伪谱模拟中不可压缩纳维-斯托克斯方程的混叠与相位移移
Aliasing and phase shifting in pseudo-spectral simulations of the incompressible Navier-Stokes equations
AI总结：
本文研究了伪谱模拟中不可压缩纳维-斯托克斯方程的混叠与相位移移问题，分析了相位移移方法在减少计算成本方面的有效性，并提供了开源框架Fluidsim中的实现。
摘要：
伪谱方法广泛用于湍流的直接数值模拟，但标准的2/3截断规则用于去混叠计算成本高，占三维情况总成本的80%。相位移移方法通过在移位网格上评估非线性项来抵消混叠误差，允许在更粗的数值网格上实现相同的物理分辨率。尽管这些方法在高分辨率湍流代码中被使用，但文献中描述有限且没有开源实现。本文对伪谱模拟不可压缩纳维-斯托克斯方程的相位移移去混叠进行了全面分析。我们从离散傅里叶空间中的二次非线性性推导了混叠机制，并解释了相位移移如何根据时间推进方案精确或近似地抵消混叠贡献。我们描述并比较了几种算法，包括Patterson Jr和Orszag（1971）和Rogallo（1981）的精确和近似RK2相位移移方案，以及一种扩展到强迫流动的算法，并讨论了它们与三维不同截断几何体的相互作用。所有算法均在开源框架Fluidsim中实现，提供了伪谱纳维-斯托克斯求解器中相位移移去混叠的首次公开实现。我们对两个测试案例进行了评估：泰勒-格林涡旋的湍流过渡和强迫各向同性湍流在Re_λ=200时的模拟。相位移移方法在相同最大波数下相比RK4与2/3截断方法可提高速度达3倍，伴随小的精度损失。
英文摘要：
Pseudo-spectral methods are widely used for direct numerical simulations of turbulence, but the standard 2/3 truncation rule for dealiasing is computationally expensive -- accounting for up to 80% of the total cost in three dimensions. Phase shifting methods provide a more efficient alternative by canceling aliasing errors the combination of nonlinear terms evaluated on shifted grids, allowing the same physical resolution to be achieved on a coarser numerical grid. Despite their use in high-resolution turbulence codes, these methods remain poorly documented in the literature and no open-source implementation exists. This paper presents a comprehensive analysis of phase-shifting dealiasing for pseudo-spectral simulations of the incompressible Navier-Stokes equations. We derive the aliasing mechanism from quadratic nonlinearities in discrete Fourier space and explain how phase-shifting cancels aliasing contributions exactly or approximately depending on the time-stepping scheme. We describe and compare several algorithms -- including the exact and approximate RK2 phase-shifting schemes of Patterson Jr and Orszag (1971) and Rogallo (1981), and an extension to forced flows -- and discuss their interaction with different truncation geometries in three dimensions. All algorithms are implemented in the open-source framework Fluidsim, providing the first publicly available implementation of phase-shifting dealiasing for pseudo spectral Navier-Stokes solvers. We evaluate the methods on two test cases: the transition to turbulence of Taylor-Green vortices and forced homogeneous isotropic turbulence at $Re_λ= 200$. Phase-shifting methods achieve speedups of up to a factor of 3 compared to RK4 with 2/3 truncation at the same maximum wavenumber, with small accuracy loss.
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