Do Lagrangian intermittency exponents follow from Eulerian ones?
In plain words
The velocity of a single fluid particle followed in time shows rare violent jumps, just as velocity differences between two points in space do. Whether the jumps along a particle path are fully predictable from the spatial ones, at all flow speeds and statistical orders, is not established.
Precise statement
For homogeneous isotropic turbulence, Lagrangian structure functions $S_{p}^{L}(\tau)=\langle\mid v(t+\tau)-v(t)\mid^p\rangle\sim \tau^{\zeta_{p}^{L}}$ for $\tau_{\eta} << \tau << T_{L}$ (Kolmogorov time to Lagrangian integral time). The multifractal bridge, which maps the Eulerian singularity spectrum $D(h)$ to Lagrangian statistics through $\tau\sim r/\delta_{r} u$, predicts $\zeta_{p}^{L}$ from the Eulerian $\zeta_p$. Determine whether this prediction holds as $\mathrm{Re}_{\lambda}\to \infty$ for p up to about 8, including the contribution of particles trapped in vortex filaments at small $\tau$. Answer: $\zeta_{p}^{L}$ with error bars compared with the bridge prediction.
What would settle it
Lagrangian particle tracking in experiments and DNS above $\mathrm{Re}_{\lambda} \sim 2000$ with converged high-order moments over at least a decade of time lags, compared with Eulerian exponents from the same flows.
Status in the literature
Eight experimental and numerical data sets at $\mathrm{Re}_{\lambda}\ 120\ \text{to}\ 740$ collapse and are captured by a multifractal description (Arneodo et al., PRL 2008); convergence at higher $\mathrm{Re}_{\lambda}$ and higher orders is not established.