Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-28T12:29:11.880Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex dynamics and boundary layer transition in flow around a rectangular cylinder with different aspect ratios at medium Reynolds number

Published online by Cambridge University Press:  29 February 2024

Jiang-Hua Li ,
Bo-Fu Wang Open the ORCID record for Bo-Fu Wang [Opens in a new window] ,
Xiang Qiu ,
Quan Zhou Open the ORCID record for Quan Zhou [Opens in a new window] ,
Shi-Xiao Fu  and
Yu-Lu Liu
Jiang-Hua Li
Affiliation:
Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, PR China School of Science, Shanghai Institute of Technology, Shanghai 201418, PR China
Bo-Fu Wang*
Affiliation:
Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, PR China
Xiang Qiu
Affiliation:
Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, PR China School of Science, Shanghai Institute of Technology, Shanghai 201418, PR China
Quan Zhou
Affiliation:
Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, PR China
Shi-Xiao Fu
Affiliation:
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 201602, PR China
Yu-Lu Liu
Affiliation:
Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, PR China School of Science, Shanghai Institute of Technology, Shanghai 201418, PR China
*
†Email address for correspondence: bofuwang@shu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The numerical investigation focuses on the flow patterns around a rectangular cylinder with three aspect ratios ($L/D=5$, $10$, $15$) at a Reynolds number of $1000$. The study delves into the dynamics of vortices, their associated frequencies, the evolution of the boundary layer and the decay of the wake. Kelvin–Helmholtz (KH) vortices originate from the leading edge (LE) shear layer and transform into hairpin vortices. Specifically, at $L/D=5$, three KH vortices merge into a single LE vortex. However, at $L/D=10$ and $15$, two KH vortices combine to form a LE vortex, with the rapid formation of hairpin vortex packets. A fractional harmonic arises due to feedback from the split LE shear layer moving upstream, triggering interaction with the reverse flow. Trailing edge (TE) vortices shed, creating a Kármán-like street in the wake. The intensity of wake oscillation at $L/D=5$ surpasses that in the other two cases. Boundary layer transition occurs after the saturation of disturbance energy for $L/D=10$ and $15$, but not for $L/D=5$. The low-frequency disturbances are selected to generate streaks inside the boundary layer. The TE vortex shedding induces the formation of a favourable pressure gradient, accelerating the flow and fostering boundary layer relaminarization. The self-similarity of the velocity defect is observed in all three wakes, accompanied by the decay of disturbance energy. Importantly, the decrease in the shedding frequency of LE (TE) vortices significantly contributes to the overall decay of disturbance energy. This comprehensive exploration provides insights into complex flow phenomena and their underlying dynamics.

JFM classification

Vortex Flows: Vortex dynamics Turbulent Flows: Turbulent transition Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 982 , 10 March 2024 , A5
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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
Balin, R. & Jansen, K.E. 2021 Direct numerical simulation of a turbulent boundary layer over a bump with strong pressure gradients. J. Fluid Mech. 918, A14.CrossRefGoogle Scholar
Basley, J., Pastur, L., Lusseyran, F., Faure, T.M. & Delprat, N. 2011 Experimental investigation of global structures in an incompressible cavity flow using time-resolved PIV. Exp. Fluids 50, 905–918.CrossRefGoogle Scholar
Blackburn, H. & Lopez, J.M. 2003 On three-dimensional quasiperiodic floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15 (8), L57–L60.CrossRefGoogle Scholar
Blackburn, H.M. & Sheard, G.J. 2010 On quasiperiodic and subharmonic floquet wake instabilities. Phys. Fluids 22 (3), 031701.CrossRefGoogle Scholar
Bolis, A. 2013 Fourier spectral/HP element method: investigation of time-stepping and parallelisation strategies. PhD thesis, Citeseer.Google Scholar
Bruno, L., Salvetti, M.V. & Ricciardelli, F. 2014 Benchmark on the aerodynamics of a rectangular 5: 1 cylinder: an overview after the first four years of activity. J. Wind Engng Ind. Aerodyn. 126, 87–106.CrossRefGoogle Scholar
Cantwell, C.D., et al. 2015 Nektar++: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219.CrossRefGoogle Scholar
Cavar, D. & Meyer, K.E. 2011 Investigation of turbulent boundary layer flow over 2D bump using highly resolved large eddy simulation. J. Fluids Engng 133, 111204.CrossRefGoogle Scholar
Cheng, H., Jiang, H., Chong, K.L., Zhou, Q., Liu, Y. & Lu, Z. 2022 The effect of surface roughness on the Lagrangian coherent structures in turbulent rayleigh–bénard convection. Phys. Fluids 34 (11), 115134.CrossRefGoogle Scholar
Chiarini, A., Gatti, D., Cimarelli, A. & Quadrio, M. 2022 a Structure of turbulence in the flow around a rectangular cylinder. J. Fluid Mech. 946, A35.CrossRefGoogle Scholar
Chiarini, A. & Quadrio, M. 2021 The turbulent flow over the barc rectangular cylinder: a DNS study. Flow Turbul. Combust. 107, 875–899.CrossRefGoogle Scholar
Chiarini, A., Quadrio, M. & Auteri, F. 2021 Linear stability of the steady flow past rectangular cylinders. J. Fluid Mech. 929, A36.CrossRefGoogle Scholar
Chiarini, A., Quadrio, M. & Auteri, F. 2022 b An almost subharmonic instability in the flow past rectangular cylinders. J. Fluid Mech. 950, A20.CrossRefGoogle Scholar
Chiarini, A., Quadrio, M. & Auteri, F. 2022 c On the frequency selection mechanism of the low-$Re$ flow around rectangular cylinders. J. Fluid Mech. 933, A44.CrossRefGoogle Scholar
Cimarelli, A., Leonforte, A. & Angeli, D. 2018 a Direct numerical simulation of the flow around a rectangular cylinder at a moderately high Reynolds number. J. Wind Engng Ind. Aerodyn. 174, 39–49.CrossRefGoogle Scholar
Cimarelli, A., Leonforte, A. & Angeli, D. 2018 b On the structure of the self-sustaining cycle in separating and reattaching flows. J. Fluid Mech. 857, 907–936.CrossRefGoogle Scholar
Cimarelli, A., Leonforte, A., de Angelis, E., Crivellini, A. & Angeli, D. 2019 a On negative turbulence production phenomena in the shear layer of separating and reattaching flows. Phys. Lett. A 383 (10), 1019–1026.CrossRefGoogle Scholar
Cimarelli, A., Leonforte, A., de Angelis, E., Crivellini, A. & Angeli, D. 2019 b Resolved dynamics and subgrid stresses in separating and reattaching flows. Phys. Fluids 31 (9), 095101.CrossRefGoogle Scholar
Coleman, G.N., Rumsey, C.L. & Spalart, P.R. 2018 Numerical study of turbulent separation bubbles with varying pressure gradient and Reynolds number. J. Fluid Mech. 847, 28–70.CrossRefGoogle ScholarPubMed
Corsini, R., Angeli, D., Stalio, E., Chibbaro, S. & Cimarelli, A. 2022 Flow solutions around rectangular cylinders: the question of spatial discretization. Wind Struct. 34 (1), 151–159.Google Scholar
Dennis, D.J. & Nickels, T.B. 2011 a Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 1. Vortex packets. J. Fluid Mech. 673, 180–217.CrossRefGoogle Scholar
Dennis, D.J. & Nickels, T.B. 2011 b Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 2. Long structures. J. Fluid Mech. 673, 218–244.CrossRefGoogle Scholar
Dixit, S.A. & Ramesh, O.N. 2008 Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers. J. Fluid Mech. 615, 445–475.CrossRefGoogle Scholar
Dong, S.C., Karniadakis, G.E. & Chryssostomidis, C. 2014 A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains. J. Comput. Phys. 261, 83–105.CrossRefGoogle Scholar
Fransson, J.H.M., Matsubara, M. & Alfredsson, P.H. 2005 Transition induced by free-stream turbulence. J. Fluid Mech. 527, 1–25.CrossRefGoogle Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Green, M.A., Rowley, C.W. & Haller, G. 2007 Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech. 572, 111–120.CrossRefGoogle Scholar
Haller, G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (4), 248–277.CrossRefGoogle Scholar
Haller, G. & Yuan, G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147 (3–4), 352–370.CrossRefGoogle Scholar
He, G.-S., Pan, C., Feng, L.-H., Gao, Q. & Wang, J.-J. 2016 Evolution of Lagrangian coherent structures in a cylinder-wake disturbed flat plate boundary layer. J. Fluid Mech. 792, 274–306.CrossRefGoogle Scholar
He, G.S., Wang, J.J. & Pan, C. 2013 Initial growth of a disturbance in a boundary layer influenced by a circular cylinder wake. J. Fluid Mech. 718, 116–130.CrossRefGoogle Scholar
Hourigan, K., Mills, R., Thompson, M.C., Sheridan, J., Dilin, P. & Welsh, M.C. 1993 Base pressure coefficients for flows around rectangular plates. J. Wind Engng Ind. Aerodyn. 49 (1–3), 311–318.CrossRefGoogle Scholar
Hourigan, K., Thompson, M.C. & Tan, B.T. 2001 Self-sustained oscillations in flows around long blunt plates. J. Fluids Struct. 15 (3–4), 387–398.CrossRefGoogle Scholar
Hunt, J.C.R. & Durbin, P.A. 1999 Perturbed vortical layers and shear sheltering. Fluid Dyn. Res. 24 (6), 375.CrossRefGoogle Scholar
Jackson, C.P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.CrossRefGoogle Scholar
Jacobs, R.G. & Durbin, P.A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.CrossRefGoogle Scholar
Jiang, H. & Cheng, L. 2018 Hydrodynamic characteristics of flow past a square cylinder at moderate Reynolds numbers. Phys. Fluids 30 (10), 104107.CrossRefGoogle Scholar
Jiang, H., Cheng, L. & An, H. 2018 Three-dimensional wake transition of a square cylinder. J. Fluid Mech. 842, 102–127.CrossRefGoogle Scholar
Kendall, J. 1998 Experiments on boundary-layer receptivity to freestream turbulence. In 36th AIAA Aerospace Sciences Meeting and Exhibit, p. 530.Google Scholar
Li, J., Wang, B., Qiu, X., Wu, J., Zhou, Q., Fu, S. & Liu, Y. 2022 a Three-dimensional vortex dynamics and transitional flow induced by a circular cylinder placed near a plane wall with small gap ratios. J. Fluid Mech. 953, A2.CrossRefGoogle Scholar
Li, J.-H., Wang, B.-F., Qiu, X., Wu, J.-Z., Zhou, Q., Fu, S.-X. & Liu, Y.-L. 2022 b The dynamics of cylinder-wake/boundary-layer interaction revealed by turbulent transports. Phys. Fluids 34 (11), 115136.CrossRefGoogle Scholar
Liu, Y. & Zhang, Q. 2015 Dynamic mode decomposition of separated flow over a finite blunt plate: time-resolved particle image velocimetry measurements. Exp. Fluids 56, 1–17.CrossRefGoogle Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
Ma, L.-Q., Feng, L.-H., Pan, C., Gao, Q. & Wang, J.-J. 2015 Fourier mode decomposition of PIV data. China Technol. Sci. 58, 1935–1948.CrossRefGoogle Scholar
Mandal, A.C. & Dey, J. 2011 An experimental study of boundary layer transition induced by a cylinder wake. J. Fluid Mech. 684 (10), 60–84.CrossRefGoogle Scholar
Manhart, M. & Friedrich, R. 2002 DNS of a turbulent boundary layer with separation. Intl J. Heat Fluid Flow 23 (5), 572–581.CrossRefGoogle Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.CrossRefGoogle Scholar
Matai, R. & Durbin, P. 2019 Large-eddy simulation of turbulent flow over a parametric set of bumps. J. Fluid Mech. 866, 503–525.CrossRefGoogle Scholar
Matsubara, M. & Alfredsson, P.H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
Mills, R., Sheridan, J. & Hourigan, K. 2002 Response of base suction and vortex shedding from rectangular prisms to transverse forcing. J. Fluid Mech. 461, 25–49.CrossRefGoogle Scholar
Mills, R., Sheridan, J., Hourigan, K. & Welsh, M.C. 1995 The mechanism controlling vortex shedding from rectangular bluff bodies. In Proceedings of the 20th Australasian Fluid Mechanics Conference, pp. 227–230.Google Scholar
Monkewitz, P.A., Huerre, P. & Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.CrossRefGoogle Scholar
Monokrousos, A., Åkervik, E., Brandt, L. & Henningson, D.S. 2010 Global three-dimensional optimal disturbances in the blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214.CrossRefGoogle Scholar
Moore, D.M., Letchford, C.W. & Amitay, M. 2019 Energetic scales in a bluff body shear layer. J. Fluid Mech. 875, 543–575.CrossRefGoogle Scholar
Morkovin, M.V. 1969 On the many faces of transition. In Viscous Drag Reduction: Proceedings of the Symposium on Viscous Drag Reduction held at the LTV Research Center, Dallas, Texas, September 24 and 25, 1968, pp. 1–31. Springer.CrossRefGoogle Scholar
Moxey, D., et al. 2020 Nektar++: enhancing the capability and application of high-fidelity spectral/hp element methods. Comput. Phys. Commun. 249, 107110.CrossRefGoogle Scholar
Na, Y. & Moin, P. 1998 Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379–405.CrossRefGoogle Scholar
Nakamura, Y. & Nakashima, M. 1986 Vortex excitation of prisms with elongated rectangular, H and [vdash] cross-sections. J. Fluid Mech. 163, 149–169.CrossRefGoogle Scholar
Nakamura, Y., Ohya, Y. & Tsuruta, H. 1991 Experiments on vortex shedding from flat plates with square leading and trailing edges. J. Fluid Mech. 222, 437–447.CrossRefGoogle Scholar
Narasimha, R. & Sreenivasan, K.R. 1973 Relaminarization in highly accelerated turbulent boundary layers. J. Fluid Mech. 61 (3), 417–447.CrossRefGoogle Scholar
Narayanan, M.A.B. & Ramjee, V. 1969 On the criteria for reverse transition in a two-dimensional boundary layer flow. J. Fluid Mech. 35 (2), 225–241.CrossRefGoogle Scholar
Naudascher, E. & Rockwell, D. 1994 Flow-induced Vibrations: An Engineering Guide. Hydraulic Structures Design Manual 7. Routledge.Google Scholar
Noack, B.R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Okajima, A. 1982 Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398.CrossRefGoogle Scholar
Ovchinnikov, V., Piomelli, U. & Choudhari, M.M. 2006 Numerical simulations of boundary-layer transition induced by a cylinder wake. J. Fluid Mech. 547, 413–441.CrossRefGoogle Scholar
Ozono, S., Ohya, Y., Nakamura, Y. & Nakayama, R. 1992 Stepwise increase in the Strouhal number for flows around flat plates. Intl J. Numer. Meth. Fluids 15 (9), 1025–1036.CrossRefGoogle Scholar
Park, D. & Yang, K.-S. 2016 Flow instabilities in the wake of a rounded square cylinder. J. Fluid Mech. 793, 915–932.CrossRefGoogle Scholar
Patel, V.C. 1965 Calibration of the Preston tube and limitations on its use in pressure gradients. J. Fluid Mech. 23 (1), 185–208.CrossRefGoogle Scholar
Patel, V.C. & Head, M.R. 1968 Reversion of turbulent to laminar flow. J. Fluid Mech. 34 (2), 371–392.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Prasanth, T.K. & Mittal, S. 2008 Vortex-induced vibrations of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 594, 463–491.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard-von kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Robichaux, J., Balachandar, S. & Vanka, S.P. 1999 Three-dimensional floquet instability of the wake of square cylinder. Phys. Fluids 11 (3), 560–578.CrossRefGoogle Scholar
Saha, A.K., Muralidhar, K. & Biswas, G. 2000 Transition and chaos in two-dimensional flow past a square cylinder. J. Engng Mech. ASCE 126 (5), 523–532.CrossRefGoogle Scholar
Sasaki, K. & Kiya, M. 1991 Three-dimensional vortex structure in a leading-edge separation bubble at moderate Reynolds numbers. Trans. ASME J. Fluids Engng 113 (3), 405–410.CrossRefGoogle Scholar
Schmid, P.J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Shadden, S.C., Astorino, M. & Gerbeau, J.-F. 2010 Computational analysis of an aortic valve jet with Lagrangian coherent structures. Chaos 20 (1), 017512.CrossRefGoogle ScholarPubMed
Shadden, S.C., Dabiri, J.O. & Marsden, J.E. 2006 Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys. Fluids 18 (4), 047105.CrossRefGoogle Scholar
Sheard, G.J., Fitzgerald, M.J. & Ryan, K. 2009 Cylinders with square cross-section: wake instabilities with incidence angle variation. J. Fluid Mech. 630, 43–69.CrossRefGoogle Scholar
So, R.M.C. & Mellor, G.L. 1975 Experiment on turbulent boundary layers on a concave wall. Aeronaut. Q. 26 (1), 25–40.Google Scholar
Sohankar, A., Norberg, C. & Davidson, L. 1999 Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers. Phys. Fluids 11 (2), 288–306.CrossRefGoogle Scholar
Spalart, P.R. & Watmuff, J.H. 1993 Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249, 337–371.CrossRefGoogle Scholar
Tamura, T., Miyagi, T. & Kitagishi, T. 1998 Numerical prediction of unsteady pressures on a square cylinder with various corner shapes. J. Wind Engng Ind. Aerodyn. 74, 531–542.CrossRefGoogle Scholar
Tan, B., Thompson, M. & Hourigan, K. 2004 Flow past rectangular cylinders: receptivity to transverse forcing. J. Fluid Mech. 515, 33–62.CrossRefGoogle Scholar
Tan, B.T., Thompson, M. & Hourigan, K. 1998 Simulated flow around long rectangular plates under cross flow perturbations. Intl J. Fluid Dyn. 2, 1.Google Scholar
Uzun, A. & Malik, M.R. 2021 Simulation of a turbulent flow subjected to favorable and adverse pressure gradients. Theor. Comput. Fluid Dyn. 35, 293–329.CrossRefGoogle Scholar
Wang, B.F., Mao, X.R. & Zaki, T.A. 2019 Low-frequency selectivity in flat-plate boundary layer with elliptic leading edge. J. Fluid Mech. 866, 239–262.CrossRefGoogle Scholar
Wang, J.-S., Feng, L.-H., Wang, J.-j. & Li, T. 2018 Görtler vortices in low-Reynolds-number flow over multi-element airfoil. J. Fluid Mech. 835, 898–935.CrossRefGoogle Scholar
Wang, J.-S. & Wang, J.-J. 2021 a Vortex dynamics for flow around the slat cove at low Reynolds numbers. J. Fluid Mech. 919, A27.CrossRefGoogle Scholar
Wang, J.-S. & Wang, J.-J. 2021 b Wake-induced transition in the low-Reynolds-number flow over a multi-element airfoil. J. Fluid Mech. 915, A28.CrossRefGoogle Scholar
Webster, D.R., DeGraaff, D.B. & Eaton, J.K. 1996 Turbulence characteristics of a boundary layer over a two-dimensional bump. J. Fluid Mech. 320, 53–69.CrossRefGoogle Scholar
Westin, K.J.A., Boiko, A.V., Klingmann, B.G.B., Kozlov, V.V. & Alfredsson, P.H. 1994 Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.CrossRefGoogle Scholar
Westin, K.J.A. & Henkes, R.A.W.M. 1997 Application of turbulence models to by-pass transition. J. Fluids Engng 119, 859–866.CrossRefGoogle Scholar
Williamson, C.H.K. 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31 (11), 3165–3168.CrossRefGoogle Scholar
Wu, X. & Squires, K.D. 1998 Numerical investigation of the turbulent boundary layer over a bump. J. Fluid Mech. 362, 229–271.CrossRefGoogle Scholar
Zaki, T.A. & Saha, S. 2009 On shear sheltering and the structure of vortical modes in single-and two-fluid boundary layers. J. Fluid Mech. 626, 111–147.CrossRefGoogle Scholar
Zhang, Q. & Liu, Y. 2015 Influence of incident vortex street on separated flow around a finite blunt plate: PIV measurement and pod analysis. J. Fluid Struct. 55, 463–483.CrossRefGoogle Scholar
Zhang, Y. & Zhou, Q. 2024 Low-Prandtl-number effects on global and local statistics in two-dimensional Rayleigh–Bénard convection. Phys. Fluids 36 (1), 015107.CrossRefGoogle Scholar
Zhang, Z., Kareem, A., Xu, F. & Jiang, H. 2023 Global instability and mode selection in flow fields around rectangular prisms. J. Fluid Mech. 955, A19.CrossRefGoogle Scholar