Dynamical origin and Reynolds-number universality of the most irregular small-scale structures in turbulence, as characterized by local Hölder exponents

The velocity field of developed turbulence contains rare 'quasisingular' events whose local regularity, measured by a calibrated pseudo-Hölder exponent field $\tilde{h}$, is far below its most probable value. Nguyen, Laval and Dubrulle [Phys. Rev. E 102, 063105 (2020)] extracted the 213 most irregular events ($\tilde{h}<1.3$ at the dissipative scale $\ell_d=1.8\eta$) from direct numerical simulations at Taylor-scale Reynolds numbers $R_\lambda=53$ and $140$, and found a statistical predominance of roll-type, nonaxisymmetric Burgers-like vortex cores in vortex-stretching topology, tied to close vortex interactions (possibly reconnections) and preceded in time by maxima of the local interscale energy transfer $D_I^\ell$. The authors state that the impact of reconnection on the singularity of events could not be confirmed, that the calibration mapping required uncontrolled extrapolation, that the Reynolds-number dependence of the calibration coefficients is unquantified, that the results are limited to modest Reynolds numbers and pseudospectral regularization, and that the upstream placement of $D_I^\ell$ maxima relative to dissipation maxima lacks a physical explanation. The open problem is to establish what these most irregular structures are and where they come from: their persistence or change at substantially higher Reynolds numbers and in well-resolved experimental data, the causal role of vortex interaction and reconnection in quasisingularity formation, a theoretical account of the observed transfer–dissipation–regularity ordering, and the extension of the characterization to inhomogeneous and anisotropic flows.

Background

Small-scale regularity of turbulent flows. In three-dimensional turbulence, kinetic energy is transferred from large eddies to small ones, down to the Kolmogorov length $\eta$ where viscosity dominates. A classical question, going back to Leray's weak solutions and Onsager's dissipation conjecture, is how irregular the velocity field can become. Local regularity is quantified by the Hölder exponent: the field is Hölder continuous of order $h$ at a point if velocity increments across distance $\ell$ scale like $\ell^h$ there; $h<1$ signals non-differentiability, and points with $h\leq1/3$ can dissipate energy even in the inviscid limit. In the multifractal formalism (Parisi–Frisch), the points carrying exponent $h$ form a set of fractal dimension $D(h)$, and intermittency is the statistics of this distribution.

Local energy transfers. Duchon and Robert derived an exact local energy balance for weak solutions of the incompressible Navier–Stokes and Euler equations, defining at each point and scale $\ell$ a local interscale energy transfer $D_I^\ell$ (the net nonlinear energy transfer across scale $\ell$; positive values correspond to a downscale cascade) and a local viscous dissipation $D_v^\ell$. These fields map the energetics of the cascade in space and time and concentrate at the same rare, intense events probed by local regularity measures.

A calibrated pseudo-Hölder field. Nguyen, Laval, Kestener, Cheskidov, Shvydkoy and Dubrulle (2019) constructed a pointwise regularity estimator for turbulent vector fields: a Mexican-hat wavelet increment $\delta W_\ell(\mathbf{u})=\ell^2\|\nabla_\ell^2\mathbf{u}\|$ is compared with multifractal thresholds $\mathcal{T}_p(\ell)\sim\ell^{h(p)}$; 'active volumes' are defined through coefficients $c_p$ chosen so that their box-counting dimensions match $D(h)$; and extended self-similarity (Benzi et al., 1993 — the empirical persistence of relative scalings beyond the inertial range) continues the calibration into the dissipative range. The result is a monotonic mapping $f_{\ell_d}$ from the wavelet increment to a pseudo-Hölder exponent $\tilde{h}$ at a chosen dissipative scale $\ell_d$, here $\ell_d=1.8\eta$.

The 2020 characterization. Nguyen, Laval and Dubrulle (2020) applied this estimator to two direct numerical simulations (DNS) of forced isotropic turbulence in a triply periodic domain at resolution $768^3$, with Taylor-scale Reynolds numbers $R_\lambda=140$ (run I) and $R_\lambda=53$ (run II) ($R_\lambda$, the Reynolds number based on the Taylor microscale, is the standard ordering parameter for homogeneous-isotropic-turbulence data). From 40 time-uncorrelated snapshots of run II they extracted 213 'most irregular' events with $\tilde{h}<1.3$ at $\ell_d=1.8\eta$ (inertial-range-equivalent exponent of order $0.1$). Their findings: (i) the event centers populate almost exclusively the vortex-stretching quadrant of the $Q$–$R$ plane of velocity-gradient-tensor invariants ($A=\nabla\mathbf{u}$ the velocity-gradient tensor; $Q=-\frac{1}{2}\mathrm{Tr}(A^2)$, $R=-\det A$; more than 75% have $R<0$); (ii) the event-averaged structure is a Burgers-vortex-like core (a Burgers vortex is an exact Navier–Stokes solution: a Gaussian axial-vorticity tube balanced by axisymmetric strain), with fitted parameters $\Gamma\tau_\nu/2\pi\nu=5$ and $\sigma/\nu=0.08$, but it is markedly nonaxisymmetric — a 'roll vortex' whose helicity changes sign across the central plane; screw vortices are rare and 'U-turn' structures, reported at higher Reynolds numbers, are absent; (iii) maxima of $D_I^\ell$ lie slightly upstream along the streamlines from the maxima of $D_v^\ell$ and from the point of minimal $\tilde{h}$, for which the authors state they have no physical explanation; (iv) in nine time-resolved events, the most singular instant occurs during the close interaction of two or more vortices (some cases involving a reconnection-like topology change), with the temporal maximum of $D_I^\ell$ preceding the minimum of $\tilde{h}$ and the maxima of vorticity and dissipation; (v) points with $\tilde{h}<2$ exist, i.e. locations where the field is not twice differentiable at $\ell_d$.

Open issues left by the source. The paper's own discussion and perspectives list the unresolved items that define this problem: all detected events are regularized by the pseudospectral cutoff at $k_{\max}$ and the Reynolds numbers are modest, so the study should be reproduced with well-resolved experimental data and much larger simulations (e.g. the $8192^3$ DNS of Yeung et al., 2015, at $R_\lambda\approx1300$); the impact of vortex reconnection on the singularity of the events could not be confirmed; the calibration mapping $f_{\ell_d}$ required a linear extrapolation beyond its calibrated range whose robustness was not tested; the variation of the coefficients $c_p$ with order $p$ 'seems to depend on the Reynolds number' and awaits quantification; and extensions to strongly inhomogeneous or anisotropic flows (wall-bounded, Richtmyer–Meshkov, Rayleigh–Taylor) require methodological adaptations because spatial statistical stationarity cannot be assumed. The vortex-reconnection cascade picture of Yao and Hussain (2020) provides the motivating dynamical hypothesis for the interaction-driven formation mechanism.

Problem Statement

The problem is to establish the physical identity, dynamical origin, and universality of the most irregular (quasisingular) small-scale structures of Navier–Stokes turbulence as characterized by local Hölder exponents — the events of minimal local regularity at dissipative scales — generalizing the $R_\lambda\leq140$ pseudospectral-DNS characterization of Nguyen, Laval and Dubrulle (2020). All components target the same objects (the low-$\tilde{h}$ events) and jointly answer what these structures are and where they come from:

(1) Universality and calibration. Determine whether the statistical predominance of roll-type, nonaxisymmetric Burgers-like cores, the fitted core parameters, the bias toward vortex-stretching topology, and the existence of $\tilde{h}<2$ locations persist at substantially higher Taylor-scale Reynolds numbers (e.g. $R_\lambda\sim400$–$1300$ DNS or well-resolved volumetric experiments) and under different small-scale regularization; this requires quantifying the Reynolds-number dependence of the active-volume coefficients $c_p$ and testing the robustness of the extrapolated mapping $f_{\ell_d}$ (e.g. with alternative calibrations or larger datasets).

(2) Dynamical origin. Establish whether close vortex interaction, and vortex reconnection in particular, is necessary, sufficient, or incidental for the formation of quasisingular events, and quantify how reconnection affects the approach to minimal $\tilde{h}$ (the source documents the association in a small time-resolved sample but cannot confirm the impact on the singularity of the event).

(3) Energetic placement. Provide a theoretical explanation of the observed ordering between local interscale transfer, dissipation, and regularity loss: why $D_I^\ell$ maxima lie upstream along streamlines from dissipation maxima and from the minimal-$\tilde{h}$ point, and why the temporal maximum of $D_I^\ell$ precedes the instant of minimal $\tilde{h}$ and the maxima of vorticity and dissipation.

(4) Generalization. Extend the local-regularity characterization to strongly inhomogeneous or anisotropic flows (e.g. wall-bounded turbulence, or Richtmyer–Meshkov- and Rayleigh–Taylor-driven turbulence), where the spatial statistical stationarity used to accumulate multifractal statistics fails.

Components (1)–(3) form the scientific core: the objective is met when the nature and universality of the most irregular structures are determined with a validated calibration (1), their formation mechanism is causally established (2), and the transfer–dissipation–regularity ordering is explained (3); component (4) is the generalization route named by the source within the same program. The source's own scope is retained; no restriction to a single Reynolds number, geometry, flow class, or method variant is intended.

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