- Editors' Suggestion
- Access by Xinjiang University
Chirality tomography: Measuring local helicity from trajectory linking
Phys. Rev. Fluids 11, 034609 – Published 25 March, 2026
DOI: https://doi.org/10.1103/8z43-zzny
Abstract
We present 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 into volumetric maps of dimensionless helicity, . 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 accurately track the evolution of domain-averaged helicity. The proportionality between and 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.
Physics Subject Headings (PhySH)
Article Text
References (59)
- J. J. Moreau, Constantes d'un îlot tourbillonnaire en fluide parfait barotrope, Comptes rendus hebdomadaires des séances de l', Académie des sciences 252, 2810 (1961).
- H. K. Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35, 117 (1969).
- H. K. Moffatt and A. Tsinober, Helicity in laminar and turbulent flow, Annu. Rev. Fluid Mech. 24, 281 (1992).
- M. A. Berger and G. B. Field, The topological properties of magnetic helicity, J. Fluid Mech. 147, 133 (1984).
- P. J. Morrison, Hamiltonian description of the ideal fluid, Rev. Mod. Phys. 70, 467 (1998).
- R. Salmon, Hamiltonian fluid mechanics, Annu. Rev. Fluid Mech. 20, 225 (1988).
- N. Padhye and P. Morrison, Fluid element relabeling symmetry, Phys. Lett. A 219, 287 (1996).
- N. Padhye and P. Morrison, Relabeling symmetries in hydrodynamics and magnetohydrodynamics, Plasma Phys. Rep. 22, 869 (1996).
- M. W. Scheeler, D. Kleckner, D. Proment, G. L. Kindlmann, and W. T. M. Irvine, Helicity conservation by flow across scales in reconnecting vortex links and knots, Proc. Natl. Acad. Sci. USA 111, 15350 (2014).
- F. Waleffe, The nature of triad interactions in homogeneous turbulence, Phys. Fluids A 4, 350 (1992).
- L. Biferale, S. Musacchio, and F. Toschi, Inverse energy cascade in three-dimensional isotropic turbulence, Phys. Rev. Lett. 108, 164501 (2012).
- Q. Chen, S. Chen, and G. L. Eyink, The joint cascade of energy and helicity in three-dimensional turbulence, Phys. Fluids 15, 361 (2003).
- A. Pouquet and G. S. Patterson, Numerical simulation of helical magnetohydrodynamic turbulence, J. Fluid Mech. 85, 305 (1978).
- T. Teitelbaum and P. D. Mininni, Effect of helicity and rotation on the free decay of turbulent flows, Phys. Rev. Lett. 103, 014501 (2009).
- M. Raffel, C. E. Willert, S. T. Wereley, and J. Kompenhans, Particle Image Velocimetry, 2nd ed. (Springer, Berlin, 2007).
- R. J. Adrian and J. Westerweel, Particle Image Velocimetry Series (Cambridge University Press, Cambridge, 2010).
- D. Schanz, S. Gesemann, and A. Schröder, Shake-The-Box: Lagrangian particle tracking at high particle image densities, Exp. Fluids 57, 70 (2016).
- A. Schröder and D. Schanz, 3D Lagrangian particle tracking in fluid mechanics, Annu. Rev. Fluid Mech. 55, 511 (2023).
- T. D. Harms, S. L. Brunton, and B. J. McKeon, Lagrangian gradient regression for the detection of coherent structures from sparse trajectory data, R. Soc. Open Sci. 11, 240586 (2024).
- J. M. Wallace, J. Balint, and L. Ong, An experimental study of helicity density in turbulent flows, Phys. Fluids 4, 2013 (1992).
- M. W. Scheeler, W. M. van Rees, H. Kedia, D. Kleckner, and W. T. M. Irvine, Complete measurement of helicity and its dynamics in vortex tubes, Science 357, 487 (2017).
- D. Ferraro, S. Servidio, A. Lauria, and R. Gaudio, A local measure of the helicity in turbulent flows, Phys. Fluids 36, 105146 (2024).
- N. Yokoi, Transport in helical fluid turbulence, in Helicities in Geophysics, Astrophysics and Beyond (American Geophysical Union, New York, 2023), pp. 25–50.
- N. Mordant, P. Metz, J. Pinton, and O. Michel, Lagrangian measurement in fully developed turbulence, AIP Conf. Proc. 622, 343 (2002).
- R. Volk, N. Mordant, G. Verhille, and J.-F. Pinton, Laser Doppler measurement of inertial particle and bubble accelerations in turbulence, Europhys. Lett. 81, 34002 (2008).
- S. Angriman, P. D. Mininni, and P. J. Cobelli, Velocity and acceleration statistics in particle-laden turbulent swirling flows, Phys. Rev. Fluids 5, 064605 (2020).
- S. Angriman, P. D. Mininni, and P. J. Cobelli, Multitime structure functions and the Lagrangian scaling of turbulence, Phys. Rev. Fluids 7, 064603 (2022).
- S. Angriman, A. Ferran, F. Zapata, P. J. Cobelli, M. Obligado, and P. D. Mininni, Clustering in laboratory and numerical turbulent swirling flows, J. Fluid Mech. 948, A30 (2022).
- S. Angriman, P. J. Cobelli, M. Bourgoin, S. G. Huisman, R. Volk, and P. D. Mininni, Broken mirror symmetry of tracer's trajectories in turbulence, Phys. Rev. Lett. 127, 254502 (2021).
- C. F. Gauss, Integral formula for linking number, in Zur Mathematischen Theorie der Electrodynamischen Wirkungen, edited by K. G. der Wissenschaften zu Göttingen (Springer, Berlin, 1833), Vol. 5, p. 605.
- E. Orlandini, M. C. Tesi, S. G. Whittington, D. W. Sumners, and E. J. J. V. Rensburg, The writhe of a self-avoiding walk, J. Phys. A: Math. Gen. 27, L333 (1994).
- M. A. Berger and C. Prior, The writhe of open and closed curves, J. Phys. A: Math. Gen. 39, 8321 (2006).
- K. Foteinopoulou, N. C. Karayiannis, M. Laso, M. Kröger, and M. L. Mansfield, Universal scaling, entanglements, and knots of model chain molecules, Phys. Rev. Lett. 101, 265702 (2008).
- E. Flapan, When Topology Meets Chemistry: A Topological Look at Molecular Chirality (Cambridge University Press, Singapore, 2000).
- M. Laso, N. C. Karayiannis, K. Foteinopoulou, M. L. Mansfield, and M. Kröger, Random packing of model polymers: Local structure, topological hindrance and universal scaling, Soft Matter 5, 1762 (2009).
- J. I. Sułkowska, E. J. Rawdon, K. C. Millett, J. N. Onuchic, and A. Stasiak, Conservation of complex knotting and slipknotting patterns in proteins, Proc. Natl. Acad. Sci. USA 109, E1715 (2012).
- L. Shen, H. Feng, F. Li, F. Lei, J. Wu, and G.-W. Wei, Knot data analysis using multiscale Gauss link integral, Proc. Natl. Acad. Sci. USA 121, e2408431121 (2024).
- E. Panagiotou, M. Kröger, and K. C. Millett, Writhe and mutual entanglement combine to give the entanglement length, Phys. Rev. E 88, 062604 (2013).
- E. Panagiotou and L. H. Kauffman, Knot polynomials of open and closed curves, Proc. R. Soc. A 476, 20200124 (2020).
- D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion, Physica D 133, 215 (1999).
- L. H. Kauffman, Knots and Physics, 3rd ed. (World Scientific, Singapore, 2001).
- Bentley and Ottmann, Algorithms for reporting and counting geometric intersections, IEEE Trans. Comput. C-28, 643 (1979).
- M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry, 2nd ed. (Springer, Berlin, 2000).
- M. Noseda and P. J. Cobelli, Lalito: Lagrangian linking tomography—Signed crossing density mapping from 3D Lagrangian trajectories (2025), https://zenodo.org/records/17070146.
- B. L. Español, M. Noseda, P. J. Cobelli, and P. D. Mininni, Effect of local flow geometry on particle pair dispersion angle, Phys. Rev. Fluids 10, 044501 (2025).
- P. D. Mininni, D. Rosenberg, R. Reddy, and A. Pouquet, A hybrid MPI–OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence, Parallel Comput. 37, 316 (2011).
- D. Rosenberg, P. D. Mininni, R. Reddy, and A. Pouquet, GPU parallelization of a hybrid pseudospectral geophysical turbulence framework using CUDA, Atmosphere 11, 178 (2020).
- G. I. Taylor and A. E. Green, Mechanism of the production of small eddies from large ones, Proc. R. Soc. Lond. 158, 499 (1937).
- M. R. Maxey and J. J. Riley, Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids 26, 883 (1983).
- R. Gatignol, The Faxén formulæ for a rigid particle in an unsteady non-uniform Stokes flow, J. Mec. Theor. Appl. 1, 143 (1983).
- T. R. Auton, J. C. R. Hunt, and M. Prud'Homme, The force exerted on a body in inviscid unsteady non-uniform rotational flow, J. Fluid Mech. 197, 241 (1988).
- J. Happel and H. Brenner, Wall effects on the motion of a single particle, in Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Springer Netherlands, Dordrecht, 1983), pp. 286–357.
- J. Boussinesq, Sur la resistance qu'oppose un fluide indefini en repos, sans pesanteur, au mouvement varie d'une sphere solide qu'il mouille sur toute sa surface, quand les vitesses restent bien continues et assez faibles pour que leurs carres et produits soient negligiables, C. R. Acad. Sci. Paris 100, 935 (1885).
- A. Basset, A Treatise on Hydrodynamics: With Numerous Examples (Deighton, Bell and Company, Cambridge, 1888), Vol. 1.
- L.-P. Wang and M. R. Maxey, Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence, J. Fluid Mech. 256, 27 (1993).
- H. Berning and T. Rösgen, Suppression of large-scale azimuthal modulations in a von Kármán flow using random forcing, Phys. Fluids 35, 075151 (2023).
- A. Craya, Contribution à L'analyse de la Turbulence Associée à Des Vitesses Moyennes, Thèse de doctorat, Université de Grenoble, 1957.
- J. R. Herring, S. A. Orszag, R. H. Kraichnan, and D. G. Fox, Decay of two-dimensional homogeneous turbulence, J. Fluid Mech. 66, 417 (1974).
- A. Alexakis and L. Biferale, Cascades and transitions in turbulent flows, Phys. Rep. 767–769, 1 (2018).