TY - JOUR
T1 - A new Green–Gauss reconstruction on unstructured meshes. Part I
T2 - Gradient reconstruction
AU - Deka, Mandeep
AU - Brahmachary, Shuvayan
AU - Thirumalaisamy, Ramakrishnan
AU - Dalal, Amaresh
AU - Natarajan, Ganesh
N1 - Funding Information:
The authors would like to sincerely thank the five anonymous reviewers whose insightful comments have helped to substantially improve the manuscript. We would also like to specifically acknowledge one of the reviewers, who brought to our notice the idea of “non-iterative piggybacking” that significantly cut down the cost of the MGG reconstruction for finite volume simulations. The financial support from DAE-BRNS, India during the course of the study is also gratefully acknowledged.
Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2020/12/1
Y1 - 2020/12/1
N2 - We describe a new and simple strategy based on the Gauss divergence theorem for obtaining centroidal gradients on unstructured meshes. Unlike the standard Green–Gauss (SGG) reconstruction which requires face values of quantities whose gradients are sought, the proposed approach reconstructs the gradients using the normal derivative(s) at the faces. The new strategy, referred to as the Modified Green–Gauss (MGG) reconstruction results in consistent gradients which are at least first-order accurate on arbitrary polygonal meshes. We show that the MGG reconstruction is linearity preserving independent of the mesh topology and retains the consistent behaviour of gradients even on meshes with large curvature and high aspect ratios. The gradient accuracy in MGG reconstruction depends on the accuracy of discretisation of the normal derivatives at faces and this necessitates an iterative approach for gradient computation on non-orthogonal meshes. Numerical studies on different mesh topologies demonstrate that MGG reconstruction gives accurate and consistent gradients on non-orthogonal meshes, with the number of iterations proportional to the extent of non-orthogonality. The MGG reconstruction is found to be consistent even on meshes with large aspect ratio and curvature with the errors being lesser than those from linear least-squares reconstruction. A non-iterative strategy in conjunction with MGG reconstruction is proposed for gradient computations in finite volume simulations that achieves the accuracy and robustness of MGG reconstruction at a cost equivalent to that of SGG reconstruction. The efficacy of this strategy for fluid flow problems is demonstrated through numerical investigations in both incompressible and compressible regimes. The MGG reconstruction may, therefore, be viewed as a novel and promising blend of least-squares and Green–Gauss based approaches which can be implemented with little effort in open-source finite-volume solvers and legacy codes.
AB - We describe a new and simple strategy based on the Gauss divergence theorem for obtaining centroidal gradients on unstructured meshes. Unlike the standard Green–Gauss (SGG) reconstruction which requires face values of quantities whose gradients are sought, the proposed approach reconstructs the gradients using the normal derivative(s) at the faces. The new strategy, referred to as the Modified Green–Gauss (MGG) reconstruction results in consistent gradients which are at least first-order accurate on arbitrary polygonal meshes. We show that the MGG reconstruction is linearity preserving independent of the mesh topology and retains the consistent behaviour of gradients even on meshes with large curvature and high aspect ratios. The gradient accuracy in MGG reconstruction depends on the accuracy of discretisation of the normal derivatives at faces and this necessitates an iterative approach for gradient computation on non-orthogonal meshes. Numerical studies on different mesh topologies demonstrate that MGG reconstruction gives accurate and consistent gradients on non-orthogonal meshes, with the number of iterations proportional to the extent of non-orthogonality. The MGG reconstruction is found to be consistent even on meshes with large aspect ratio and curvature with the errors being lesser than those from linear least-squares reconstruction. A non-iterative strategy in conjunction with MGG reconstruction is proposed for gradient computations in finite volume simulations that achieves the accuracy and robustness of MGG reconstruction at a cost equivalent to that of SGG reconstruction. The efficacy of this strategy for fluid flow problems is demonstrated through numerical investigations in both incompressible and compressible regimes. The MGG reconstruction may, therefore, be viewed as a novel and promising blend of least-squares and Green–Gauss based approaches which can be implemented with little effort in open-source finite-volume solvers and legacy codes.
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U2 - 10.1016/j.jcp.2018.10.023
DO - 10.1016/j.jcp.2018.10.023
M3 - Article
AN - SCOPUS:85055639834
SN - 0021-9991
VL - 422
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 108325
ER -