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 ($\int_0^\infty r g(r)\,dr = 0$), 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 ($k<sup>4$) or Saffman ($k<sup>2$) 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.