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Statistics of velocity gradient and vortex sheet structures in polymeric turbulent von Kármán swirling flow

Published online by Cambridge University Press:  15 October 2025

Feng Wang Open the ORCID record for Feng Wang [Opens in a new window] ,
Yi-Bao Zhang Open the ORCID record for Yi-Bao Zhang [Opens in a new window] ,
Ping-Fan Yang Open the ORCID record for Ping-Fan Yang [Opens in a new window]  and
Heng-Dong Xi Open the ORCID record for Heng-Dong Xi [Opens in a new window]
Feng Wang
Affiliation:
Institute of Extreme Mechanics, School of Aeronautics, National Key Laboratory of Aircraft Configuration Design, Key Laboratory for Extreme Mechanics of Aircraft of Ministry of Industry and Information Technology, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, PR China
Yi-Bao Zhang
Affiliation:
Institute of Extreme Mechanics, School of Aeronautics, National Key Laboratory of Aircraft Configuration Design, Key Laboratory for Extreme Mechanics of Aircraft of Ministry of Industry and Information Technology, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, PR China New Cornerstone Science Laboratory, Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing, 100084, PR China
Ping-Fan Yang*
Affiliation:
Institute of Extreme Mechanics, School of Aeronautics, National Key Laboratory of Aircraft Configuration Design, Key Laboratory for Extreme Mechanics of Aircraft of Ministry of Industry and Information Technology, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, PR China
Heng-Dong Xi*
Affiliation:
Institute of Extreme Mechanics, School of Aeronautics, National Key Laboratory of Aircraft Configuration Design, Key Laboratory for Extreme Mechanics of Aircraft of Ministry of Industry and Information Technology, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, PR China
*
Corresponding authors: Ping-Fan Yang, yangpingfan@nwpu.edu.cn; Heng-Dong Xi, hengdongxi@nwpu.edu.cn
Corresponding authors: Ping-Fan Yang, yangpingfan@nwpu.edu.cn; Heng-Dong Xi, hengdongxi@nwpu.edu.cn
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Abstract

Investigations into the effects of polymers on small-scale statistics and flow patterns were conducted in a turbulent von Kármán swirling (VKS) flow. We employed the tomographic particle image velocimetry technique to obtain full information on three-dimensional velocity data, allowing us to effectively resolve dissipation scales. Under varying Reynolds numbers ($R_\lambda =168{-}235$) and polymer concentrations ($\phi =0{-}25\ {\textrm{ppm}}$), we measured the velocity gradient tensor (VGT) and related quantities. Our findings reveal that the ensemble average and probability density function (PDF) of VGT invariants, which represent turbulent dissipation and enstrophy along with their generation terms, are suppressed as polymer concentration increases. Notably, the joint PDFs of the invariants of VGT, which characterise local flow patterns, exhibited significant changes. Specifically, the third-order invariants, especially the local vortex stretching, are greatly suppressed, and strong events of dissipation and enstrophy coexist in space. The local flow pattern tends to be two-dimensional, where the eigenvalues of the rate-of-strain tensor satisfy a ratio $1:0:-1$, and the vorticity aligns with the intermediate eigenvector of the rate-of-strain tensor, while it is perpendicular to the other two. We find that these statistics observations can be well described by the vortex sheet model. Moreover, we find that these vortex sheet structures align with the symmetry axis of the VKS system, and orient randomly in the horizontal plane. Further investigation, including flow visualisation and conditional statistics on vorticity, confirms the presence of vortex sheet structures in turbulent flows with polymer additions. Our results establish a link between single-point statistics and small-scale flow topology, shedding light on the previously overlooked small-scale structures in polymeric turbulence.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Isotropic turbulence Non-Newtonian Flows: Viscoelasticity

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JFM Papers
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Journal of Fluid Mechanics , Volume 1021 , 25 October 2025 , A20
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© The Author(s), 2025. Published by Cambridge University Press

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References

Andreotti, B. 1997 Studying Burgers’ models to investigate the physical meaning of the alignments statistically observed in turbulence. Phys. Fluids 9, 735–742.10.1063/1.869228CrossRefGoogle Scholar
Ashurst, W.T., Kerstein, A.R., Kerr, R.M. & Gibson, C.H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30, 2343–2353.CrossRefGoogle Scholar
Benzi, R., De Angelis, E., Govindarajan, R. & Procaccia, I. 2003 Shell model for drag reduction with polymer additives in homogeneous turbulence. Phys. Rev. E 68, 016308.10.1103/PhysRevE.68.016308CrossRefGoogle ScholarPubMed
Betchov, R. 1956 An inequality concerning the production of vorticity in isotropic turbulence. J. Fluid Mech. 1, 497–504.10.1017/S0022112056000317CrossRefGoogle Scholar
Blackburn, H.M., Mansour, N.N. & Cantwell, B.J. 1996 Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech. 310, 269–292.10.1017/S0022112096001802CrossRefGoogle Scholar
Boffetta, G., Mazzino, A., Musacchio, S. & Vozella, L. 2010 Polymer heat transport enhancement in thermal convection: the case of Rayleigh–Taylor turbulence. Phys. Rev. Lett. 104, 184501.CrossRefGoogle ScholarPubMed
Bonn, D., Couder, Y., Van Dam, P.H.J. & Douady, S. 1993 From small scales to large scales in three-dimensional turbulence: the effect of diluted polymers. Phys. Rev. E 47, R28.10.1103/PhysRevE.47.R28CrossRefGoogle ScholarPubMed
Buaria, D., Bodenschatz, E. & Pumir, A. 2020 Vortex stretching and enstrophy production in high Reynolds number turbulence. Phys. Rev. Fluids 5, 104602.10.1103/PhysRevFluids.5.104602CrossRefGoogle Scholar
Buaria, D. & Pumir, A. 2022 Vorticity–strain rate dynamics and the smallest scales of turbulence. Phys. Rev. Lett. 128, 094501.CrossRefGoogle ScholarPubMed
Buaria, D., Pumir, A., Bodenschatz, E. & Yeung, P.K. 2019 Extreme velocity gradients in turbulent flows. New J. Phys. 21, 043004.10.1088/1367-2630/ab0756CrossRefGoogle Scholar
Burnishev, Y. & Steinberg, V. 2016 Influence of polymer additives on turbulence in von Karman swirling flow between two disks. II. Phys. Fluids 28, 033101.CrossRefGoogle Scholar
Buxton, O.R.H., Breda, M. & Chen, X. 2017 Invariants of the velocity-gradient tensor in a spatially developing inhomogeneous turbulent flow. J. Fluid Mech. 817, 1–20.10.1017/jfm.2017.93CrossRefGoogle Scholar
Cadot, O., Bonn, D. & Douady, S. 1998 Turbulent drag reduction in a closed flow system: boundary layer versus bulk effects. Phys. Fluids 10, 426–436.CrossRefGoogle Scholar
Cai, W.-H., Li, F.-C. & Zhang, H.-N. 2010 DNS study of decaying homogeneous isotropic turbulence with polymer additives. J. Fluid Mech. 665, 334.10.1017/S0022112010003939CrossRefGoogle Scholar
Cantwell, B.J. 1992 Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4, 782–793.10.1063/1.858295CrossRefGoogle Scholar
Carter, D.W. & Coletti, F. 2018 Small-scale structure and energy transfer in homogeneous turbulence. J. Fluid Mech. 854, 505–543.10.1017/jfm.2018.616CrossRefGoogle Scholar
Casciola, C.M. & De Angelis, E. 2007 Energy transfer in turbulent polymer solutions. J. Fluid Mech. 581, 419–436.CrossRefGoogle Scholar
de Chaumont Quitry, A. & Ouellette, N.T. 2016 Concentration effects on turbulence in dilute polymer solutions far from walls. Phys. Rev. E 93, 063116.10.1103/PhysRevE.93.063116CrossRefGoogle ScholarPubMed
Chong, M.S., Soria, J., Perry, A.E., Chacin, J., Cantwell, B.J. & Na, Y. 1998 Turbulence structures of wall-bounded shear flows found using DNS data. J. Fluid Mech. 357, 225–247.CrossRefGoogle Scholar
Cocconi, G., De Angelis, E., Frohnapfel, B., Baevsky, M. & Liberzon, A. 2017 Small scale dynamics of a shearless turbulent/non-turbulent interface in dilute polymer solutions. Phys. Fluids 29, 075102.10.1063/1.4991921CrossRefGoogle Scholar
Crawford, A.M. 2004 Particle tracking measurements in fully developed turbulence: water and dilute polymer solutions. PhD thesis. Cornell University.Google Scholar
Crawford, A.M., Mordant, N., Xu, H. & Bodenschatz, E. 2008 Fluid acceleration in the bulk of turbulent dilute polymer solutions. New J. Phys. 10, 123015.10.1088/1367-2630/10/12/123015CrossRefGoogle Scholar
De Angelis, E., Casciola, C.M., Benzi, R. & Piva, R. 2005 Homogeneous isotropic turbulence in dilute polymers. J. Fluid Mech. 531, 1–10.CrossRefGoogle Scholar
Debue, P., Kuzzay, D., Saw, E.-W., Daviaud, F., Dubrulle, B., Canet, L., Rossetto, V. & Wschebor, N. 2018 Experimental test of the crossover between the inertial and the dissipative range in a turbulent swirling flow. Phys. Rev. Fluids 3, 024602.10.1103/PhysRevFluids.3.024602CrossRefGoogle Scholar
Den Toonder, J.M.J., Hulsen, M.A., Kuiken, G.D.C. & Nieuwstadt, F.T.M. 1997 Drag reduction by polymer additives in a turbulent pipe flow: numerical and laboratory experiments. J. Fluid Mech. 337, 193–231.10.1017/S0022112097004850CrossRefGoogle Scholar
van Doorn, E., White, C.M. & Sreenivasan, K.R. 1999 The decay of grid turbulence in polymer and surfactant solutions. Phys. Fluids 11, 2387–2393.10.1063/1.870100CrossRefGoogle Scholar
Douady, S., Couder, Y. & Brachet, M.E. 1991 Direct observation of the intermittency of intense vorticity filaments in turbulence. Phys. Rev. Lett. 67, 983.10.1103/PhysRevLett.67.983CrossRefGoogle ScholarPubMed
Dubief, Y., White, C.M., Terrapon, V.E., Shaqfeh, E.S.G., Moin, P. & Lele, S.K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280.10.1017/S0022112004000291CrossRefGoogle Scholar
Elsinga, G.E. & Marusic, I. 2010 Universal aspects of small-scale motions in turbulence. J. Fluid Mech. 662, 514–539.10.1017/S0022112010003381CrossRefGoogle Scholar
Fathali, M. & Khoei, S. 2019 Spectral energy transfer in a viscoelastic homogeneous isotropic turbulence. Phys. Fluids 31, 095105.10.1063/1.5112161CrossRefGoogle Scholar
Fiscaletti, D., Ragni, D., Overmars, E.F.J., Westerweel, J. & Elsinga, G.E. 2022 Tomographic long-distance $\unicode{x03BC}$ PIV to investigate the small scales of turbulence in a jet at high Reynolds number. Exp. Fluids 63, 1–16.10.1007/s00348-021-03359-5CrossRefGoogle Scholar
Fouxon, A. & Lebedev, V. 2003 Spectra of turbulence in dilute polymer solutions. Phys. Fluids 15, 2060–2072.10.1063/1.1577563CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press.10.1017/CBO9781139170666CrossRefGoogle Scholar
Ganapathisubramani, B., Lakshminarasimhan, K. & Clemens, N.T. 2007 Determination of complete velocity gradient tensor by using cinematographic stereoscopic PIV in a turbulent jet. Exp. Fluids 42, 923–939.10.1007/s00348-007-0303-5CrossRefGoogle Scholar
Ganapathisubramani, B., Lakshminarasimhan, K. & Clemens, N.T. 2008 Investigation of three-dimensional structure of fine scales in a turbulent jet by using cinematographic stereoscopic particle image velocimetry. J. Fluid Mech. 598, 141–175.10.1017/S0022112007009706CrossRefGoogle Scholar
de Gennes, P.G. 1986 Towards a scaling theory of drag reduction. Physica A 140, 9–25.10.1016/0378-4371(86)90200-1CrossRefGoogle Scholar
Gomes-Fernandes, R., Ganapathisubramani, B. & Vassilicos, J.C. 2014 Evolution of the velocity-gradient tensor in a spatially developing turbulent flow. J. Fluid Mech. 756, 252–292.10.1017/jfm.2014.452CrossRefGoogle Scholar
Gotoh, T., Watanabe, T. & Saito, I. 2023 Kinematic effects on probability density functions of energy dissipation rate and enstrophy in turbulence. Phys. Rev. Lett. 130, 254001.10.1103/PhysRevLett.130.254001CrossRefGoogle ScholarPubMed
Guimarães, M.C., Pimentel, N. & Pinho, F.T. & da Silva, C.B. 2020 Direct numerical simulations of turbulent viscoelastic jets. J. Fluid Mech. 899, A11.10.1017/jfm.2020.402CrossRefGoogle Scholar
Horiuti, K., Matsumoto, K. & Fujiwara, K. 2013 Remarkable drag reduction in non-affine viscoelastic turbulent flows. Phys. Fluids 25, 015106.10.1063/1.4774239CrossRefGoogle Scholar
Jiménez, J., Wray, A.A., Saffman, P.G. & Rogallo, R.S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.10.1017/S0022112093002393CrossRefGoogle Scholar
Johnson, P.L. & Wilczek, M. 2024 Multiscale velocity gradients in turbulence. Annu. Rev. Fluid Mech. 56, 463–490.10.1146/annurev-fluid-121021-031431CrossRefGoogle Scholar
Kalelkar, C., Govindarajan, R. & Pandit, R. 2005 Drag reduction by polymer additives in decaying turbulence. Phys. Rev. E 72, 017301.10.1103/PhysRevE.72.017301CrossRefGoogle ScholarPubMed
Kerr, R.M. 1985 Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–58.10.1017/S0022112085001136CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
La Porta, A., Voth, G.A., Crawford, A.M., Alexander, J. & Bodenschatz, E. 2001 Fluid particle accelerations in fully developed turbulence. Nature 409, 1017–1019.10.1038/35059027CrossRefGoogle ScholarPubMed
Lacassagne, T., Simoëns, S., El Hajem, M., Lyon, A. & Champagne, J.-Y. 2019 Oscillating grid turbulence in shear-thinning polymer solutions. Phys. Fluids 31, 083102.10.1063/1.5113551CrossRefGoogle Scholar
Lawson, J.M. & Dawson, J.R. 2014 A scanning PIV method for fine-scale turbulence measurements. Exp. Fluids 55, 1–19.10.1007/s00348-014-1857-7CrossRefGoogle Scholar
Liberzon, A., Guala, M., Kinzelbach, W. & Tsinober, A. 2006 On turbulent kinetic energy production and dissipation in dilute polymer solutions. Phys. Fluids 18, 125101.CrossRefGoogle Scholar
Liberzon, A., Guala, M., Lüthi, B., Kinzelbach, W. & Tsinober, A. 2005 Turbulence in dilute polymer solutions. Phys. Fluids 17, 031707.CrossRefGoogle Scholar
Lin, C.-Y., Owolabi, B.E. & Lin, C.-A. 2022 Polymer–turbulence interactions in a complex flow and implications for the drag reduction phenomenon. Phys. Fluids 34, 043106.CrossRefGoogle Scholar
Lumley, J.L. 1969 Drag reduction by additives. Annu. Rev. Fluid Mech. 1, 367–384.CrossRefGoogle Scholar
Lund, T.S. & Rogers, M.M. 1994 An improved measure of strain state probability in turbulent flows. Phys. Fluids 6, 1838–1847.CrossRefGoogle Scholar
Lundgren, T.S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193–2203.10.1063/1.863957CrossRefGoogle Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219–245.10.1146/annurev-fluid-122109-160708CrossRefGoogle Scholar
Moisy, F. & Jiménez, J. 2004 Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 513, 111–133.10.1017/S0022112004009802CrossRefGoogle Scholar
Mortimer, L.F. & Fairweather, M. 2022 Prediction of polymer extension, drag reduction, and vortex interaction in direct numerical simulation of turbulent channel flows. Phys. Fluids 34, 073318.10.1063/5.0094978CrossRefGoogle Scholar
Nguyen, M.Q., Delache, A., Simoëns, S., Bos, W.J.T. & El Hajem, M. 2016 Small scale dynamics of isotropic viscoelastic turbulence. Phys. Rev. Fluids 1, 083301.10.1103/PhysRevFluids.1.083301CrossRefGoogle Scholar
Ouellette, N.T., Xu, H. & Bodenschatz, E. 2009 Bulk turbulence in dilute polymer solutions. J. Fluid Mech. 629, 375–385.10.1017/S0022112009006697CrossRefGoogle Scholar
Ouellette, N.T., Xu, H., Bourgoin, M. & Bodenschatz, E. 2006 Small-scale anisotropy in Lagrangian turbulence. New J. Phys. 8, 102.10.1088/1367-2630/8/6/102CrossRefGoogle Scholar
Peng, S.-H., Zhang, Y.-B. & Xi, H.-D. 2023 Effects of polymer additives on the entrainment of turbulent water jet. Phys. Fluids 35, 045110.Google Scholar
Perlekar, P., Mitra, D. & Pandit, R. 2006 Manifestations of drag reduction by polymer additives in decaying, homogeneous, isotropic turbulence. Phys. Rev. Lett. 97, 264501.CrossRefGoogle ScholarPubMed
Perlekar, P., Mitra, D. & Pandit, R. 2010 Direct numerical simulations of statistically steady, homogeneous, isotropic fluid turbulence with polymer additives. Phys. Rev. E 82, 066313.10.1103/PhysRevE.82.066313CrossRefGoogle ScholarPubMed
Peters, T. & Schumacher, J. 2007 Two-way coupling of finitely extensible nonlinear elastic dumbbells with a turbulent shear flow. Phys. Fluids 19, 065109.10.1063/1.2735562CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
ur Rehman, S., Lee, J. & Lee, C. 2022 Effect of Weissenberg number on polymer-laden turbulence. Phys. Rev. Fluids 7, 064303.10.1103/PhysRevFluids.7.064303CrossRefGoogle Scholar
Rosti, M.E., Perlekar, P. & Mitra, D. 2023 Large is different: nonmonotonic behavior of elastic range scaling in polymeric turbulence at large Reynolds and Deborah numbers. Sci. Adv. 9, eadd3831.10.1126/sciadv.add3831CrossRefGoogle Scholar
Scarano, F. 2012 Tomographic PIV: principles and practice. Meas. Sci. Technol. 24, 012001.10.1088/0957-0233/24/1/012001CrossRefGoogle Scholar
She, Z.-S., Jackson, E. & Orszag, S.A. 1990 Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344, 226–228.10.1038/344226a0CrossRefGoogle Scholar
Siggia, E.D. 1981 Numerical study of small-scale intermittency in three-dimensional turbulence. J. Fluid Mech. 107, 375–406.10.1017/S002211208100181XCrossRefGoogle Scholar
da Silva, C.B. & Pereira, J.C.F. 2008 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids 20, 055101.10.1063/1.2912513CrossRefGoogle Scholar
Sinhuber, M., Ballouz, J.G. & Ouellette, N.T. 2018 Probing the strain-rotation balance in non-Newtonian turbulence with inertial particles. Phys. Rev. Fluids 3, 082602.CrossRefGoogle Scholar
Soria, J., Sondergaard, R., Cantwell, B.J., Chong, M.S. & Perry, A.E. 1994 A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids 6, 871–884.CrossRefGoogle Scholar
Sreenivasan, K.R. & Antonia, R.A. 1997 The phenomenology of small-scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472.CrossRefGoogle Scholar
Sreenivasan, K.R. & White, C.M. 2000 The onset of drag reduction by dilute polymer additives, and the maximum drag reduction asymptote. J. Fluid Mech. 409, 149–164.10.1017/S0022112099007818CrossRefGoogle Scholar
Terrapon, V.E., Dubief, Y., Moin, P., Shaqfeh, E.S.G. & Lele, S.K. 2004 Simulated polymer stretch in a turbulent flow using Brownian dynamics. J. Fluid Mech. 504, 61–71.10.1017/S0022112004008250CrossRefGoogle Scholar
Toms, B.A. 1948 Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. In Proceedings of the First International Congress on Rheology (ed. J.M. Burgers), vol. 2, pp. 135–141. North-Holland, Amsterdam.Google Scholar
Tong, P., Goldburg, W.I., Huang, J.S. & Witten, T.A. 1990 Anisotropy in turbulent drag reduction. Phys. Rev. Lett. 65, 2780.10.1103/PhysRevLett.65.2780CrossRefGoogle ScholarPubMed
Townsend, A.A. 1951 On the fine-scale structure of turbulence. Proc. R. Soc. Lond. A 208, 534–542.Google Scholar
Valente, P.C., Da Silva, C.B. & Pinho, F.T. 2014 The effect of viscoelasticity on the turbulent kinetic energy cascade. J. Fluid Mech. 760, 39–62.10.1017/jfm.2014.585CrossRefGoogle Scholar
Valente, P.C., da Silva, C.B. & Pinho, F.T. 2016 Energy spectra in elasto-inertial turbulence. Phys. Fluids 28, 075108.10.1063/1.4955102CrossRefGoogle Scholar
Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. 43, 837–842.10.1051/jphys:01982004306083700CrossRefGoogle Scholar
Vieillefosse, P. 1984 Internal motion of a small element of fluid in an inviscid flow. Physica A 125, 150–162.10.1016/0378-4371(84)90008-6CrossRefGoogle Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.CrossRefGoogle Scholar
Vincent, A. & Meneguzzi, M. 1994 The dynamics of vorticity tubes in homogeneous turbulence. J. Fluid Mech. 258, 245–254.10.1017/S0022112094003319CrossRefGoogle Scholar
Vonlanthen, R. & Monkewitz, P.A. 2013 Grid turbulence in dilute polymer solutions: PEO in water. J. Fluid Mech. 730, 76–98.10.1017/jfm.2013.316CrossRefGoogle Scholar
Voth, G.A., La Porta, A., Crawford, A.M., Alexander, J. & Bodenschatz, E. 2002 Measurement of particle accelerations in fully developed turbulence. J. Fluid Mech. 469, 121–160.10.1017/S0022112002001842CrossRefGoogle Scholar
Wang, F., Zhang, Y.-B. & Xi, H.-D. 2025 a Energy transfer in von Kármán swirling flow of dilute polymer solutions. Phys. Fluids 37, 035142.10.1063/5.0259847CrossRefGoogle Scholar
Wang, F., Zhang, Y.-B. & Xi, H.-D. 2025 b Tomo-PIV measurement of small-scale structures in Newtonian and polymeric turbulence. Exp. Fluids 66, 1–15.CrossRefGoogle Scholar
Warholic, M.D., Massah, H. & Hanratty, T.J. 1999 Influence of drag-reducing polymers on turbulence: effects of Reynolds number, concentration and mixing. Exp. Fluids 27, 461–472.10.1007/s003480050371CrossRefGoogle Scholar
Warwaruk, L. & Ghaemi, S. 2024 Local flow topology of a polymer-laden turbulent boundary layer. J. Fluid Mech. 983, A22.CrossRefGoogle Scholar
Watanabe, T. & Gotoh, T. 2010 Coil-stretch transition in an ensemble of polymers in isotropic turbulence. Phys. Rev. E 81, 066301.CrossRefGoogle Scholar
Watanabe, T. & Gotoh, T. 2013 Hybrid Eulerian–Lagrangian simulations for polymer–turbulence interactions. J. Fluid Mech. 717, 535–575.CrossRefGoogle Scholar
Watanabe, T. & Gotoh, T. 2014 Power-law spectra formed by stretching polymers in decaying isotropic turbulence. Phys. Fluids 26, 035110.10.1063/1.4869102CrossRefGoogle Scholar
White, C.M. & Mungal, M.G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235–256.10.1146/annurev.fluid.40.111406.102156CrossRefGoogle Scholar
White, C.M., Somandepalli, V.S.R. & Mungal, M.G. 2004 The turbulence structure of drag-reduced boundary layer flow. Exp. Fluids 36, 62–69.10.1007/s00348-003-0630-0CrossRefGoogle Scholar
Wieneke, B. 2008 Volume self-calibration for 3D particle image velocimetry. Exp. Fluids 45, 549–556.10.1007/s00348-008-0521-5CrossRefGoogle Scholar
Xi, H.-D., Bodenschatz, E. & Xu, H. 2013 Elastic energy flux by flexible polymers in fluid turbulence. Phys. Rev. Lett. 111, 024501.10.1103/PhysRevLett.111.024501CrossRefGoogle ScholarPubMed
Xu, F., Liu, X.-S., Li, X.-M. & Xia, K.-Q. 2025 Restoration of axisymmetric flow structure in turbulent thermal convection by polymer additives. Phys. Rev. Lett. 134, 084001.10.1103/PhysRevLett.134.084001CrossRefGoogle ScholarPubMed
Xu, F., Zhang, L. & Xia, K.-Q. 2024 Experimental measurement of spatio-temporally resolved energy dissipation rate in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 984, A8.10.1017/jfm.2024.164CrossRefGoogle Scholar
Zhang, Y.-B., Bodenschatz, E., Xu, H. & Xi, H.-D. 2021 Experimental observation of the elastic range scaling in turbulent flow with polymer additives. Sci. Adv. 7, eabd3525.10.1126/sciadv.abd3525CrossRefGoogle ScholarPubMed
Zhang, Y.-B. & Xi, H.-D. 2022 Measured energy injection, transfer, and dissipation rates in the bulk of dilute polymeric turbulent flow: the concentration and Weissenberg number effects. Phys. Fluids 34, 075114.CrossRefGoogle Scholar
Zimmermann, R., Xu, H., Gasteuil, Y., Bourgoin, M., Volk, R., Pinton, J.-F. & Bodenschatz, E. 2010 The Lagrangian exploration module: an apparatus for the study of statistically homogeneous and isotropic turbulence. Rev. Sci. Instrum. 81, 055112.10.1063/1.3428738CrossRefGoogle Scholar