AI 中文总结
该研究基于傅里叶空间中非线性项的显式三元分解，为三维不可压缩纳维 - 斯托克斯方程的能量传递开发确定性公式，推导出跨尺度非线性能量通量表示，从确定性角度重审湍流惯性范围，表明柯尔莫哥洛夫\(-5/3\)标度与三元能量传递机制一致。
AI 中文摘要
我们基于傅里叶空间中非线性项的显式三元分解，为三维不可压缩纳维 - 斯托克斯方程中的能量传递开发了一种确定性、尺度分辨公式。通过对速度场进行系统的二元定位，我们推导出跨尺度的非线性能量通量的精确表示，并根据定义明确的尺度分量之间的相互作用进行组织。在适当的光滑性假设下，我们得到绝对收敛的三元展开和定量界限，区分尺度空间中的局部和非局部贡献。该框架提供了关于三元相互作用如何介导能量传递以及尺度局部性如何作为非线性的结构属性出现的透明且完全显式的描述。在此公式基础上，我们从确定性角度重新审视经典的湍流惯性范围图景。我们表明，在尺度不变通量假设下，柯尔莫哥洛夫\(-5/3\)标度在结构层面与三元能量传递机制形式上一致。结果不依赖统计假设，而是源于纳维 - 斯托克斯方程的结构属性与能量通量的尺度分辨表示相结合。本工作因此提供了三元相互作用分析、二元尺度分解和经典湍流现象学的连贯综合，提供了一个确定性框架，阐明了柯尔莫哥洛夫型标度约束如何在基础方程的尺度分辨结构中出现。
英文摘要
We develop a deterministic, scale-resolved formulation of energy transfer in the three-dimensional incompressible Navier-Stokes equations based on an explicit triadic decomposition of the nonlinear term in Fourier space. Using a systematic dyadic localization of the velocity field, we derive an exact representation of the nonlinear energy flux across scales and organize it in terms of interactions between well-defined scale components. Under suitable smoothness assumptions, we obtain an absolutely convergent triadic expansion and quantitative bounds that distinguish local and nonlocal contributions in scale space. This framework provides a transparent and fully explicit description of how energy transfer is mediated by triadic interactions and how scale locality emerges as a structural property of the nonlinearity. Building on this formulation, we revisit the classical inertial-range picture of turbulence from a deterministic perspective. We show that, under a scale-invariant flux assumption, the Kolmogorov $-5/3$ scaling is formally consistent with the triadic energy-transfer mechanism at a structural level. The result does not rely on statistical assumptions, but instead follows from the structural properties of the Navier-Stokes equations combined with a scale-resolved representation of the energy flux. The present work thus provides a coherent synthesis of triadic interaction analysis, dyadic scale decomposition, and classical turbulence phenomenology, offering a deterministic framework that clarifies how Kolmogorov-type scaling constraints arise in the scale-resolved structure of the underlying equations.
Comments9 pages, published in Physica D: Nonlinear Phenomena
Journal refPhysica D: Nonlinear Phenomena, 2026, 135338
DOI：10.1016/j.physd.2026.135338
