Symmetry-preserving discretization of Navier-Stokes on unstructured grids: collocated vs staggered

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.F.X.T., N.V. and A.O. have been financially supported by the Ministerio de Economía y Competitividad, Spain, ANUMESOL project (ENE2017-88697-R). F.X.T. and A.O. are supported by the Generalitat de Catalunya RIS3CAT-FEDER, FusionCAT project (001-P-001722). N.V. was supported by an FI AGAUR-Generalitat de Catalunya fellowship (2017FI B 00616). Calculations were performed on the IBM MareNostrum 4 supercomputer at the BSC. The authors thankfully acknowledge these institutions.Peer ReviewedPostprint (published version

Supra-conservative finite-volume methods for the simulation of subsonic flow

It is demonstrated how finite-volume methods can be designed such that, next to the primary invariants (mass, momentum and internal energy), they also conserve secondary invariants (kinetic energy), i.e., they are supra-conservative. Key ingredient is a consistency between the discrete divergence terms in the constituting equations and the discrete pressure gradient. The requirements hold for any discretization method with a volume-consistent scaling.<br/

Supraconservative finite-volume methods for the Euler equations of subsonic compressible flow

It has been found advantageous for finite-volume discretizations of flow equations to possess additional (secondary) invariants, next to the (primary) invariants from the constituting conservation laws. The paper presents general (necessary and sufficient) requirements for a method to convectively preserve discrete kinetic energy. The key ingredient is a close discrete consistency between the convective term in the momentum equation and the terms in the other conservation equations (mass, internal energy). As examples, the Euler equations for subsonic (in)compressible flow are discretized with such supra-conservative finite-volume methods on structured as well as unstructured grids

Numerical analysis of conservative unstructured discretisations for low Mach flows

This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. https://authorservices.wiley.com/author-resources/Journal-Authors/licensing-and-open-access/open-access/self-archiving.htmlUnstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low-Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell centred velocity reconstructions the standard first-order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd-even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable-density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy preserving scheme is applied to the momentum equations, namely the Symmetry-Preserving (SP) scheme. Furthermore, a new approach to define the far-neighbouring nodes of the QUICK scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self-igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable-density flows. Furthermore, the QUICK schemes shows close to second order behaviour on unstructured meshes and the SP is reliably used in all computations.Peer ReviewedPostprint (author's final draft

An energy-preserving unconditionally stable fractional step method on collocated grids

Preservation of energy is fundamental in order to avoid the introduction of unphysical energy that can lead to unstable simulations. In this work, an energy-preserving unconditionally stable fractional step method on collocated grids is presented as a method which guarantees both preservation of energy and stability of our simulation. Using an algebraic (matrix-vector) representation of the classical incompressible Navier-Stokes equations mimicking the continuous properties of the differential operators, conservation of energy is formally proven. Furthermore, the appearence of unphysical velocities in highly distorted meshes is also adressed. This problem comes from the interpolation of the pressure gradient from faces to cells in the velocity correction equation, and can be corrected by using a proper interpolation.This work has been financially supported by the project RETOtwin [PDC2021-120970-I00] funded by MCIN/AEI/10.13039/501100011033 and European Union Next Generation EU/PRTR. D. Santos acknowledges a FI AGAUR-Generalitat de Catalunya fellowship (2020FI B 00839). The authors thankfully acknowledge these institutions.Peer ReviewedPostprint (published version

On the Interpolation Problem for the Poisson Equation on Collocated Meshes

The appearence of unphysical velocities in highly distorted meshes is a common problem in many simulations. In collocated meshes, this problem arises from the interpolation of the pressure gradient from faces to cells. Using an algebraic form for the classical incompressible Navier-Stokes equations, this problem is adressed. Starting from the work of F. X. Trias et. al. [FX.Trias et al. JCP 258: 246-267, 2014], a new approach for studying the Poisson equation obtained using the Fractional Step Method is found, such as a new interpolator is proposed in order to found a stable solution, which avoid the appearence of these unpleasant velocities. The stability provided by the interpolator is formally proved for cartesian meshes and its rotations, using fully-explicit time discretizations. The construction of the Poisson equation is supported on mimicking the symmetry properties of the differential operators and the Fractional Step Method. Then it is reinterpreted using a recursive application of the Fractional Step Method in order to study the system as an stationary iterative solver. Furthermore, a numerical analysis for unstructured mesh is also provided.This work has been financially supported by the Ministerio de Economía y Competitividad, Spain (project ref. ENE2017-88697-R). D. Santos acknowledges a FI AGAUR-Generalitat de Catalunya fellowship (2020FI B 00839), and N. Valle also acknowledges a FI AGAUR-Generalitat de Catalunya fellowship (2017FI B 00616). The authors thankfully acknowledge these institutions.Peer ReviewedPostprint (published version