三维不可压缩 Navier-Stokes 方程的混合离散外微积分离散化与 Fourier 变换
A Hybrid Discrete Exterior Calculus Discretization and Fourier Transform of the Incompressible Navier-Stokes Equations in 3D
AI总结：
针对三维不可压缩 NSE 传统求解成本高的问题，提出平面 DEC 与第三维 Fourier 展开结合的混合离散方法，并通过顶盖驱动空腔和 TGV 流验证其与文献结果相当。
摘要：
流体流动问题的模拟，尤其是由 Navier-Stokes 方程（NSE）控制的不可压缩流动，在一系列科学和工程应用中具有基础性意义。用于在三维（3D）网格上求解这些方程的传统数值方法，通常以其保守性质一般、计算强度高以及资源需求巨大而著称。凭借其捕捉单纯形网格内在几何和拓扑性质的能力，离散外微积分（DEC）提供了微分形式的离散对应物，并能够在网格上对偏微分方程（PDE）进行离散化。我们提出了一种基于 DEC 和 Fourier 变换（FT）的三维不可压缩 Navier-Stokes 方程混合离散化方法。在平面维度中，采用 Jagad 等人[1]提出的、针对曲面单纯形网格上不可压缩 Navier-Stokes 方程的既有保守原始变量 DEC 离散化，而在第三维中应用 Fourier 展开。三维顶盖驱动空腔和粘性三维 Taylor-Green 涡（TGV）流动测试案例表明，使用这种混合方法得到的模拟结果与文献相当。
英文摘要：
The simulation of fluid flow problems, specifically incompressible flows governed by the Navier-Stokes equations (NSE), holds fundamental significance in a range of scientific and engineering applications. Traditional numerical methods employed for solving these equations on three-dimensional (3D) meshes are commonly known for their moderate conservation properties, high computational intensity and substantial resource demands. Relying on its ability to capture the intrinsic geometric and topological properties of simplicial meshes, discrete exterior calculus (DEC) provides a discrete analog to differential forms and enables the discretization of partial differential equations (PDEs) on meshes.We present a hybrid discretization approach for the 3D incompressible Navier-Stokes equations based on DEC and Fourier transform (FT). An existing conservative primitive variable DEC discretization of incompressible Navier-Stokes equations over surface simplicial meshes developed by Jagad et al. [1] is considered in the planar dimension while the Fourier expansion is applied in the third dimension. The test cases of three-dimensional lid-driven cavity and viscous Taylor-Green three-dimensional vortex (TGV) flows show that the simulation results using this hybrid approach are comparable to literature.
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