Causal coupling of intermittent local energy transfer and vortical structures, quasi-singular shear-layer events, and their universality in the turbulent von Kármán flow

In a combined experimental and direct-numerical study of the turbulent von Kármán flow (swirling flow between counter-rotating impellers), Faller and co-workers (J. Fluid Mech. 914, A2, 2021) showed that local interscale energy transfer and vorticity are both strongly intermittent and correlated, that the multifractal spectrum of the transfer field is milder than that of velocity or vorticity increments, and that the shear layer exhibits a highly correlated tail suggestive of self-similar quasi-singular structures reminiscent of Euler-equation singularities. They left four issues open: the causal direction between local energy transfer and vorticity intensification (their maxima are spatially displaced); whether the shear-layer vortices are self-similar quasi-singular structures that dominate high-order scaling; whether the milder transfer intermittency is a resolution artifact or a genuine physical distinction; and whether these properties are universal or specific to the von Kármán geometry. Recent work has made substantial partial progress — time-resolved 4D particle-tracking measurements of extreme transfer events by the same collaboration (2025), a contrasting null correlation in homogeneous isotropic turbulence (2023), perturbation-based causal analysis in isotropic turbulence (2023), and an active theory of strain self-amplification — but none of the four questions is settled; the problem remains open at its original generality.

Background

Turbulence intermittency denotes the concentration of small-scale turbulent activity — velocity gradients, dissipation, and inter-scale energy flux — into rare, violent, spatially clumped events, producing measurable deviations of structure-function scaling from Kolmogorov's 1941 self-similar theory (K41). A structure function $S_p(\ell)=\langle|\delta u_\ell|^p\rangle$ averages the $p$-th power of velocity increments over separations $\ell$; intermittency makes its scaling exponents depart from the K41 line $\zeta_p=p/3$. In the multifractal framework, the set of points where increments scale as $\delta u_\ell\sim \ell^h$ has a fractal dimension $C(h)$, recovered from the exponents by a Legendre transform; a broader or more asymmetric $C(h)$ means stronger intermittency. The phenomenon has been recognized since Batchelor and Townsend's 1949 hot-wire measurements, and Kolmogorov's refined similarity hypothesis (1962) tied intermittent velocity statistics to the intermittency of local dissipation.

The von Kármán flow — fluid stirred between two counter-rotating impellers in a closed cylinder — is a canonical closed turbulent flow. It is globally inhomogeneous and anisotropic, with a turbulent shear layer (a cylindrical mixing layer separating the two impeller-driven recirculation cells) and a stagnation point at mid-height. It is one of few configurations where matched high-Reynolds-number experiments (stereoscopic and tomographic particle image velocimetry, SPIV and TPIV, measuring planar or volumetric velocity fields) and high-resolution direct numerical simulation (DNS; here a spectral/finite-element computation with about $3\times10^8$ grid points — 250 azimuthal Fourier modes plus finite elements in the meridional plane, equivalent grid $400\times800\times509$ — at Taylor-scale Reynolds number $R_\lambda\approx72$, the standard turbulence-strength measure built on the root-mean-square velocity fluctuation and the Taylor microscale, with 21 statistically uncorrelated snapshots) can be realized in the same geometry.

A key modern tool in this program is the local interscale energy transfer $D_\ell$: following Duchon and Robert's (2000) weak-solution energy balance and Dubrulle's (2019) construction, spatial filtering of the incompressible Navier–Stokes energy budget yields, at each point and scale $\ell$, the energy flux from scales larger than $\ell$ to smaller ones, generalizing the Kármán–Howarth relation to non-homogeneous flows. Earlier measurements in the same facility had characterized extreme events of this 'inertial dissipation' (Saw et al. 2016). Faller et al. (2021) computed wavelet structure functions of the velocity ($S_W$), of the solenoidal (vortical) part of the flow ($S_\Omega$), and of the local transfer ($S_D$), their multifractal spectra, and joint statistics of transfer and vorticity $\omega=\nabla\times u$. Their findings: (a) velocity, vortical and transfer structure functions all obey a generalized extended scaling (extended self-similarity: power-law scaling of $S_p$ against $S_3$ rather than against $\ell$ itself, which extends the observable scaling range), and the velocity and vorticity spectra nearly coincide, supporting a local refined similarity hypothesis; (b) the transfer spectrum $C_D$ is milder — more peaked around its most probable exponent — than the velocity spectrum $C_W$; (c) in joint probability densities, extreme energy transfers occur at only moderate vorticity while extreme vorticity is essentially independent of transfer, so the maxima of the two fields are spatially displaced (as previously observed in the same experiment); (d) the DNS joint density displays a highly correlated linear tail associated with shear-layer vortices, at a location reminiscent of where Luo and Hou (2014) computed a potentially self-similar blowing-up solution of the (inviscid) axisymmetric Euler equations in a comparable geometry; (e) conditioned on the sign of the transfer, upscale (negative) transfers are less intermittent than downscale ones; and (f) the transfer statistics obey a log-universality, collapsing across Reynolds numbers when rescaled by $\log R_\lambda$, generalizing earlier results of Geneste et al. (2019).

These findings sit inside a live debate on the mechanism of the turbulent cascade: whether intense small-scale dynamics are organized around vortex tubes (intense vorticity stretched by ambient strain) or around strain sheets, and whether the cascade is driven by vortex stretching (intensification of vorticity by the strain field through which it is advected) or by strain self-amplification (nonlinear self-intensification of the strain field; Johnson 2020; Buaria and Pumir 2026). What the correlation measurements cannot reveal is the direction of the interaction, the dynamical nature of the extreme objects, and whether the picture survives outside this particular flow — the gaps the source paper itself identified.

Problem Statement

Determine the dynamical mechanism — and its universality — underlying the observed intermittency of the local energy cascade in the turbulent von Kármán flow, by resolving the four coupled issues left open by Faller et al. (2021):

1. Causal direction (the core). Establish whether local interscale energy transfer drives the intensification of vorticity, or whether intense vortical structures generate the extreme transfers — that is, settle, from time-resolved three-dimensional data (lagged conditional statistics or controlled perturbations), the dynamical relation between the two fields whose maxima are observed to be spatially displaced.

2. Nature of the extreme structures. Establish whether the shear-layer vortices that produce the highly correlated tail of the transfer–vorticity joint statistics are self-similar quasi-singular structures — in the sense of approaching Euler-equation blow-up dynamics of the Luo–Hou type — that dominate the scaling of large-order structure functions, or whether they are ordinary intense vortices, and what their lifecycle in space and time is.

3. Origin of the milder transfer intermittency. Decide whether the multifractal spectrum of the local energy transfer is milder than that of the velocity because finite spatial resolution misses the largest transfer events, or because of a genuine physical difference between the intermittency of vorticity and of interscale transfer at the smallest scales.

4. Universality. Determine whether the transfer–vorticity correlations, the dominance (or not) of quasi-singular structures, and the log-universality of the transfer statistics extend to turbulent flows lacking the von Kármán shear layer and stagnation point, or are specific to this geometry.

The components are linked: 1 and 2 are the dynamical and structural faces of the same mechanism; 3 tests whether the measured transfer statistics can be taken at face value when building that mechanism; 4 tests whether the mechanism is a universal feature of turbulence. A resolution identifies the mechanism and states, with evidence, the answer to each of the four components; progress on 2–4 alone, without settling the causal question, does not complete the problem.

References

1. H. Faller, D. Geneste, T. Chaabo, A. Cheminet, V. Valori, Y. Ostovan, L. Cappanera, Ch. Cuvier, F. Daviaud, J.-M. Foucaut, J.-L. Guermond, J.-Ph. Laval, C. Nore, V. Padilla, C. Wiertel, B. Dubrulle (2021). On the nature of intermittency in a turbulent von Kármán flow. J. Fluid Mech. 914, A2. DOI 10.1017/jfm.2020.908. https://doi.org/10.1017/jfm.2020.908
2. Benjamin Musci, Berengere Dubrulle, Jean LeBris, Damien Geneste, Pierre Braganca, Jean-Marc Foucaut, Christophe Cuvier, Adam Cheminet (2025). Experimental measurement of the vorticity–strain alignment around extreme energy transfer events. J. Fluid Mech. 1016, A53. DOI 10.1017/jfm.2025.10447 (preprint arXiv:2503.22627). https://doi.org/10.1017/jfm.2025.10447
3. Ryan McKeown, Alain Pumir, Shmuel M. Rubinstein, Michael P. Brenner, Rodolfo Ostilla-Mónico (2023). Energy transfer and vortex structures: visualizing the incompressible turbulent energy cascade. New J. Phys. 25, 103029. DOI 10.1088/1367-2630/acffeb (preprint arXiv:2204.09023). https://doi.org/10.1088/1367-2630/acffeb
4. Miguel P. Encinar, Javier Jiménez (2023). Identifying causally significant features in three-dimensional isotropic turbulence. J. Fluid Mech. 965, A20. DOI 10.1017/jfm.2023.407 (preprint arXiv:2302.04630, posted under the earlier title 'Identifying causal significance in three-dimensional isotropic turbulence'). https://doi.org/10.1017/jfm.2023.407
5. P. Debue, V. Valori, C. Cuvier, F. Daviaud, J.-M. Foucaut, J.-P. Laval, C. Wiertel, V. Padilla, B. Dubrulle (2021). Three-dimensional analysis of precursors to non-viscous dissipation in an experimental turbulent flow. J. Fluid Mech. 914, A9. DOI 10.1017/jfm.2020.574. https://doi.org/10.1017/jfm.2020.574
6. Valentina Valori, Paul Debue, Adam Cheminet, Tarek Chaabo, Yaşar Ostovan, Christophe Cuvier, Jean-Marc Foucaut, Jean-Philippe Laval, Cécile Wiertel, Vincent Padilla, François Daviaud, Bérengère Dubrulle (2021). On Lagrangian Intermittency from 4-D Particle Tracking Velocimetry measurements of a turbulent von Kármán flow. arXiv:2104.04222. https://arxiv.org/abs/2104.04222
7. Dhawal Buaria, Alain Pumir (2026). Turbulence intermittency and velocity gradients. J. Fluid Mech. 1034, P1. DOI 10.1017/jfm.2026.11494. https://doi.org/10.1017/jfm.2026.11494
8. Bérengère Dubrulle (2019). Beyond Kolmogorov cascades. J. Fluid Mech. 867, P1. DOI 10.1017/jfm.2019.98. https://doi.org/10.1017/jfm.2019.98
9. Jean Duchon, Raoul Robert (2000). Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13 (1), 249–255. DOI 10.1088/0951-7715/13/1/312. https://doi.org/10.1088/0951-7715/13/1/312
10. Guo Luo, Thomas Y. Hou (2014). Potentially singular solutions of the 3D axisymmetric Euler equations. Proc. Natl. Acad. Sci. USA 111 (36), 12968–12973. DOI 10.1073/pnas.1405238111. https://doi.org/10.1073/pnas.1405238111
11. P. L. Johnson (2020). Energy transfer from large to small scales in turbulence by multiscale nonlinear strain and vorticity interactions. Phys. Rev. Lett. 124, 104501. DOI 10.1103/PhysRevLett.124.104501. https://doi.org/10.1103/PhysRevLett.124.104501
12. E.-W. Saw, D. Kuzzay, D. Faranda, A. Guittonneau, F. Daviaud, C. Wiertel-Gasquet, V. Padilla, B. Dubrulle (2016). Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow. Nat. Commun. 7, 12466. DOI 10.1038/ncomms12466. https://doi.org/10.1038/ncomms12466
13. Damien Geneste, Hugues Faller, Florian Nguyen, Vishwanath Shukla, Jean-Philippe Laval, Francois Daviaud, Ewe-Wei Saw, Bérengère Dubrulle (2019). About Universality and Thermodynamics of Turbulence. Entropy 21 (3), 326. DOI 10.3390/e21030326. https://doi.org/10.3390/e21030326
14. Bérengère Dubrulle (2022). Multi-Fractality, Universality and Singularity in Turbulence. Fractal Fract. 6 (10), 613. DOI 10.3390/fractalfract6100613. https://doi.org/10.3390/fractalfract6100613
15. Alexey Cheskidov, Roman Shvydkoy (2023). Volumetric Theory of Intermittency in Fully Developed Turbulence. Arch. Ration. Mech. Anal. 247 (3), 45. DOI 10.1007/s00205-023-01878-5. https://doi.org/10.1007/s00205-023-01878-5
16. Siddhartha Mukherjee, Sugan Durai Murugan, Ritwik Mukherjee, Samriddhi Sankar Ray (2024). Turbulent Flows Are Not Uniformly Multifractal. Phys. Rev. Lett. 132, 184002. DOI 10.1103/PhysRevLett.132.184002. https://doi.org/10.1103/PhysRevLett.132.184002
17. H. S. Larssen, J. C. Vassilicos (2023). Turbulent energy transfers in physical and scale spaces at dissipative length scales. J. Fluid Mech. 976, A27. DOI 10.1017/jfm.2023.918. https://doi.org/10.1017/jfm.2023.918
18. Daniel J. Ruth, Filippo Coletti (2024). Structure and energy transfer in homogeneous turbulence below a free surface. J. Fluid Mech. 1001, A46. DOI 10.1017/jfm.2024.1017. https://doi.org/10.1017/jfm.2024.1017
19. A. Pouquet, D. Rosenberg, R. Marino, P. Mininni (2023). Intermittency Scaling for Mixing and Dissipation in Rotating Stratified Turbulence at the Edge of Instability. Atmosphere 14 (9), 1375. DOI 10.3390/atmos14091375. https://doi.org/10.3390/atmos14091375
20. M. Creff, H. Faller, B. Dubrulle, J.-L. Guermond, C. Nore (2024). Tracking dynamo mechanisms from local energy transfers: Application to the von Kármán sodium dynamo. Phys. Plasmas 31, 022306. DOI 10.1063/5.0174251. https://doi.org/10.1063/5.0174251
21. Benjamin Musci, Sebastien Aumaitre, Enzo Francisco, Adam Cheminet, Berengere Dubrulle (2026). The Excess Dissipation of Energy in a Turbulent Boundary-Layer and its Departure from Log-Normality. arXiv:2607.02974. DOI 10.48550/arXiv.2607.02974. https://arxiv.org/abs/2607.02974
22. A. N. Kolmogorov (1962). A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13 (1), 82–85. DOI 10.1017/S0022112062000518. https://doi.org/10.1017/S0022112062000518
23. G. K. Batchelor, A. A. Townsend (1949). The nature of turbulent motion at large wave-numbers. Proc. R. Soc. Lond. A 199 (1057), 238–255. DOI 10.1098/rspa.1949.0136. https://doi.org/10.1098/rspa.1949.0136