In this paper, we consider the recently introduced EMAC formulation for the incompressible Navier-Stokes (NS) equations, which is the only known NS formulation that conserves energy, momentum and angular momentum when the divergence constraint is only weakly enforced. Since its introduction, the EMAC formulation has been successfully used for a wide variety of fluid dynamics problems. We prove that discretizations using the EMAC formulation are potentially better than those built on the commonly used skew-symmetric formulation, by deriving a better longer time error estimate for EMAC: while the classical results for schemes using the skew-symmetric formulation have Gronwall constants dependent on $\exp(C\cdot Re\cdot T)$ with $Re$ the Reynolds number, it turns out that the EMAC error estimate is free from this explicit exponential dependence on the Reynolds number. Additionally, it is demonstrated how EMAC admits smaller lower bounds on its velocity error, since {incorrect treatment of linear momentum, angular momentum and energy induces} lower bounds for $L^2$ velocity error, and EMAC treats these quantities more accurately. Results of numerical tests for channel flow past a cylinder and 2D Kelvin-Helmholtz instability are also given, both of which show that the advantages of EMAC over the skew-symmetric formulation increase as the Reynolds number gets larger and for longer simulation times.

中文翻译：

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使用 EMAC 公式的不可压缩 Navier-Stokes 模拟具有更长的时间精度

在本文中，我们考虑了最近引入的不可压缩 Navier-Stokes (NS) 方程的 EMAC 公式，这是唯一已知的 NS 公式，当发散约束仅弱执行时，它可以保存能量、动量和角动量。自推出以来，EMAC 公式已成功用于解决各种流体动力学问题。我们证明了使用 EMAC 公式的离散化可能比建立在常用偏对称公式上的离散化更好，方法是为 EMAC 推导出更好的较长时间误差估计：而使用偏对称公式的方案的经典结果具有依赖于 Gronwall 常数的在 $\exp(C\cdot Re\cdot T)$ 上，$Re$ 是雷诺数，事实证明，EMAC 误差估计不受这种对雷诺数的显式指数依赖性的影响。此外，还证明了 EMAC 如何承认其速度误差的下限较小，因为{对线性动量、角动量和能量的不正确处理会导致}$L^2$ 速度误差的下限，并且 EMAC 更准确地处理这些量。还给出了通道流过圆柱体和二维开尔文-亥姆霍兹不稳定性的数值测试结果，这两个结果都表明，随着雷诺数变大和模拟时间变长，EMAC 优于偏对称公式的优势也会增加。角动量和能量会导致 $L^2$ 速度误差的下限，而 EMAC 更准确地处理这些量。还给出了通道流过圆柱体和二维开尔文-亥姆霍兹不稳定性的数值测试结果，这两个结果都表明，随着雷诺数变大和模拟时间更长，EMAC 相对于偏对称公式的优势会增加。角动量和能量会导致 $L^2$ 速度误差的下限，而 EMAC 更准确地处理这些量。还给出了通道流过圆柱体和二维开尔文-亥姆霍兹不稳定性的数值测试结果，这两个结果都表明，随着雷诺数变大和模拟时间变长，EMAC 优于偏对称公式的优势也会增加。