Closed-form transform for the von Kármán–Pao correlation

Derive a closed-form expression in standard special functions for the longitudinal autocorrelation function associated with the full von Kármán–Pao energy spectrum, including its fractional-exponent viscous cutoff.

Background

The paper derives an exact modified-Bessel-function representation for the longitudinal autocorrelation generated by the pure von Kármán spectrum. When the Pao viscous factor, whose exponent is proportional to (kη)4/3(k\eta)^{4/3}, is included, the resulting transform combines algebraic spectral factors with a stretched-exponential cutoff. The authors state that this prevents a closed-form representation in standard hypergeometric or Bessel functions, while relying instead on asymptotic matching and arbitrary-precision numerical quadrature to study positivity. Thus, obtaining an analytic closed form for the full von Kármán–Pao transform remains unresolved in the paper.

References

Consequently, unlike the pure von Kármán spectrum, $f_{\text{VKP}}(r)$ cannot be expressed in closed form to the best of our knowledge.

— On the Non-Negativity of Longitudinal Velocity Autocorrelations in Homogeneous Isotropic Turbulence: Kinematic Constraints, Convexity, and Spectral Broadening  (2609.01251 - Govindaraju, 1 Sep 2026) in Appendix, subsection “The von Kármán–Pao Spectrum with Viscous Cutoff: Asymptotics and Numerical Investigation,” paragraph “Absence of Closed-Form Special Functions”