Symmetry-Preserving Discretization of Navier-Stokes on Unstructured Grids: Collocated Vs Staggered

Abstract

The essence of turbulence are the smallest scales of motion. They result from a subtle balance between convective transport and diffusive dissipation. Mathematically, these terms are governed by two differential operators differing in symmetry: the convective operator is skew-symmetric, whereas the diffusive is symmetric and positive-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows around complex configurations. With this in mind, a fully-conservative discretization method for general unstructured grids was proposed [Trias et al., J.Comp.Phys. 258, 246-267, 2014]: it exactly preserves the symmetries of the underlying differential operators on a collocated mesh. However, any pressure-correction method on collocated grids suffer from the same drawbacks: the cell-centered velocity field is not exactly incompressible and some artificial dissipation is inevitable introduced. On the other hand, for staggered velocity fields, the projection onto a divergence-free space is a well-posed problem: given a velocity field, it can be uniquely decomposed into a solenoidal vector and the gradient of a scalar (pressure) field. This can be easily done without introducing any dissipation as it should be from a physical point-of-view. In this work, we explore the possibility to build up staggered formulations based on collocated discrete operators.

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