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Vortex-to-velocity reconstruction for wall-bounded turbulence via the field-based linear stochastic estimation

Published online by Cambridge University Press:  12 July 2021

Chengyue Wang Open the ORCID record for Chengyue Wang [Opens in a new window] ,
Qi Gao Open the ORCID record for Qi Gao [Opens in a new window] ,
Biao Wang ,
Chong Pan  and
Jinjun Wang Open the ORCID record for Jinjun Wang [Opens in a new window]
Chengyue Wang
Affiliation:
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai519082, PR China Fluid Mechanics Key Laboratory of Education Ministry, Beijing University of Aeronautics and Astronautics, Beijing100191, PR China
Qi Gao*
Affiliation:
School of Aeronautics and Astronautics, Zhejiang University, Hangzhou310027, PR China Fluid Mechanics Key Laboratory of Education Ministry, Beijing University of Aeronautics and Astronautics, Beijing100191, PR China
Biao Wang*
Affiliation:
Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai519082, PR China
Chong Pan
Affiliation:
Fluid Mechanics Key Laboratory of Education Ministry, Beijing University of Aeronautics and Astronautics, Beijing100191, PR China
Jinjun Wang
Affiliation:
Fluid Mechanics Key Laboratory of Education Ministry, Beijing University of Aeronautics and Astronautics, Beijing100191, PR China
*
† Email addresses for correspondence: [email protected], [email protected]
† Email addresses for correspondence: [email protected], [email protected]
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Abstract

Representing complex flows by evolving vortex structures is an important principle in many investigations of wall-bounded turbulence. The practice of this principle benefits from the bi-directional transformation between the velocity field and the corresponding vortex field. While the velocity-to-vortex transformation could be implemented by various vortex identification criteria, few efforts have been devoted to the inverse process. This work develops a linear reconstruction method, which allows an effective reconstruction for the velocity field of wall turbulence based on a given vortex field. The vortex field is defined as a vector field by combining the swirl strength and the real eigenvector of the velocity gradient tensor. The reconstructed velocity fields are calculated by convolution operations on the vortex fields, with the kernel functions derived by the field-based linear stochastic estimation. The method can effectively recover the turbulent motions in a large scale range, showing clear advantages over the Biot–Savart formula in the near-wall region. The method is also employed to investigate the inducing effects of vortices at different heights. The wall-bounding effect on the induced motions is observed from the contribution spectra of vortices. The higher-order moments of the reconstructed streamwise velocity component present larger deviations from the original data, which is discussed and explained reasonably. At last, the vortex fields filtered by prescribed thresholds are employed to reconstruct the velocity fields. It is found that the strongest vortex components occupying 5 % of the total volume can reasonably recover the main flow features including both the near-wall streaks and the large-scale motions.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent boundary layers Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 922 , 10 September 2021 , A18
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© The Author(s), 2021. Published by Cambridge University Press

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References

Adrian, R.J. 1994 Stochastic estimation of conditional structure: a review. Flow Turbul. Combust. 53 (3), 291–303.Google Scholar
Adrian, R.J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.CrossRefGoogle Scholar
Adrian, R.J., Meinhart, C.D. & Tomkins, C.D. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.CrossRefGoogle Scholar
del Álamo, J.C., Jiménez, J., Zandonade, P. & Moser, R.D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.CrossRefGoogle Scholar
Baars, W.J., Hutchins, N. & Marusic, I. 2016 Spectral stochastic estimation of high-Reynolds-number wall-bounded turbulence for a refined inner-outer interaction model. Phys. Rev. Fluids 1, 054406.CrossRefGoogle Scholar
Baars, W.J. & Marusic, I. 2020 a Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra. J. Fluid Mech. 882, A25.CrossRefGoogle Scholar
Baars, W.J. & Marusic, I. 2020 b Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 2. Integrated energy and A1. J. Fluid Mech. 882, A26.CrossRefGoogle Scholar
Bandyopadhyay, P. 1980 Large structure with a characteristic upstream interface in turbulent boundary layers. Phys. Fluids 23 (11), 2326–2327.CrossRefGoogle Scholar
Bandyopadhyay, P.R. 1989 Effect of abrupt pressure gradients on the structure of turbulent boundary layers. In Proceedings of the 10th Australasian Fluid Mechanics Conference (ed. Perry, A.E. et al. ), vol. 1, pp. 1.1–1.4. University of Melbourne.Google Scholar
Bandyopadhyay, P.R. 2020 Vortex bursting near a free surface. J. Fluid Mech. 888, A27.CrossRefGoogle Scholar
Bandyopadhyay, P.R. & Hellum, A.M. 2014 Modeling how shark and dolphin skin patterns control transitional wall-turbulence vorticity patterns using spatiotemporal phase reset mechanisms. Sci. Rep. 4 (1), 6650.CrossRefGoogle ScholarPubMed
Bandyopadhyay, P.R. & Watson, R.D. 1988 Structure of rough-wall turbulent boundary layers. Phys. Fluids 31 (7), 1877–1883.CrossRefGoogle Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.Google Scholar
Berk, T. & Ganapathisubramani, B. 2019 Effects of vortex-induced velocity on the development of a synthetic jet issuing into a turbulent boundary layer. J. Fluid Mech. 870, 651–679.CrossRefGoogle Scholar
Bernard, P.S. 2013 Vortex dynamics in transitional and turbulent boundary layers. AIAA J. 51 (8), 1828–1842.CrossRefGoogle Scholar
Bernard, P.S., Thomas, J.M. & Handler, R.A. 1993 Vortex dynamics and the production of Reynolds stress. J. Fluid Mech. 253, 385–419.CrossRefGoogle Scholar
Borrell, G., Sillero, J.A. & Jiménez, J. 2013 A code for direct numerical simulation of turbulent boundary layers at high Reynolds numbers in BG/P supercomputers. Comput. Fluids 80, 37–43.CrossRefGoogle Scholar
Bradshaw, P. 1967 The turbulence structure of equilibrium boundary layers. J. Fluid Mech. 29 (4), 625–645.CrossRefGoogle Scholar
Caprace, D.-G., Winckelmans, G. & Chatelain, P. 2020 An immersed lifting and dragging line model for the vortex particle-mesh method. Theor. Comput. Fluid Dyn. 34 (1–2), 21–48.CrossRefGoogle Scholar
Chakraborty, P., Balachandar, S. & Adrian, R.J. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535 (4), 189–214.CrossRefGoogle Scholar
Chen, X. & Sreenivasan, K.R. 2021 Reynolds number scaling of the peak turbulence intensity in wall flows. J. Fluid Mech. 908, R3.CrossRefGoogle Scholar
Chong, M.S., Perry, A.E. & Cantwell, B.J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids 2 (5), 765–777.CrossRefGoogle Scholar
Christensen, K.T. & Adrian, R.J. 2001 Statistical evidence of hairpin vortex packets in wall turbulence. J. Fluid Mech. 431, 433–443.CrossRefGoogle Scholar
Corrsin, S. 1962 Turbulent dissipation fluctuations. Phys. Fluids 5 (10), 1301.CrossRefGoogle Scholar
Das, S.K., Tanahashi, M., Shoji, K. & Miyauchi, T. 2006 Statistical properties of coherent fine eddies in wall-bounded turbulent flows by direct numerical simulation. Theor. Comput. Fluid Dyn. 20 (2), 55–71.CrossRefGoogle Scholar
Deng, S., Pan, C., Wang, J. & He, G. 2018 On the spatial organization of hairpin packets in a turbulent boundary layer at low-to-moderate Reynolds number. J. Fluid Mech. 844, 635–668.CrossRefGoogle Scholar
Einstein, H.A. & Li, H. 1958 The viscous sublayer along a smooth boundary. Trans. Am. Soc. Civil Engrs 123, 293–313.CrossRefGoogle Scholar
Fernholz, H.H. & Finleyt, P.J. 1996 The incompressible zero-pressure-gradient turbulent boundary layer: an assessment of the data. Prog. Aerosp. Sci. 32 (4), 245–311.CrossRefGoogle Scholar
Gad-el-Hak, M. & Bandyopadhyay, P.R. 1994 Reynolds number effects in wall-bounded turbulent flows. Appl. Mech. Rev. 47 (8), 307–365.CrossRefGoogle Scholar
Ganapathisubramani, B., Longmire, E.K. & Marusic, I. 2003 Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46.CrossRefGoogle Scholar
Ganapathisubramani, B., Longmire, E.K. & Marusic, I. 2006 Experimental investigation of vortex properties in a turbulent boundary layer. Phys. Fluids 18 (5), 1–464.CrossRefGoogle Scholar
Gao, Q., Ortiz-Dueñas, C. & Longmire, E.K. 2007 Circulation signature of vortical structures in turbulent boundary layers. In 16th Australasian Fluid Mechanics Conference Crown Plaza, Gold Coast, Australia.Google Scholar
Gao, Q., Ortiz-Dueñas, C. & Longmire, E.K. 2011 Analysis of vortex populations in turbulent wall-bounded flows. J. Fluid Mech. 678 (8), 87–123.CrossRefGoogle Scholar
Gao, Y. & Liu, C. 2018 Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30, 085107.CrossRefGoogle Scholar
Gentle, J.E. 1998 Numerical Linear Algebra for Applications in Statistics. Springer.CrossRefGoogle Scholar
Hama, F. 1954 Boundary layer characteristics for smooth and rough surfaces. In Trans. 1954 Annual Meeting of the Society of Naval Architects and Marine Engineers. SNAME.Google Scholar
Hambleton, W.T., Hutchins, N. & Marusic, I. 2006 Simultaneous orthogonal-plane particle image velocimetry measurements in a turbulent boundary layer. J. Fluid Mech. 560 (560), 53–64.CrossRefGoogle Scholar
Head, M.R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107 (107), 297–338.CrossRefGoogle Scholar
Herpin, S., Stanislas, M., Jean, M.F. & Coudert, S. 2013 Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 716 (2), 5–50.CrossRefGoogle Scholar
Hu, R., Yang, X.I.A. & Zheng, X. 2020 Wall-attached and wall-detached eddies in wall-bounded turbulent flows. J. Fluid Mech. 885, A30.CrossRefGoogle Scholar
Hunt, J.C.R., Wray, A.A. & Moin, P. 1988 Eddies, streams, and convergence zones in turbulent flows. Report No. CTR-S88, pp. 193–208. Center for Turbulence Research.Google Scholar
Hutchins, N., Hambleton, W.T. & Marusic, I. 2005 Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365 (1852), 647–664.Google ScholarPubMed
Jeong, J.J.J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 332 (1), 339–363.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389 (389), 335–359.CrossRefGoogle 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.CrossRefGoogle Scholar
Jodai, Y. & Elsinga, G.E. 2016 Experimental observation of hairpin auto-generation events in a turbulent boundary layer. J. Fluid Mech. 795, 611–633.CrossRefGoogle Scholar
Kang, S.J., Tanahashi, M. & Miyauchi, T. 2007 Dynamics of fine scale eddy clusters in turbulent channel flows. J. Turbul. 8, N52.CrossRefGoogle Scholar
Kline, S.J., Reynolds, W.C., Schraub, F.A. & Runstadler, P.W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30 (4), 741–773.CrossRefGoogle Scholar
Koumoutsakos, P. & Leonard, A. 1995 High-resolution simulations of the flow around an impulsively started cylinder using vortex methods. J. Fluid Mech. 296 (1), 1–38.CrossRefGoogle Scholar
Kress, R. 2014 Tikhonov Regularization, pp. 323–349. Springer.Google Scholar
Kuo, A.Y.-S. & Corrsin, S. 1972 Experiment on the geometry of the fine-structure regions in fully turbulent fluid. J. Fluid Mech. 56 (3), 447–479.CrossRefGoogle Scholar
Lee, J., Lee, J., Choi, J. & Sung, H. 2014 Spatial organization of large- and very-large-scale motions in a turbulent channel flow. J. Fluid Mech. 749, 818–840.CrossRefGoogle Scholar
Lee, J.-H. & Sung, H.J. 2009 Structures in turbulent boundary layers subjected to adverse pressure gradients. J. Fluid Mech. 639, 101–131.CrossRefGoogle Scholar
Leonard, A. 1985 Computing three-dimensional incompressible flows with vortex elements. Annu. Rev. Fluid Mech. 17 (1), 523–559.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432–471.CrossRefGoogle Scholar
Lund, T.S., Wu, X. & Squires, K.D. 1998 Generation of turbulent inflow data for spatially-developing boundary layer simulations. J. Comput. Phys. 140 (2), 233–258.CrossRefGoogle Scholar
Lundgren, T.S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25 (12), 2193–2203.CrossRefGoogle Scholar
Marusic, I. 2001 On the role of large-scale structures in wall turbulence. Phys. Fluids 13 (13), 735–743.CrossRefGoogle Scholar
Marusic, I. & Adrian, R.J. 2012 The eddies and scales of wall turbulence. In Turbulence (ed. K.R. Sreenivasan, P.A. Davidson & Y. Kaneda), pp. 176–220. Cambridge University Press.CrossRefGoogle Scholar
Marusic, I., Baars, W.J. & Hutchins, N. 2017 Scaling of the streamwise turbulence intensity in the context of inner-outer interactions in wall turbulence. Phys. Rev. Fluids 2, 100502.CrossRefGoogle Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 Predictive model for wall-bounded turbulent flow. Science 329 (5988), 193–196.CrossRefGoogle ScholarPubMed
Marusic, I. & Monty, J.P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 49–74.CrossRefGoogle Scholar
Marusic, I. & Perry, A.E. 1995 A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support. J. Fluid Mech. 298 (298), 389–407.CrossRefGoogle Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2011 A predictive inner–outer model for streamwise turbulence statistics in wall-bounded flows. J. Fluid Mech. 681, 537–566.CrossRefGoogle Scholar
Monkewitz, P.A., Chauhan, K.A. & Nagib, H.M. 2007 Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19 (11), 319–316.CrossRefGoogle Scholar
Natrajan, V.K., Wu, Y. & Christensen, K.T. 2007 Spatial signatures of retrograde spanwise vortices in wall turbulence. J. Fluid Mech. 574, 155–167.CrossRefGoogle Scholar
Ong, L. & Wallace, J.M. 1998 Joint probability density analysis of the structure and dynamics of the vorticity field of a turbulent boundary layer. J. Fluid Mech. 367 (367), 291–328.CrossRefGoogle Scholar
Panton, R.L. 2001 Overview of the self-sustaining mechanisms of wall turbulence. Prog. Aerosp. Sci. 37 (4), 341–383.CrossRefGoogle Scholar
Perry, A.E. & Chong, M.S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173–217.CrossRefGoogle Scholar
Perry, A.E. & Chong, M.S. 1987 A description of eddying motions and flow patterns using critical-point concepts. Annu. Rev. Fluid Mech. 19 (1), 125–155.CrossRefGoogle Scholar
Perry, A.E., Henbest, S. & Chong, M.S. 1986 A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165(165), 163–199.CrossRefGoogle Scholar
Perry, A.E. & Marusic, I. 1995 A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis. J. Fluid Mech. 298 (298), 361–388.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2010 On the dynamical relevance of coherent vortical structures in turbulent boundary layers. J. Fluid Mech. 648, 325–349.CrossRefGoogle Scholar
Pullin, D.I. & Saffman, P.G. 1993 On the Lundgren–Townsend model of turbulent fine scales. Phys. Fluids 5 (1), 126–145.CrossRefGoogle Scholar
Ramirez, D., Via, J. & Santamaria, I. 2008 A generalization of the magnitude squared coherence spectrum for more than two signals: definition, properties and estimation. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Las Vegas, Nevada, USA.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Sillero, J.A., Jiménez, J. & Moser, R.D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to δ+≈2000. Phys. Fluids 25 (10), 105102.CrossRefGoogle Scholar
Sillero, J.A., Jiménez, J. & Moser, R.D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to δ+≈2000. Phys. Fluids 26 (10), 105109.CrossRefGoogle Scholar
de Silva, C.M., Hutchins, N. & Marusic, I. 2016 a Uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 786, 309–331.CrossRefGoogle Scholar
de Silva, C.M., Kevin, A.K., Baidya, R., Hutchins, N. & Marusic, I. 2018 Large coherence of spanwise velocity in turbulent boundary layers. J. Fluid Mech. 847, 161–185.CrossRefGoogle Scholar
de Silva, C.M., Woodcock, J.D., Hutchins, N. & Marusic, I. 2016 b Influence of spatial exclusion on the statistical behavior of attached eddies. Phys. Rev. Fluids 1, 022401.CrossRefGoogle Scholar
Simens, M.P., Jiménez, J., Hoyas, S. & Mizuno, Y. 2009 A high-resolution code for turbulent boundary layers. J. Comput. Phys. 228 (11), 4218–4231.CrossRefGoogle Scholar
Stanislas, M., Perret, L. & Foucaut, J.M. 2008 Vortical structures in the turbulent boundary layer: a possible route to a universal representation. J. Fluid Mech. 602, 327–382.CrossRefGoogle Scholar
Stoica, P. & Moses, R. 2005 Spectral Analysis of Signals. Prentice Hall.Google Scholar
Synge, J.L. & Lin, C.C. 1943 On a statistical model of isotropic turbulence. Trans. R. Soc. Can. 37, 45.Google Scholar
Tanahashi, M., Kang, S.J., Miyamoto, T., Shiokawa, S. & Miyauchi, T. 2004 Scaling law of fine scale eddies in turbulent channel flows up to Re τ = 800. Intl J. Heat Fluid Flow 25 (3), 331–340.CrossRefGoogle Scholar
Tennekes, H. 1968 Simple model for the small-scale structure of turbulence. Phys. Fluids 11 (3), 669–671.CrossRefGoogle Scholar
Theodorsen, T. 1952 Mechanism of turbulence. In Proceedings of the Midwestern Conference on Fluid Mechanics, Ohio State University.Google Scholar
Tian, S., Gao, Y., Dong, X. & Liu, C. 2018 Definitions of vortex vector and vortex. J. Fluid Mech. 849, 312–339.CrossRefGoogle Scholar
Tinney, C.E., Coiffet, F., Delville, J., Hall, A.M., Jordan, P. & Glauser, M.N. 2006 On spectral linear stochastic estimation. Exp. Fluids 41 (5), 763–775.CrossRefGoogle Scholar
Townsend, A.A. 1951 On the fine-scale structure of turbulence. Proc. R. Soc. Lond. 208 (1095), 534–542.Google Scholar
Townsend, A.A. 1956 The structure of turbulent shear flow. Q. J. R. Meteorol. Soc. 83 (357), 411–412.Google Scholar
Wang, C., Gao, Q., Wang, J., Wang, B. & Pan, C. 2019 Experimental study on dominant vortex structures in near-wall region of turbulent boundary layer based on tomographic particle image velocimetry. J. Fluid Mech. 874, 426–454.CrossRefGoogle Scholar
Wang, W., Pan, C., Gao, Q. & Wang, J. 2018 Wall-normal variation of spanwise streak spacing in turbulent boundary layer with low-to-moderate Reynolds number. Entropy 21 (1), 24.CrossRefGoogle ScholarPubMed
Wang, W., Pan, C. & Wang, J. 2021 Energy transfer structures associated with large-scale motions in a turbulent boundary layer. J. Fluid Mech. 906, A14.CrossRefGoogle Scholar
Wang, H.P., Wang, S.Z. & He, G.W. 2017 The spanwise spectra in wall-bounded turbulence. Acta Mech. Sinica 34 (3), 452–461.CrossRefGoogle Scholar
Woodcock, J.D. & Marusic, I. 2015 The statistical behaviour of attached eddies. Phys. Fluids 27 (1), 97–120.CrossRefGoogle Scholar
Wu, Y. & Christensen, K.T. 2006 Population trends of spanwise vortices in wall turbulence. J. Fluid Mech. 568, 55–76.CrossRefGoogle Scholar
Xiong, S. & Yang, Y. 2019 Identifying the tangle of vortex tubes in homogeneous isotropic turbulence. J. Fluid Mech. 874, 952–978.CrossRefGoogle Scholar
Yang, X.I.A., Marusic, I. & Meneveau, C. 2016 Moment generating functions and scaling laws in the inertial layer of turbulent wall-bounded flows. J. Fluid Mech. 791, R2.CrossRefGoogle Scholar
Yoon, M., Hwang, J., Yang, J. & Sung, H.J. 2020 Wall-attached structures of streamwise velocity fluctuations in an adverse-pressure-gradient turbulent boundary layer. J. Fluid Mech. 885, A12.CrossRefGoogle Scholar
Zhou, J. 1997 Self-sustaining formation of packets of hairpin vortices in a turbulent wall layer. PhD thesis, University of Illinois.Google Scholar
Zhou, J., Adrian, R.J., Balachandar, S. & Kendall, T.M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.CrossRefGoogle Scholar
Zhu, L. & Xi, L. 2019 Vortex axis tracking by iterative propagation (VATIP): a method for analysing three-dimensional turbulent structures. J. Fluid Mech. 866, 169–215.CrossRefGoogle Scholar