Estimate of the truncation error of a finite volume discretisation of the Navier-Stokes equations on colocated grids

TL;DR

This paper introduces a method to estimate the truncation error in finite volume discretisations of the incompressible Navier-Stokes equations on colocated grids, using a solution restriction technique and a novel momentum interpolation variant.

Contribution

It applies a known error estimation concept specifically to colocated finite volume schemes with a new momentum interpolation approach for pressure-independent mass fluxes.

Findings

Method accurately estimates truncation errors in 2D flows.

Numerical experiments validate the proposed methodology.

Extension to 3D flows is feasible.

Abstract

A methodology is proposed for the calculation of the truncation error of finite volume discretisations of the incompressible Navier-Stokes equations on colocated grids. The truncation error is estimated by restricting the solution obtained on a given grid to a coarser grid and calculating the image of the discrete Navier-Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed methodology is not a new concept but its application to colocated finite volume discretisations of the incompressible Navier-Stokes equations is made possible by the introduction of a variant of the momentum interpolation technique for mass fluxes where the pressure-part of the mass fluxes is not dependent on the coefficients of the linearised momentum equations. The theory presented is supported by a number of numerical experiments. The methodology is developed for two-dimensional flows, but extension to three-dimensional cases should not pose problems.