不可压缩Navier-Stokes方程的强涡量拟能稳定积分器
Strongly enstrophy-stable integrators for the incompressible Navier-Stokes equations
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中文总结 AI 辅助
本文提出一种混合有限元离散格式，通过辅助变量在强意义上保持能量与涡量拟能演化律，从而在欠分辨网格上实现与Reynolds数无关的稳定，并在二维和三维数值实验中验证了其强稳定效果。
中文摘要 AI 辅助
我们针对不可压缩Navier-Stokes方程提出了一种混合有限元离散格式，该格式以比先前离散格式更强的意义保留了能量和涡量拟能的演化规律。在二维情形下，涡量拟能的演化规律仅允许热力学孤立系统发生耗散，从而得到与Reynolds数无关的速度梯度界，该界自然地稳定了格式，即使在严重欠分辨的网格上也是如此。在三维情形下，该格式同时保留了耗散以及通过涡旋拉伸产生的涡量拟能生成。我们通过系统地在离散化中引入辅助变量来强制执行这些演化规律。虽然这些格式的协调实现需要具有增强正则性的离散Stokes复形，但我们引入了（i）等价重新参数化和（ii）惩罚公式，它们仅需要标准离散de Rham复形中典型的旋度和散度协调空间。该格式处理不同类型的边界条件和弯曲域。通过对剪切流、球形涡旋和障碍物绕流的数值模拟，展示了所提出格式的稳健稳定性质。我们在数值上观察到，以这种方式保留涡量拟能的离散演化对数值解具有强烈的稳定效应，尤其是在二维情形下。
英文摘要
We propose a mixed finite element discretisation for the incompressible Navier-Stokes equations that preserves the evolution laws of both energy and enstrophy, in a stronger sense than previous discretisations. In two dimensions, the evolution law for enstrophy only permits dissipation for thermodynamically isolated systems, leading to a Reynolds-number-independent bound on the velocity gradient that naturally stabilises the scheme, even on severely under-resolved meshes. In three dimensions, the scheme preserves both dissipation and the generation of enstrophy through vortex stretching. We enforce these evolution laws by systematically introducing auxiliary variables into the discretisation. While conforming implementations of these schemes require discrete Stokes complexes with enhanced regularity, we introduce both (i) equivalent reparametrisations and (ii) penalty formulations that require only the typical curl- and div-conforming spaces from the standard discrete de Rham complex. The scheme handles different types of boundary conditions and curved domains. The robust stabilisation properties of the proposed scheme are demonstrated through numerical simulations of a shear flow, a spherical vortex, and flow past an obstacle. We observe numerically that preserving the discrete evolution of enstrophy in this way has a strong stabilising effect on the numerical solution, especially in two dimensions.
发表机构
- Mathematical Institute, University of Oxford（牛津大学数学研究所）
- Mathematical Institute, Charles University（查理大学数学研究所）
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