Implementation and Evaluation of a 1-Exact Central Divergence Operator for Convective Flow in a Cell-Centered Finite Volume Method

Abstract

In scale-resolving simulations, central schemes are used as alternatives to the more dis- sipative upwind methods. On deformed grids in a cell-centered metric, they suffer from inaccuracies due to the scheme’s sensitivity to the face centroids. This is especially prevalent when constant reconstruction is used. A common method used for gradient computation, the Green-Gauss Gradient (GGG) method, shows a similar behavior. When the gradient computation is 1-exact, meaning analytically accurate on first-order polynomials, the problems are reduced significantly. To combat the observed problems faced by the central method, specifically by the commonly used Jameson-Schmidt-Turkel (JST) scheme, its parallels to the central divergence operator are exploited. The correction method to achieve 1-exactness of the gradient is translated onto the central discrete divergence operator. This is motivated by the fact that, firstly, the gradient and divergence are adjoint operators analytically, and secondly, the structure of the Green-Gauss gradient computation mirrors a central divergence operator. Then the 1-exact divergence is used to create a corrected JST scheme. This corrected JST scheme, used for the discretization of the convective flux, is then thoroughly evaluated using a 2D vortex transport case and a multitude of distorted grids. Over the course of this evaluation, the correction is shown to significantly improve the result on such distorted grids on which the default JST scheme exhibited significant numerical instabilities and produced unusable results. The produced results are found to be considerably less grid dependent through the use of the 1-exact correction. Furthermore, the correction method is shown to increase the convergence rate to the expected second order in many cases. The correction is then experimentally applied to the skew-symmetric Kok scheme. There, too, the correction was shown to improve the results significantly. Concluding, the corrected JST scheme is examined using the Taylor-Green vortex (TGV). There the correction method is confirmed to work in 3D. Additionally, its use in enabling the JST scheme as a low-dissipation alternative on distorted grids is emphasized. For this, features of the TGV are observed to be more accurately computed on severely deformed grids, nearing the quality of the solution on the Cartesian grid.