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SplitFXM（SplitFXM）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对均匀各向同性湍流纵向速度自相关的非负性问题，通过谱映射与波利亚判据给出凸性充分条件，构造窄带反例否定普适非负性，揭示谱带宽与非线性作用对成熟湍流中该量非负性的调控机制。
AI 中文摘要
均匀各向同性湍流统计理论中的一个基本问题是，纵向速度自相关$f(r)$是否始终保持非负，以及能否从运动学可实现性等第一性原理出发证明这一点。不可压缩性会迫使横向相关$g(r)$出现负环（$\textstyle\boldsymbol{\rm \textbf{∫}}_0^∞ r g(r)\text{d}r = 0$），但$f(r)$的符号行为却十分微妙。我们证明，由于三维球面投影核存在符号变化，基于博赫纳（Bochner）定理与舍恩伯格（Schoenberg）定理的正定性无法从结构上强制逐点正性。通过余弦变换将问题映射到一维谱$E_{11}(k_1)$并应用波利亚（Pólya）判据，我们发现$E_{11}(k_1)$的凸性是$f(r) \boldsymbol{\rm \textbf{≥}} 0$的严格充分条件。纯柯尔莫哥洛夫（Kolmogorov）$-5/3$标度律是严格凸的，但具有巴彻勒（Batchelor）$k^4$红外标度或萨夫曼（Saffman）$k^2$红外标度的物理可实现谱，在$k_1 = 0$附近必然是凹的。对于冯·卡门-鲍（von Kármán--Pao）模型这类成熟的宽谱，高精度数值积分证实凸性的惯性区占主导，使尾部严格为正，同时表明表观的负凹陷是离散傅里叶变换的伪影。此外，我们构建了一个光滑、无散、有限能量的窄带初始场，它会产生真实的负环（$\textstyle\boldsymbol{\rm \textbf{min}}_r f(r) \boldsymbol{\rm \textbf{≈}} -0.083$），可作为普适非负性猜想的反例。研究表明谱带宽是核心控制物理参数：非线性三波相互作用会在一个涡周转时间内迅速展宽窄带场，系统性地抑制负偏移，从而在成熟湍流中维持$f(r) \boldsymbol{\rm \textbf{≥}} 0$。
英文摘要
A fundamental question in the statistical theory of homogeneous isotropic turbulence is whether the longitudinal velocity autocorrelation $f(r)$ remains non-negative and if it can be shown from first principles such as kinematic realizability. While incompressibility forces the transverse correlation $g(r)$ to exhibit a negative loop ($\int_0^\infty r g(r)\,dr = 0$), the sign behavior of $f(r)$ is subtle. We show that positive-definiteness via Bochner's and Schoenberg's theorems structurally fails to enforce pointwise positivity because the 3D spherical projection kernels are sign-changing. By mapping the problem to the one-dimensional spectrum $E_{11}(k_1)$ via the cosine transform and applying Pólya's criterion, convexity of $E_{11}(k_1)$ provides a rigorous sufficient condition for $f(r) \ge 0$. While pure Kolmogorov $-5/3$ scaling is strictly convex, physically realizable spectra with Batchelor ($k^4$) or Saffman ($k^2$) infrared scaling are necessarily concave near $k_1 = 0$. For mature broadband spectra such as the von Kármán--Pao model, high-precision quadrature confirms that the convex inertial bulk dominates, yielding strictly positive tails and demonstrating that apparent negative dips are discrete Fourier artifacts. Also, we construct a smooth, divergence-free, finite-energy narrowband initial field that produces a genuine negative loop ($\min_r f(r) \approx -0.083$), serving as a counterexample to universal non-negativity. Spectral bandwidth is shown as the governing physical parameter, with nonlinear triad interactions rapidly broadening narrowband fields over an eddy turnover time, systematically suppressing negative excursions and preserving $f(r) \ge 0$ in mature turbulence.
Comments34 pages, 8 figures
