FLUID In the literature: open

Origin of anomalous scaling exponents in three-dimensional turbulence

In plain words

In turbulence, velocity differences between two points grow with their separation, and the simplest theory (Kolmogorov 1941) predicts how; measurements show rare violent events that break this prediction more and more strongly for higher statistical moments. Nobody can yet calculate the observed deviations from the equations of fluid motion.

Precise statement

System: statistically stationary, homogeneous, isotropic turbulence of the 3D incompressible Navier-Stokes equations at $\mathrm{Re}_{\lambda} \to \infty$. Quantity: inertial-range exponents $\zeta_{p}$ in S_p(r) = <|delta_r u|^p> ~ r^zeta_p, which deviate from the Kolmogorov 1941 value $p/3$ (measured $\zeta_{6} \sim 1.78$, approximately). Answer: a derivation of zeta_p from the Navier-Stokes equations, or from an approximation with controlled error, that reproduces measured values for p up to about 10 and identifies the flow structures responsible.

What would settle it

A first-principles calculation of $\zeta_{p}$ that matches high-resolution DNS and experiment within error bars for $p\ \text{up to}\ 10$.

Status in the literature

Unverified note

Multifractal and log-Poisson models fit the data with adjustable inputs; no first-principles derivation exists as of 2026.