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A Modified Convective Formulation in Navier–Stokes Simulations

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Abstract

For the discretization of the convective term in the Navier–Stokes equations, the commonly used convective formulation (CONV) does not preserve kinetic energy if the divergence constraint is only discretely enforced. In this paper, we apply the skew-symmetrization technique in Cockburn et al. (Math Comput 74(251):1067–1095, 2005. https://doi.org/10.1090/S0025-5718-04-01718-1) to conforming finite element methods, which restores energy conservation for CONV. The main idea is to replace the discrete advective velocity with an \(H({\text {div}})\)-conforming divergence-free one in CONV. We prove that the modified convective formulation also conserves linear momentum, helicity, 2D enstrophy and total vorticity under some appropriate senses. The popular CNLE (linearly extrapolated Crank–Nicolson) method also conserves them when this modified formulation is used. Under the assumption \(\varvec{u}\in L^{2}(0,T;\varvec{W}^{1,\infty }(\Omega )),\) it can be shown that the Gronwall constant does not explicitly depend on the Reynolds number in the error estimates. Numerical experiments show that the linearized and modified convective formulation has similar performance with the EMAC formulation and outperforms the usual skew-symmetric formulation (SKEW).

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Data Availability

All data generated or analysed during this study are included in this manuscript. Partial work was completed during the first author’s visit at Weierstrass Institute, 10117 Berlin, Germany.

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Acknowledgements

We are very grateful to Ziyan Li from City University of Hong Kong for many valuable discussions on coding implementation.

Funding

This work was supported by the National Natural Science Foundation of China (Grant 12131014). The first author was also supported by the China Scholarship Council (Grant 202106220106).

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  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Xu Li & Hongxing Rui

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Li, X., Rui, H. A Modified Convective Formulation in Navier–Stokes Simulations. J Sci Comput 96, 69 (2023). https://doi.org/10.1007/s10915-023-02286-3

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