On the Non-Negativity of Longitudinal Velocity Autocorrelations in Homogeneous Isotropic Turbulence: Kinematic Constraints, Convexity, and Spectral Broadening
Abstract
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 (∫0∞ 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 E11(k1) via the cosine transform and applying Pólya's criterion, convexity of E11(k1) provides a rigorous sufficient condition for f(r) ≥ 0. While pure Kolmogorov −5/3 scaling is strictly convex, physically realizable spectra with Batchelor (k4) or Saffman (k2) infrared scaling are necessarily concave near k1 = 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 (minr f(r) ≈ −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) ≥ 0 in mature turbulence.
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BibTeX
@misc{govindaraju2026nonnegativity,
title = {{On the Non-Negativity of Longitudinal Velocity Autocorrelations in Homogeneous Isotropic Turbulence: Kinematic Constraints, Convexity, and Spectral Broadening}},
author = {Pavan B. Govindaraju},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01251},
eprint = {2609.01251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01251},
}
RIS
TY - UNPB
TI - On the Non-Negativity of Longitudinal Velocity Autocorrelations in Homogeneous Isotropic Turbulence: Kinematic Constraints, Convexity, and Spectral Broadening
AU - Pavan B. Govindaraju
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01251
UR - https://arxiv.org/abs/2609.01251
AB - 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 (∫_0^∞ 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) ≥ 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) ≈ −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) ≥ 0 in mature turbulence.
ER -