基于离散外算子的表面单纯形网格上不可压缩纳维-斯托克斯方程的离散化
Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes
AI总结：
本文基于离散外算子开发了不可压缩纳维-斯托克斯方程的保守离散化方法，采用代数离散化内积运算符和组合离散化楔积运算符，通过结构化三角网格和非结构化网格验证了该方法的二阶和一阶精度，实现了质量和涡量的守恒。
摘要：
本文基于离散外算子开发了不可压缩纳维-斯托克斯方程的保守离散化方法。我们的方法特征在于使用代数离散化内积运算符和组合离散化楔积运算符。首先将控制方程改写为外微分算子、霍奇星算子和楔积算子的外微分形式，然后通过DEC框架中的相应离散算子进行离散化。数值实验表明，使用结构化三角网格时，该方案具有二阶精度，而使用非结构化网格时则为一阶精度。通过构造，该方法在机器精度范围内实现了质量和涡量的守恒。无粘流测试案例中，动能相对误差随着网格尺寸和时间步长的减小而以二阶方式收敛。
英文摘要：
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.
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