# Approximation of the convective flux in the incompressible Navier-Stokes equations using local boundary-value problems

### Uittreksel

We present the spatial discretization of the nonlinear convective flux in the incompressible Navier-Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier-Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection-diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations.

Originele taal-2 Engels 523-536 14 Journal of Computational and Applied Mathematics 340 https://doi.org/10.1016/j.cam.2018.01.030 Gepubliceerd - 1 jan 2018

### Vingerafdruk

Incompressible Navier-Stokes Equations
Navier Stokes equations
Boundary value problems
Boundary Value Problem
Fluxes
Approximation
Source Terms
Discretization
Face
Grid
Staggered Grid
Convection-diffusion
Cell
Weighted Average
Finite Volume
Transverse
Inclusion
Oscillation

### Citeer dit

@article{1feefbdfc10a4919b12d41825aaae7a7,
title = "Approximation of the convective flux in the incompressible Navier-Stokes equations using local boundary-value problems",
abstract = "We present the spatial discretization of the nonlinear convective flux in the incompressible Navier-Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier-Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection-diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations.",
keywords = "Cell-face velocities, Convective terms, Finite-volume method, Incompressible Navier-Stokes equations",
author = "N. Kumar and {ten Thije Boonkkamp}, {J. H.M.} and B. Koren",
year = "2018",
month = "1",
day = "1",
doi = "10.1016/j.cam.2018.01.030",
language = "English",
volume = "340",
pages = "523--536",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",

}

In: Journal of Computational and Applied Mathematics, Vol. 340, 01.01.2018, blz. 523-536.

TY - JOUR

T1 - Approximation of the convective flux in the incompressible Navier-Stokes equations using local boundary-value problems

AU - Kumar, N.

AU - ten Thije Boonkkamp, J. H.M.

AU - Koren, B.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We present the spatial discretization of the nonlinear convective flux in the incompressible Navier-Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier-Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection-diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations.

AB - We present the spatial discretization of the nonlinear convective flux in the incompressible Navier-Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier-Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection-diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations.

KW - Cell-face velocities

KW - Convective terms

KW - Finite-volume method

KW - Incompressible Navier-Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=85043322743&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2018.01.030

DO - 10.1016/j.cam.2018.01.030

M3 - Article

AN - SCOPUS:85043322743

VL - 340

SP - 523

EP - 536

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -