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.
Highlights
Structure preserving numerical methods that satisfy the mathematical and physical properties of the governing equations are increasingly becoming popular.1–3 One of such methods is discrete exterior calculus (DEC)
The theory of exterior calculus and differential forms dates back to the pioneering work of Poincaré5, Cartan6, and Goursat7
DEC made its way to the applications such as computer graphics and computer vision13–19 to mitigate the effects of numerical viscosity that causes detrimental visual consequences and fluid flows20–23 to preserve the physical properties of the governing equations, i.e., to conserve secondary quantities, such as kinetic energy and/or discretely preserving Kelvin’s circulation theorem
Summary
Structure preserving numerical methods that satisfy the mathematical and physical properties of the governing equations are increasingly becoming popular. One of such methods is discrete exterior calculus (DEC). As this theory advanced, DEC made its way to the applications such as computer graphics and computer vision to mitigate the effects of numerical viscosity that causes detrimental visual consequences and fluid flows to preserve the physical properties of the governing equations, i.e., to conserve secondary quantities, such as kinetic energy and/or discretely preserving Kelvin’s circulation theorem. DEC has low numerical diffusion and superior conservation of physical quantities such as vorticity and kinetic energy and preserves vortical structures over a long period of time.21,27 We conjecture that these properties potentially make DEC favorable for turbulence simulations in three dimensions. The present method is very suitable for simulating flows on two-dimensional surfaces, which are relevant to applications such as geophysical flows, flow over complex terrains, atomic physics, bio-membrane flows, laboratory flows in a soap film, and electromagnetically driven thin-layer flow.
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