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Flow-induced vibrations of a D-section prism at a low Reynolds number

Published online by Cambridge University Press:  06 May 2022

Weilin Chen
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300350, PR China
Chunning Ji*
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300350, PR China Key Laboratory of Earthquake Engineering Simulation and Seismic Resilience of China Earthquake Administration, Tianjin University, Tianjin 300350, PR China
Md. Mahbub Alam
Center for Turbulence Control, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, PR China
Dong Xu
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300350, PR China
Hongwei An
School of Engineering, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia
Feifei Tong
School of Engineering, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia
Yawei Zhao
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300350, PR China
Email address for correspondence:


This paper presents the response and the wake modes of a freely vibrating D-section prism with varying angles of attack ($\alpha = 0^\circ \text {--}180^\circ$) and reduced velocity ($U^* = 2\text {--}20$) by a numerical investigation. The Reynolds number, based on the effective diameter, is fixed at 100. The vibration of the prism is allowed only in the transverse direction. We found six types of response with increasing angle of attack: typical vortex-induced vibration (VIV) at $\alpha = 0^\circ \text {--}35^\circ$; extended VIV at $\alpha = 40^\circ \text {--}65^\circ$; combined VIV and galloping at $\alpha = 70^\circ \text {--}80^\circ$; narrowed VIV at $\alpha = 85^\circ \text {--}150^\circ$; transition response, from narrowed VIV to pure galloping, at $\alpha = 155^\circ \text {--}160^\circ$; and pure galloping at $\alpha = 165^\circ \text {--}180^\circ$. The typical and narrowed VIVs are characterized by linearly increasing normalized vibration frequency with increasing $U^*$, which is attributed to the stationary separation points of the boundary layer. On the other hand, in the extended VIV, the vortex shedding frequency matches the natural frequency in a large $U^*$ range with increasing $\alpha$ generally. The galloping is characterized by monotonically increasing amplitude with enlarging $U^*$, with the largest amplitude being $A^* = 3.2$. For the combined VIV and galloping, the vibration amplitude is marginal in the VIV branch while it significantly increases with $U^*$ in the galloping branch. In the transition from narrowed VIV to pure galloping, the vibration frequency shows a galloping-like feature, but the amplitude does not monotonically increase with increasing $U^*$. Moreover, a partition of the wake modes in the $U^*$$\alpha$ parametric plane is presented, and the flow physics is elucidated through time variations of the displacement, drag and lift coefficients and vortex dynamics. The angle-of-attack range of galloping is largely predicted by performing a quasi-steady analysis of the galloping instability. Finally, the effects of $m^*$ and ${\textit {Re}}$, the roles of afterbody and the roles of separation point in determining vibration responses and vortex shedding frequency are further discussed.

JFM Papers
© The Author(s), 2022. Published by Cambridge University Press

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Alam, M.M. 2022 A note on flow-induced force measurement of oscillating cylinder by loadcell. Ocean Engng 245, 110538.CrossRefGoogle Scholar
Alam, M.M. & Meyer, J.P. 2013 Global aerodynamic instability of twin cylinders in cross flow. J. Fluids Struct. 41 (8), 135145.CrossRefGoogle Scholar
Bearman, P.W. 1984 Vortex shedding from oscillating bluff bodies. Annu. Rev. Fluid Mech. 16, 195222.CrossRefGoogle Scholar
Bearman, P.W. 2011 Circular cylinder wakes and vortex-induced vibrations. J. Fluids Struct. 27 (5–6), 648658.CrossRefGoogle Scholar
Bearman, P.W. & Davies, M.E. 1977 The flow about oscillating bluff structures. In Proceedings of the International Conference on Wind Effects on Buildings and Structures (ed. K.J. Eaton), pp. 285–295. Cambridge University Press.Google Scholar
Bearman, P.W., Gartshore, I.S., Maull, D.J. & Parkinson, G.V. 1987 Experiments on flow-induced vibration of a square-section cylinder. J. Fluids Struct. 1 (1), 1934.CrossRefGoogle Scholar
Bhatt, R. & Alam, M.M. 2018 Vibrations of a square cylinder submerged in a wake. J. Fluid Mech. 853, 301332.CrossRefGoogle Scholar
Bhinder, A.P.S., Sarkar, S. & Dalal, A. 2012 Flow over and forced convection heat transfer around a semi-circular cylinder at inci ce. Intl J. Heat Mass Transfer 55 (19–20), 51715184.CrossRefGoogle Scholar
Bishop, R.E.D. & Hassan, A.Y. 1964 The lift and drag forces on a circular cylinder oscillating in a flowing fluid. Proc. R. Soc. 277 (1368), 5175.Google Scholar
Blevins, R.D. 1990 Flow-Induced Vibration. Von Nostrand Reinhold.Google Scholar
Bourguet, R. 2020 Two-degree-of-freedom flow-induced vibrations of a rotating cylinder. J. Fluid Mech. 897, A31.CrossRefGoogle Scholar
Bourguet, R. & Lo Jacono, D. 2014 Flow-induced vibrations of a rotating cylinder. J. Fluid Mech. 740, 342380.CrossRefGoogle Scholar
Brooks, P.N.H. 1960 Experimental investigation of the aeroelastic instability of bluff two-dimensional cylinders. PhD thesis, University of British Columbia.Google Scholar
Cagney, N. & Balabani, S. 2019 The role of the separation point in streamwise vortex-induced vibrations. J. Fluids Struct. 86, 316328.CrossRefGoogle Scholar
Chen, W., Ji, C., Alam, M.M., Williams, J. & Xu, D. 2020 a Numerical simulations of flow past three circular cylinders in equilateral-triangular arrangements. J. Fluid Mech. 891, A14.CrossRefGoogle Scholar
Chen, W., Ji, C., Alam, M.M., Xu, D. & ZHANG, Z. 2022 Three-dimensional flow past a circular cylinder in proximity to a stationary wall. Ocean Engng 247, 110783.CrossRefGoogle Scholar
Chen, W., Ji, C., Williams, J., Xu, D., Yang, L. & Cui, Y. 2018 Vortex-induced vibrations of three tandem cylinders in laminar cross-flow: vibration response and galloping mechanism. J. Fluids Struct. 78, 215238.CrossRefGoogle Scholar
Chen, W., Ji, C. & Xu, D. 2019 a Vortex-induced vibrations of two side-by-side circular cylinders with two degrees of freedom in laminar cross-flow. Comput. Fluids 193, 104288.CrossRefGoogle Scholar
Chen, W., Ji, C., Xu, W., Liu, S. & Campbell, J. 2015 Response and wake patterns of two side-by-side elastically supported circular cylinders in uniform laminar cross-flow. J. Fluids Struct. 55, 218236.CrossRefGoogle Scholar
Chen, W., Ji, C., Xu, D. & Williams, J. 2019 b Two-degree-of-freedom vortex-induced vibrations of a circular cylinder in the vicinity of a stationary wall. J. Fluids Struct. 91, 102728.CrossRefGoogle Scholar
Chen, W., Ji, C., Xu, D., Zhang, Z. & Wei, Y. 2020 b Flow-induced vibrations of an equilateral triangular prism at various angles of attack. J. Fluids Struct. 97, 103099.CrossRefGoogle Scholar
Chen, W., Zhao, Y., Ji, C., Srinil, N. & Song, L. 2021 Experimental observation of multiple responses of an oscillating D-section prism. Phys. Fluids 33, 091701.CrossRefGoogle Scholar
Cimbala, J.M., Nagib, H.M. & Roshko, A. 1988 Large structure in the far wakes of two-dimensional bluff bodies. J. Fluid Mech. 190, 265298.CrossRefGoogle Scholar
Cui, Z., Zhao, M., Teng, B. & Cheng, L. 2015 Two-dimensional numerical study of vortex-induced vibration and galloping of square and rectangular cylinders in steady flow. Ocean Engng 106, 189206.CrossRefGoogle Scholar
Derakhshandeh, J.F. & Alam, M.M. 2019 A review of bluff body wakes. Ocean Engng 182, 475488.CrossRefGoogle Scholar
Ding, L., Zhang, L., Wu, C., Mao, X. & Jiang, D. 2015 Flow induced motion and energy harvesting of bluff bodies with different cross sections. Energy Convers. Manage. 91, 416426.CrossRefGoogle Scholar
Esmaeili, M., Rabiee, A.H. & Bayandor, P. 2020 Numerical simulation of fluid–structure interaction and vortex induced vibration of the circular and truncated cylinders. J. Hydraul. 15 (2), 1530.Google Scholar
Feng, C. 1968 The measurement of vortex induced effects in flow past stationary and oscillating circular and D-section cylinders. PhD thesis, University of British Columbia.Google Scholar
Govardhan, R. & Williamson, C.H.K. 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85130.CrossRefGoogle Scholar
Inoue, O. & Yamazaki, T. 1999 Secondary vortex streets in two-dimensional cylinder wakes. Fluid Dyn. Res. 25 (1), 118.CrossRefGoogle Scholar
Jauvtis, N. & Williamson, C.H.K. 2004 The effect of two degrees of freedom on vortex-induced vibration at low mass and damping. J. Fluid Mech. 509, 2362.CrossRefGoogle Scholar
Ji, C., Munjiza, A. & Williams, J.J.R. 2012 A novel iterative direct-forcing immersed boundary method and its finite volume applications. J. Comput. Phys. 231 (4), 17971821.CrossRefGoogle Scholar
Ji, C., Xiao, Z., Wang, Y. & Wang, H. 2011 Numerical investigation on vortex-induced vibration of an elastically mounted circular cylinder at low Reynolds number using the fictitious domain method. Intl J. Comput. Fluid Dyn. 25 (4), 207221.CrossRefGoogle Scholar
Jiang, H. 2020 Separation angle for flow past a circular cylinder in the subcritical regime. Phys. Fluids 32 (1), 014106.Google Scholar
Jiang, H. & Cheng, L. 2019 Transition to the secondary vortex street in the wake of a circular cylinder. J. Fluid Mech. 867, 691722.CrossRefGoogle Scholar
Jukes, T.N. & Choi, K.S. 2009 Control of unsteady flow separation over a circular cylinder using dielectric-barrier-discharge surface plasma. Phys. Fluids 21 (9), 094106.CrossRefGoogle Scholar
Khalak, A. & Williamson, C.H.K. 1999 Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 13 (7–8), 813851.CrossRefGoogle Scholar
Kim, S. & Alam, M.M. 2015 Characteristics and suppression of flow-induced vibrations of two side-by-side circular cylinders. J. Fluids Struct. 54, 629642.CrossRefGoogle Scholar
Kim, S., Alam, M.M., Sakamoto, H. & Zhou, Y. 2009 Flow-induced vibrations of two circular cylinders in tandem arrangement. Part 1: characteristics of vibration. J. Wind Engng Ind. Aerodyn. 97 (5–6), 304311.CrossRefGoogle Scholar
Kumar, B. & Mittal, S. 2012 On the origin of the secondary vortex street. J. Fluid Mech. 711, 641666.CrossRefGoogle Scholar
Kumar, D., Singh, A.K. & Sen, S. 2018 Identification of response branches for oscillators with curved and straight contours executing VIV. Ocean Engng 164, 616627.CrossRefGoogle Scholar
Lanchester, F.W. 1907 Aerodynamics. Constable & Co.Google Scholar
Leontini, J.S., Stewart, B.E., Thompson, M.C. & Hourigan, K. 2006 a Wake state and energy transitions of an oscillating cylinder at low Reynolds number. Phys. Fluids 18 (6), 067101.CrossRefGoogle Scholar
Leontini, J.S., Thompson, M.C. & Hourigan, K. 2006 b The beginning of branching behaviour of vortex-induced vibration during two-dimensional flow. J. Fluids Struct. 22 (6–7), 857864.CrossRefGoogle Scholar
Lian, J., Yan, X., Liu, F., Zhang, J., Ren, Q. & Yang, X. 2017 Experimental investigation on soft galloping and hard galloping of triangular prisms. Appl. Sci. 7 (2), 198.CrossRefGoogle Scholar
Lighthill, J. 1986 Fundamentals concerning wave loading on offshore structures. J. Fluid Mech. 173, 667681.CrossRefGoogle Scholar
Massai, T., Zhao, J., Lo Jacono, D., Bartoli, G. & Sheridan, J. 2018 The effect of angle of attack on flow-induced vibration of low-side-ratio rectangular cylinders. J. Fluids Struct. 82, 375393.CrossRefGoogle Scholar
Mei, V.C. & Currie, I.G. 1969 Flow separation on a vibrating circular cylinder. Phys. Fluids 12, 2248.CrossRefGoogle Scholar
Meneghini, J.R., Saltara, F., Fregonesi, R. & Yamamoto, C. 2005 Vortex-induced vibration on flexible cylinders. In Numerical Models in Fluid–Structure Interaction (ed. S.K. Chakrabarti), WIT Press.CrossRefGoogle Scholar
Menon, K. & Mittal, R. 2021 On the initiation and sustenance of flow-induced vibration of cylinders: insights from force partitioning. J. Fluid Mech. 907, A37.CrossRefGoogle Scholar
Naudascher, E. & Rockwell, D. 2005 Flow-Induced Vibrations: An Engineering Guide. Courier Corporation.Google Scholar
Navrose, & Mittal, S. 2016 Lock-in in vortex-induced vibration. J. Fluid Mech. 794, 565594.CrossRefGoogle Scholar
Nemes, A., Zhao, J., Lo Jacono, D. & Sheridan, J. 2012 The interaction between flow-induced vibration mechanisms of a square cylinder with varying angles of attack. J. Fluid Mech. 710, 102130.CrossRefGoogle Scholar
Novak, M. & Tanaka, H. 1974 Effect of turbulence on galloping instability. ASCE J. Engng Mech. Div. 100 (1), 2747.CrossRefGoogle Scholar
Païdoussis, M.P., Price, S.J. & De Langre, E. 2010 Fluid–Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University Press.CrossRefGoogle Scholar
Parkinson, G. 1989 Phenomena and modelling of flow-induced vibrations of bluff bodies. Prog. Aerosp. Sci. 26 (2), 169224.CrossRefGoogle Scholar
Parkinson, G.V. 1963 Aeroelastic galloping in one degree of freedom. In Symposium Wind Effects on Buildings and Structures, pp. 582–609. National Physical Laboratory.Google Scholar
Parkinson, G.V. & Smith, J.D. 1964 The square prism as an aeroelastic non-linear oscillator. Q. J. Mech. Appl. Maths 17 (2), 225239.CrossRefGoogle Scholar
Parthasarathy, N., Dhiman, A. & Sarkar, S. 2017 Flow and heat transfer over a row of multiple semi-circular cylinders: selection of optimum number of cylinders and effects of gap ratios. Eur. Phys. J. Plus. 132 (12), 532.CrossRefGoogle Scholar
Peskin, C.S. 1972 Flow patterns around heart valves: a digital computer method for solving the equations of motion. PhD thesis, Yeshiva University.CrossRefGoogle Scholar
Prasanth, T.K. & Mittal, S. 2008 Vortex-induced vibrations of a circular cylinder at low Reynolds numbers. J. Fluid Mech. 594, 463491.CrossRefGoogle Scholar
Qin, B., Alam, M.M. & Zhou, Y. 2017 Two tandem cylinders of different diameters in cross-flow: flow-induced vibration. J. Fluid Mech. 829, 621658.CrossRefGoogle Scholar
Sarpkaya, T. 2004 A critical review of the intrinsic nature of vortex-induced vibrations. J. Fluids Struct. 19 (4), 389447.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2017 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Sen, S. & Mittal, S. 2011 Free vibration of a square cylinder at low Reynolds numbers. J. Fluids Struct. 27 (5–6), 875884.CrossRefGoogle Scholar
Sen, S., Mittal, S. & Biswas, G. 2009 Steady separated flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech. 620, 89119.CrossRefGoogle Scholar
Seyed-Aghazadeh, B., Carlson, D.W. & Modarres-Sadeghi, Y. 2017 Vortex-induced vibration and galloping of prisms with triangular cross-sections. J. Fluid Mech. 817, 590618.CrossRefGoogle Scholar
Sharma, G., Garg, H. & Bhardwaj, R. 2022 Flow-induced vibrations of elastically-mounted C- and D-section cylinders. J. Fluids Struct. 109, 103501.CrossRefGoogle Scholar
Shi, X., Alam, M. & Bai, H. 2020 Wakes of elliptical cylinders at low Reynolds number. Intl J. Heat Fluid Flow 82, 108553.CrossRefGoogle Scholar
Singh, S.P. & Mittal, S. 2005 Vortex-induced oscillations at low Reynolds numbers: hysteresis and vortex-shedding modes. J. Fluids Struct. 20 (8), 10851104.CrossRefGoogle Scholar
Tamimi, V., Naeeni, S.T.O., Zeinoddini, M., Seif, M.S. & Dolatshahi Pirooz, M. 2019 Effects of after-body on the FIV of a right-angle triangular cylinder in comparison to circular, square, and diamond cross-sections. Ships Offshore Struct. 14 (6), 589599.CrossRefGoogle Scholar
Thompson, M.C., Radi, A., Rao, A., Sheridan, J. & Hourigan, K. 2014 Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate. J. Fluid Mech. 751, 570600.CrossRefGoogle Scholar
Van der Vorst, H.A. 1992 Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (2), 631644.CrossRefGoogle Scholar
Wang, H., Zhao, D., Yang, W. & Yu, G. 2015 Numerical investigation on flow-induced vibration of a triangular cylinder at a low Reynolds number. Fluid Dyn. Res. 47, 015501.CrossRefGoogle Scholar
Weaver, D.S. & Veljkovic, I. 2005 Vortex shedding and galloping of open semi-circular and parabolic cylinders in cross-flow. J. Fluids Struct. 21 (1), 6574.CrossRefGoogle Scholar
Williamson, C.H.K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413455.CrossRefGoogle Scholar
Williamson, C.H.K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355381.CrossRefGoogle Scholar
Wu, X., Ge, F. & Hong, Y. 2012 A review of recent studies on vortex-induced vibrations of long slender cylinders. J. Fluids Struct. 28, 292308.CrossRefGoogle Scholar
Wu, M., Wen, C., Yen, R., Weng, M. & Wang, A. 2004 Experimental and numerical study of the separation angle for flow around a circular cylinder at low Reynolds number. J. Fluid Mech. 515, 233260.CrossRefGoogle Scholar
Zhang, J., Xu, G., Liu, F., Lian, J. & Yan, X. 2016 Experimental investigation on the flow induced vibration of an equilateral triangle prism in water. Appl. Ocean Res. 61, 92100.CrossRefGoogle Scholar
Zhao, M. 2015 Flow-induced vibrations of square and rectangular cylinders at low Reynolds number. Fluid Dyn. Res. 47 (2), 025502.CrossRefGoogle Scholar
Zhao, M., Cheng, L. & Zhou, T. 2013 Numerical simulation of vortex-induced vibration of a square cylinder at a low Reynolds number. Phys. Fluids 25 (2), 023603.CrossRefGoogle Scholar
Zhao, J., Hourigan, K. & Thompson, M.C. 2018 Flow-induced vibration of D-section cylinders: an afterbody is not essential for vortex-induced vibration. J. Fluid Mech. 851, 317343.CrossRefGoogle Scholar
Zhao, J., Leontini, J.S., Lo Jacono, D. & Sheridan, J. 2014 Fluid–structure interaction of a square cylinder at different angles of attack. J. Fluid Mech. 747, 688721.CrossRefGoogle Scholar
Zheng, Q. & Alam, M.M. 2019 Evolution of the wake of three inline square prisms. Phys. Rev. Fluids 4 (10), 104701.CrossRefGoogle Scholar