  Sur les résidus des intégrales doubles , 1887 .

[57]  Diana Adler,et al.  Differential Forms With Applications To The Physical Sciences , 2016 .

[58]  J. Blair Perot,et al.  Differential forms for scientists and engineers , 2014, J. Comput. Phys..

[59]  Jörn Behrens,et al.  Toward goal-oriented R-adaptive models in geophysical fluid dynamics using a generalized discretization approach , 2013 .

[60]  E. Grinspun.  Discrete differential geometry : An applied introduction , 2008, SIGGRAPH 2008.

[61]  M. Shashkov,et al.  Compatible spatial discretizations , 2006 .

[62]  S. Majumdar,et al.  Laminar flow past a circular cylinder at reynolds number varying from 50 to 5000 , 2005 .

[63]  Jason Frank,et al.  Conservation Properties of Smoothed Particle Hydrodynamics Applied to the Shallow Water Equation , 2001 .

[64]  P. Moin,et al.  Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows , 1997 .

[65]  Georges de Rham.  Variétés différentiables : formes, courants, formes harmoniques , 1955 .

[66]  R. Becker,et al.  The classical theory of electricity and magnetism , 1932 .

[67]  E. Goursat,et al.  Sur certains systèmes d'équations aux différentiels totales et sur une généralisation du problème de Pfaff , 1915 .

[68]  E. Cartan,et al.  Sur certaines expressions différentielles et le problème de Pfaff , 1899 .