On the temporal and spatial fourth‐order finite volume velocity de‐averaging for unsteady incompressible flows simulation

Abstract

PurposeThis paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured uniform grids. The main purpose of the paper is to assess the accuracy enhancement with the inclusion of a high‐order reconstruction of the point‐wise velocity field on a fourth‐order accurate numerical scheme for the solution of the unsteady incompressible Navier‐Stokes equations.Design/methodology/approachThe present finite volume method uses a fractional time‐step for decoupling velocity and pressure. A Runge‐Kutta integration scheme is implemented for integrating the momentum equation along with a polynomial interpolation and Simpson formula for space‐integration. The formulation is based on step‐by‐step de‐averaging process applied to the velocity field.FindingsThe reconstruction of the point‐wise velocity field on a higher‐order basis is essential to obtain solutions that effectively stand for a fourth‐order approximation of the point‐wise one. Results are provided for the Taylor vortex decay problem and for co‐ and counter‐rotating vortices to assess the increase in accuracy promoted by the inclusion of the high‐order de‐averaging procedure.Research limitations/implicationsHigh‐order reconstruction of the point‐wise velocity field should be considered in high‐order finite volume methods for the solution of the unsteady incompressible form of the Navier‐Stokes equations on structured grids.Practical implicationsThe inclusion of a high‐order reconstruction of the point‐wise velocity field is a simple and effective method of enhancing the accuracy of a finite volume code for the computational fluid dynamics analysis.Originality/valueThe paper develops an improved version of a fourth‐order accurate finite volume projection method with the inclusion of a high‐order reconstruction step.