Chirality tomography: measuring local helicity from trajectory linking

TL;DR

Helicity is a fundamental 3D invariant tied to flow chirality, but measuring it in turbulence is difficult due to vorticity requirements. The authors introduce chirality tomography, a Lagrangian, voxel-based approach that reconstructs local helicity density from trajectory linking, using the mean crossing proxy $\mathcal{K}$ averaged over projections. They derive and validate a local proportionality between the linking rate and the coarse-grained helicity density, with a geometric scale $\ell_h$ and effective time $\tau_h^{\text{eff}}$, demonstrated on Taylor–Green DNS and von Kármán experiments, yielding 3D helicity maps and time-resolved helicity proxies. The method is robust to voxel geometry and modest particle inertia, but does not apply in laminar or time-modulated flows, highlighting both its practical value and regime limits. Overall, this work bridges trajectory topology and helicity, providing a practical diagnostic for turbulent flows and a platform for local helicity reconstruction from Lagrangian data.

Abstract

We present the first three-dimensional helicity maps of fully developed turbulence obtained through chirality tomography, a Lagrangian voxel-based method that reconstructs helicity density from particle trajectories. Our approach builds on an empirically established relation between helicity and trajectory linking, converting local counts of signed crossings $K$ into volumetric maps of dimensionless helicity, $H(\mathbf{x})$. We demonstrate that the entanglement of particle trajectories, quantified by the mean signed crossing number, provides a robust proxy for helicity, not only at the global scale, but also locally in space and time. Our method can reveal local spatial heterogeneities in helicity and relate them to large-scale flow organization, enabling the reconstruction of spatially resolved chiral structures. Applied to von Kármán experiments and Taylor-Green direct numerical simulations, the method reveals iso-helicity surfaces and coherent chiral features, while time series of $K$ accurately track the evolution of domain-averaged helicity. The proportionality between $K$ and $H$ remains robust across different voxel geometries and different values of particle inertia, but is not held in laminar or time-modulated flows. This study shows that chirality tomography provides a practical helicity diagnostic in turbulent flows, while establishing a direct bridge between trajectory-level topology and a fundamental dynamical invariant of turbulence.

Figures (10)

• Figure 1: Illustration of the relation between entanglement and crossings in 2D projections. The figure is organized in two rows and three columns; each column corresponds to a different pair of space curves (e.g., particle trajectories). Panels (a)–(c) (top row) show 3D views, with black arrows indicating the traversal direction along each curve. Panels (d)–(f) (bottom row) display, for the corresponding pairs, an arbitrarily chosen orthographic 2D projection onto the $y$–$z$ plane as viewed from $x>0$ toward the origin. This is one among infinitely many possible projection directions. Whenever the projected curves exhibit an apparent crossing, its sign is assigned according to the convention sketched below the panels. Panel (a) shows the Hopf link (two linked circles). For closed curves, a single generic projection such as that shown in (d) already determines $\mathop{\mathrm{Lk}}\nolimits$ via the signed crossing number $C(\hat{\mathbf n})$ (see text); in this view both crossings have the same (negative) sign, yielding $\mathrm{Lk}=-1$. Panels (b) and (c) show open curves. In (e), the selected projection presents crossings of equal (positive) sign, consistent with an intertwined configuration. In (f), the apparent crossings compensate with opposite signs, corresponding to a non-entangled pair. For open curves, a single projection is not sufficient: a meaningful estimate of the Gauss linking integral requires averaging crossing information over many projection directions (see Sec. \ref{['sec:S2_z']}).
• Figure 2: Schematics of the experimental and numerical configurations. (a) Von Kármán experimental setup; $f_0$ denotes the rotation frequency of the impellers. The arrows inside the cell illustrate the principal features of the mean large-scale flow. (b) Geometry used for the Taylor-Green simulations. The periodic domain, a cube of side $(2\pi L_0)$, contains eight subcells of volume $(\pi L_0)^3$, each of which hosts large-scale structures reminiscent of those in the VK flow. This correspondence is illustrated by the highlighted subcell. Figure adapted from Espanol2025.
• Figure 3: Tomographic slice of the Taylor--Green turbulent flow: reconstructed $\mathcal{K}$ and ground-truth $\mathcal{H}$. (a) Time-averaged mean linking number, computed from $N = 10^4$ trajectories and mapped onto $10^3$ cubic voxels; slice on a plane parallel to $x$--$y$ at $z \simeq \pi/5$. (b) Dimensionless helicity obtained from the DNS fields, coarse-grained to the same voxel geometry and averaged over the same interval; slice at the same location. The characteristic four-cell organization of the flow is visible in both panels.
• Figure 4: Mean linking number as a proxy for helicity in the TG flow. (a) Time-averaged values of $\mathcal{K}$ obtained from a subdivision of the domain into $10^3$ cubic voxels. Red symbols correspond to $N=10^4$ trajectories observed for a time window of length $\Delta T = T_0$, using $20$ random projections; error bars denote the standard error across ten independent subsamples. Blue symbols show the same analysis with shorter histories, $\Delta T=0.2 \, T_0$, and $N=10^3$ particles. Insets display the probability density functions of the residuals. (b) Spatial average of $\mathcal{K}$ computed with the same parameters as in the red dataset of the left panel, compared with the domain-averaged dimensionless helicity $\mathcal{H}$ (dotted black line), rescaled by the fitted proportionality constant. (c) Spread of the PDF of residuals for different spatial resolutions, $\mathbf{N}$. The deviation increases as the number of voxels grows, since the available statistics are finite. Conversely, no significant reduction of the spread is observed when using $\mathbf{N} = (10, \, 10, \, 10)$ voxels or fewer. (d) Comparison of $\mathcal{K}(t)$ obtained with different temporal windows. Shorter time windows provide improved temporal resolution, as the entanglement of trajectories is averaged over shorter intervals; however, the associated uncertainty increases due to reduced statistics, even though the mean value remains close to the non-dimensional helicity.
• Figure 5: Three-dimensional tomographic reconstructions of helicity in Taylor--Green (numerical) and von Kármán (experimental) turbulent flows. (a) Volumetric rendering of the reconstructed field $\mathcal{K}/\alpha$ for the TG dataset. Individual subcells can be clearly distinguished, with helicity alternating sign between neighboring regions, a characteristic feature of the TG flow. (b) Isosurfaces at $\mathcal{H}=0.4$ in the full TG domain. (c) Isosurfaces at $\mathcal{K}=0.07$ for the VK experiment in the full observation volume (units in meters). (d) Zoom of the isosurfaces shown in (b), restricted to the subcell $[\pi, \, 2\pi] \times [0, \, \pi] \times [0, \, \pi]$.