Turbulent Flows Are Not Uniformly Multifractal

Understanding turbulence rests delicately on the conflict between Kolmogorov’s 1941 theory of nonintermittent, space-filling energy dissipation characterized by a unique scaling exponent and the overwhelming evidence to the contrary of intermittency, multiscaling, and multifractality. Strangely, multifractality is not typically envisioned as a local flow property, variations in which might be clues exposing inroads into the fundamental unsolved issues of anomalous dissipation and finite time blowup. We present a simple construction of local multifractality and find that much of the dissipation field remains surprisingly monofractal à la Kolmogorov. Multifractality appears as small islands in this calm sea, its strength growing logarithmically with the local fluctuations in energy dissipation—a seemingly universal feature. These results suggest new ways to understand how singularities could arise and provide a fresh perspective on anomalous dissipation and intermittency. The simplicity and adaptability of our approach also holds great promise in applications ranging from climate sciences to medical data analysis.

Figure 1

Spatially varying dissipation $\epsilon$ at a representative cross section from the Johns Hopkins Turbulence Database for a flow with ${\mathrm{Re}}_{\lambda }\approx 610$. Superimposed is a regular tiling as a pictorial guide to show the subdivisions used for local multifractal analysis (made with Processing [41, 42]). Generalized dimensions $D_q$ [inset (A)] and the singularity spectrum $f_{\alpha }-\alpha$ [inset (B)] calculated over the full 2D cross section show the essential global multifractal nature of the dissipation field, as is usually reported.

Figure 2

A cross section of the $D_2$ (correlation dimension) field, obtained from a local multifractal analysis, which coarsens a $4096\times 4096\times 8$ data slice to $512\times 512$ tiles. The stark variation in $D_2$ over space, which remained hitherto unseen, highlights coherent regions of $D_2$ nestling in a fluctuating (and nonrandom) field.

Figure 3

(a) Local generalized dimensions $D_q$ vs $q$, for randomly sampled tiles, and (b) the corresponding local $f_{\alpha }-\alpha$ spectra of singularity strengths, show strong variation in the multifractal properties over space. (c) The resultant $\mathrm{\Phi }$ field shows large regions of the flow are almost monofractal with $\mathrm{\Phi }\approx 0$, with pockets of $\mathrm{\Phi }\sim \mathcal{O}(1)$ with its PDF [(d), inset)] shifting toward higher $\mathrm{\Phi }$ when sampled in regions of higher $\mathrm{\Delta }\epsilon ={\epsilon }_{\max }-{\epsilon }_{\min }$. (d) The mean value of $\mathrm{\Phi }$ is shown to grow as the logarithm of $\mathrm{\Delta }\epsilon$. (e) The joint distribution of $\mathrm{\Phi }$ and the mean dissipation $\overline{\epsilon }$ in each tile show that most likely $\mathrm{\Phi }$ values grow logarithmically (dashed white line) with $\overline{\epsilon }$. (f) A volume rendering of $\overline{\epsilon }\ge 1$ is superimposed with the $\mathrm{\Phi }\le 0.5$ field, which being spatially exclusive, clearly illustrates that the most monofractal flow regions are coincident with mild dissipation.