Wind and Structures

  • 제37권3호
  • /
  • Pages.191-209
  • /
  • 2023
  • /
  • 1226-6116(pISSN)
  • /
  • 1598-6225(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Applied Koopmanistic interpretation of subcritical prism wake physics using the dynamic mode decomposition

  • Cruz Y. Li (Department of Civil Engineering, Chongqing University) ;
  • Xisheng Lin (Department of Civil and Environmental Engineering, the Hong Kong University of Science and Technology) ;
  • Gang Hu (School of Civil and Environmental Engineering, Harbin Institute of Technology) ;
  • Lei Zhou (Department of Civil and Environmental Engineering, the Hong Kong University of Science and Technology) ;
  • Tim K.T. Tse (Department of Civil Engineering, Chongqing University) ;
  • Yunfei Fu (Department of Civil and Environmental Engineering, the Hong Kong University of Science and Technology)
  • 투고 : 2022.08.18
  • 심사 : 2023.04.20
  • 발행 : 2023.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

This work investigates the subcritical free-shear prism wake at Re=22,000 by the Koopman analysis using the Dynamic Mode Decomposition (DMD) algorithm. The Koopman model linearized nonlinearities in the stochastic, homogeneous anisotropic turbulent wake, generating temporally orthogonal eigen tuples that carry meaningful, coherent structures. Phenomenological analysis of dominant modes revealed their physical interpretations: Mode 1 renders the mean-field dynamics, Modes 2 describes the roll-up of the Strouhal vortex, Mode 3 describes the Bloor-Gerrard vortex resulting from the Kelvin-Helmholtz instability inside shear layers, its superposition onto the Strouhal vortex, and the concurrent flow entrainment, Modes 6 and 10 describe the low-frequency shedding of turbulent separation bubbles (TSBs) and turbulence production, respectively, which contribute to the beating phenomenon in the lift time history and the flapping motion of shear layers, Modes 4, 5, 7, 8, and 9 are the relatively trivial harmonic excitations. This work demonstrates the Koopman analysis' ability to provide insights into free-shear flows. Its success in subcritical turbulence also serves as an excellent reference for applications in other nonlinear, stochastic systems.

키워드

과제정보

We give a special thanks to the IT Office of the Department of Civil and Environmental Engineering at the Hong Kong University of Science and Technology. Its support for installing, testing, and maintaining our high-performance servers is indispensable for the current project.

참고문헌

  1. Andersen, M.S. and Bossen, N.S. (2021), "Modal decomposition of the pressure field on a bridge deck under vortex shedding using POD, DMD and ERA with correlation functions as Markov parameters", J. Wind Eng. Ind. Aerod., 215, 104699. https://doi.org/10.1016/J.JWEIA.2021.104699.
  2. Arbabi, H. and Mezic, I. (2017), "Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator", SIAM J. Appl. Dyn. Syst., 16(4), 2096-2126. https://doi.org/10.1137/17M1125236.
  3. Bai, H. and Alam, Md. M. (2018), "Dependence of square cylinder wake on Reynolds number", Phys. Fluids, 30(1), 015102. https://doi.org/10.1063/1.4996945.
  4. Bloor, M.S. (1964), "The transition to turbulence in the wake of a circular cylinder", J. Fluid Mech., 19(2), 290-304. https://doi.org/10.1017/S0022112064000726.
  5. Bollt, E.M. and Santitissadeekorn, N. (2013), "Applied and computational measurable dynamics", Appl. Comput. Measur. Dyn., https://doi.org/10.1137/1.9781611972641.
  6. Brunton, S.L., Noack, B.R. and Koumoutsakos, P. (2020), "Machine learning for fluid mechanics", Annu. Rev. Fluid Mech., 52(1), 477-508. https://doi.org/10.1146/annurev-fluid-010719-060214.
  7. Budisic, M., Mohr, R. and Mezic, I. (2012), Applied Koopmanism. Chaos, 22(4), 047510. https://doi.org/10.1063/1.4772195.
  8. Cao, Y. and Tamura, T. (2015), "Numerical investigations into effects of three-dimensional wake patterns on unsteady aerodynamic characteristics of a circular cylinder at Re=1.3×105", J. Fluids Struct., 59, 351-369. https://doi.org/10.1016/j.jfluidstructs.2015.10.001.
  9. Cao, Y., Tamura, T. and Kawai, H. (2020), "Spanwise resolution requirements for the simulation of high-Reynolds-number flows past a square cylinder", Comput. Fluids, 196, 104320. https://doi.org/10.1016/j.compfluid.2019.104320.
  10. Carlsson, H., Carlsson, C., Fuchs, L. and Bai, X.S. (2014), "Large eddy simulation and extended dynamic mode decomposition of flow-flame interaction in a lean premixed low swirl stabilized flame", Flow, Turbul. Combust., 93(3), 505-519. https://doi.org/10.1007/s10494-014-9560-6.
  11. Celik, I.B., Cehreli, Z.N. and Yavuz, I. (2005), "Index of resolution quality for large eddy simulations", J. Fluids Eng. Transact. ASME, 127(5), 949-958. https://doi.org/10.1115/1.1990201.
  12. Cesur, A., Carlsson, C., Feymark, A., Fuchs, L. and Revstedt, J. (2014), "Analysis of the wake dynamics of stiff and flexible cantilever beams using POD and DMD", Comput. Fluids, 101, 27-41. https://doi.org/https://doi.org/10.1016/j.compfluid.2014.05.012.
  13. Chen, K.K., Tu, J.H. and Rowley, C.W. (2012), "Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier Analyses", J. Nonline. Sci. 22(6), 887-915. https://doi.org/10.1007/s00332-012-9130-9.
  14. Chen, Z., Bai, J., Wang, S., Xue, X., Li, K., Tse, K.T., Li, C.Y. and Lin, C. (2023), "The role of transverse inclination on the flow phenomenology around cantilevered prisms and the tripole wake mode", J. Fluids Struct., 118, 103837. https://doi.org/10.1016/j.jfluidstructs.2023.103837.
  15. Chen, Z., Fu, X., Xu, Y., Li, C.Y., Kim, B. and Tse, K.T. (2021), "A perspective on the aerodynamics and aeroelasticity of tapering: Partial reattachment", J. Wind Eng. Ind. Aerod., 212(October 2020), 104590. https://doi.org/10.1016/j.jweia.2021.104590.
  16. Chen, Z., Huang, H., Tse, T.K.T., Xu, Y. and Li, C.Y. (2020), "Characteristics of unsteady aerodynamic forces on an aeroelastic prism: A comparative study", J. Wind Eng. Ind. Aerod., 205, 104325. https://doi.org/10.1016/j.jweia.2020.104325.
  17. Chen, Z., Li, D., Li, C.Y. and Fu, X. (2022), "Research on aerodynamic mechanism of single high-rise building based on twisted wind field in mountainous area. In I. Calotescu, A. Chitez, C. Cosoiu, & A. C. Vladut (Eds.), 8th EUROPEAN-AFRICAN CONFERENCE ON WIND ENGINEERING (8EACWE) Proceedings (pp. 199-202). Editura Conspress.
  18. Chen, Z., Tse, T.K.T., Li, Y.C., Kwok, K.C.S. and Kareem, A. (2020), "A mathematical model for VIV-galloping force acting on slender prisms", The 4th Hong Kong Wind Engineering Society Workshop, 59-60.
  19. Chen, Z., Zhang, L., Li, K., Xue, X., Zhang, X., Kim, B. and Li, C.Y. (2023), "Machine-learning prediction of aerodynamic damping for buildings and structures undergoing flow-induced vibrations", J. Build. Eng., 63, 105374. https://doi.org/10.1016/j.jobe.2022.105374.
  20. Chen, Z.-S., Wang, Y., Wang, S., Huang, H., Tse, K.T., Li, C.Y. and Lin, C. (2022), "Decoupling bi-directional fluid-structure interactions by the Koopman theory: Actualizing one-way subcases and the role of crosswind structure motion", Phys. Fluids, 34(9), 095103. https://doi.org/10.1063/5.0101749.
  21. Chu, Q., Liu, H., Xia, S., Dong, J., Lei, M., Tse, T.K.T., Teng, L., Li, C.Y. and Fu, Y. (2022), "Numerical and experimental study on the member performance and stability bearing capacity of wheel coupler formwork supports", Appl. Sci., 12(20), 10452. https://doi.org/10.3390/app122010452.
  22. Dawson, S.T.M., Hemati, M.S., Williams, M.O. and Rowley, C.W. (2016), "Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition", Experiment. Fluids, 57(3). https://doi.org/10.1007/S00348-016-2127-7/FIGURES/12.
  23. Ducoin, A., Loiseau, J.-Ch. and Robinet, J.-Ch. (2016), "Numerical investigation of the interaction between laminar to turbulent transition and the wake of an airfoil", Europ. J. Mech. - B/Fluids, 57, 231-248. https://doi.org/10.1016/j.euromechflu.2016.01.005.
  24. Erichson, N.B. and Donovan, C. (2016), "Randomized low-rank Dynamic Mode Decomposition for motion detection", Comput. Vision Image Understanding, 146, 40-50. https://doi.org/10.1016/j.cviu.2016.02.005.
  25. Erichson, N.B., Mathelin, L., Kutz, J.N. and Brunton, S.L. (2019), "Randomized dynamic mode decomposition", SIAM J. Appl. Dyn. Syst., 18(4), 1867-1891. https://doi.org/10.1137/18M1215013.
  26. Fan, X., Zhang, X., Weerasuriya, A.U., Hang, J., Zeng, L., Luo, Q., Li, C.Y. and Chen, Z. (2022), "Numerical investigation of the effects of environmental conditions, droplet size, and social distancing on droplet transmission in a street canyon", Build. Environ., 221, 109261. https://doi.org/10.1016/j.buildenv.2022.109261.
  27. Franke, J., Hellsten, A., Schlunzen, H. and Carissimo, B. (2007), "Best practice guideline for the CFD simulation of flows in the urban environment", University of Hamburg, 44(May).
  28. Froyland, G., Gottwald, G.A. and Hammerlindl, A. (2014), "A computational method to extract macroscopic variables and their dynamics in multiscale systems", SIAM J. Appl. Dyn. Syst., 13(4), 1816-1846. https://doi.org/10.1137/130943637.
  29. Fu, Y., Li, L., Qin, X., Li, C.Y. and Tse, T.K.T. (2022), "Numerical study of reactive air pollutant dispersion in near-field wake", In I. Calotescu, A. Chitez, C. Cosoiu, & A. C. Vladut (Eds.), 8th EUROPEAN-AFRICAN CONFERENCE ON WIND ENGINEERING (8EACWE) Proceedings, 507-510.
  30. Fu, Y., Lin, X., Li, L., Chu, Q., Liu, H., Zheng, X., Liu, C.-H., Chen, Z., Lin, C., Tse, T.K.T. and Li, C.Y. (2023), "A POD-DMD augmented procedure to isolating dominant flow field features in a street canyon", Phys. Fluids, 35(2), 025112. https://doi.org/10.1063/5.0133375.
  31. Gerrard, J.H. (1966), "The mechanics of the formation region of vortices behind bluff bodies", J. Fluid Mech., 25(2), 401-413. https://doi.org/10.1017/S0022112066001721.
  32. Gerrard, J.H. (1978), "The wakes of cylindrical bluff bodies at low Reynolds number", Philosop. Transact. Royal Soc., 288(1354), 351-382. https://doi.org/10.1098/rsta.1978.0020.
  33. Gomez, F., Blackburn, H.M., Rudman, M., McKeon, B.J., Luhar, M., Moarref, R. and Sharma, A.S. (2014), "On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow", Phys. Fluids, 26(10), 101703. https://doi.org/10.1063/1.4900768.
  34. He, G., Wang, J. and Pan, C. (2013), "Initial growth of a disturbance in a boundary layer influenced by a circular cylinder wake", J. Fluid Mech., 718, 116-130. https://doi.org/10.1017/jfm.2012.599.
  35. He, Y., Zhang, L., Chen, Z. and Li, C.Y. (2022), "A framework of structural damage detection for civil structures using a combined multi-scale convolutional neural network and echo state network", Eng. Comput., 1, 1-19. https://doi.org/10.1007/s00366-021-01584-4.
  36. Helmholtz, H. (1858), "uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen", Journal Fur Die Reine Und Angewandte Mathematik, 55, 25-55.
  37. Hemati, M.S., Rowley, C.W., Deem, E.A. and Cattafesta, L.N. (2017), "De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets", Theoretic. Comput. Fluid Dyn., 31(4), 349-368. https://doi.org/10.1007/s00162-017-0432-2.
  38. Hong, S., Huang, G., Yang, Y. and Liu, Z. (2018), "Introduction of DMD method to study the dynamic structures of a three-dimensional centrifugal compressor with and without flow control", Energies, 11(11). https://doi.org/10.3390/en11113098.
  39. Hu, G., Tse, K.T. and Kwok, K.C.S. (2016), "Aerodynamic mechanisms of galloping of an inclined square cylinder", J. Wind Eng. Ind. Aerod., 148, 6-17. https://doi.org/10.1016/j.jweia.2015.10.011.
  40. Hussain, A.K.M.F. (1981), Coherent structures and studies of perturbed and unperturbed jets. (ed.), Berlin, Fed. Rep. Germany, Springer-Verlag, 1981, Session 3-Experiments, 252-291. (Lecture, 252-291. https://doi.org/10.1007/3-540-10289-2_30.
  41. Hussain, A.K.M.F. (1986), "Coherent structures and turbulence", J. Fluid Mech., 173, 303-356. https://doi.org/10.1017/S0022112086001192.
  42. Iousef, S., Montazeri, H., Blocken, B. and van Wesemael, P.J.V. (2017), "On the use of non-conformal grids for economic LES of wind flow and convective heat transfer for a wall-mounted cube", Build. Environ., 119, 44-61. https://doi.org/10.1016/j.buildenv.2017.04.004.
  43. Jardin, T. and Bury, Y. (2012), "Lagrangian and spectral analysis of the forced flow past a circular cylinder using pulsed tangential jets", J. Fluid Mech., 696, 285-300. https://doi.org/10.1017/jfm.2012.35.
  44. Jovanovic, M.R., Schmid, P.J. and Nichols, J.W. (2014), "Sparsity-promoting dynamic mode decomposition", Phys. Fluids, 26(2), 24103. https://doi.org/10.1063/1.4863670.
  45. Kiya, M. and Sasaki, K. (1983), "Structure of a turbulent separation bubble", J. Fluid Mech., 137, 83-113. https://doi.org/10.1017/S002211208300230X
  46. Koopman, B.O. (1931), "Hamiltonian systems and transformation in Hilbert Space", Proceedings of the National Academy of Sciences, 17(5), 315-318. https://doi.org/10.1073/PNAS.17.5.315.
  47. Kou, J. and Zhang, W. (2017), "An improved criterion to select dominant modes from dynamic mode decomposition", Europ. J. Mech., B/Fluids, 62, 109-129. https://doi.org/10.1016/j.euromechflu.2016.11.015.
  48. Kundu, P.K. (2004). Fluid Mechanics, Elsevier Academic Press.
  49. Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L., Dmd, T., Armstrong, C., Theory, S., Expansions, E., Dmd, T., Fu, X., Brunton, S. L., Dmdc, F., These, D., Kutz, J.N., Koopman, D., Factors, R., Twain, M., Ex-, N.Y.S., Union, E. and Proctor, J.L. (2016), "Dynamic mode decomposition: Data-driven modeling of complex systems. In society for industrial and applied mathematics, 32(4). https://doi.org/10.1137/1.9781611974508.
  50. Le Clainche, S. and Vega, J.M. (2017), "Higher order dynamic mode decomposition", SIAM J. Appl. Dyn. Syst., 16(2), 882-925. https://doi.org/10.1137/15M1054924.
  51. Li, C.Y., Chen, Z., Lin, X., Weerasuriya, A.U., Zhang, X., Fu, Y. and Tse, T.K.T. (2022), "The linear-time-invariance notion to the Koopman analysis: The architecture, pedagogical rendering, and fluid-structure association", Phys. Fluids, 34(12), 125136. https://doi.org/10.1063/5.0124914.
  52. Li, C.Y., Chen, Z. and Tse, T.K.T. (2022), "Associating structure surface pressure with corresponding flow field excitation-the data-driven answer to fluid-structure interaction", In I. Calotescu, A. Chitez, C. Cosoiu, & A. C. Vladut (Eds.), 8th EUROPEAN-AFRICAN CONFERENCE ON WIND ENGINEERING (8EACWE) Proceedings, 103-106.
  53. Li, C.Y., Chen, Z., Tse, T.K.T., Weerasuriya, A.U., Zhang, X., Fu, Y. and Lin, X. (2021), "Establishing direct phenomenological connections between fluid and structure by the Koopman-Linearly Time-Invariant analysis", Phys. Fluids, 33(12), 121707. https://doi.org/10.1063/5.0075664.
  54. Li, C.Y., Chen, Z., Tse, T.K.T., Weerasuriya, A.U., Zhang, X., Fu, Y. and Lin, X. (2022a), "A parametric and feasibility study for data sampling of the dynamic mode decomposition: Spectral insights and further explorations", Phys. Fluids, 34(3), 035102. https://doi.org/10.1063/5.0082640.
  55. Li, C.Y., Chen, Z., Tse, T.K.T., Weerasuriya, A.U., Zhang, X., Fu, Y. and Lin, X. (2022b), "A parametric and feasibility study for data sampling of the dynamic mode decomposition: range, resolution, and universal convergence states", Nonlinear Dyn., 107(4), 3683-3707. https://doi.org/10.1007/s11071-021-07167-8.
  56. Li, C.Y., Chen, Z., Tse, T.K.T., Weerasuriya, A.U., Zhang, X., Fu, Y. and Lin, X. (2022c), "Best practice for the dynamic mode decomposition in wind engineering applications", In I. Calotescu, A. Chitez, C. Cosoiu, & A. C. Vladut (Eds.), 8th EUROPEAN-AFRICAN CONFERENCE ON WIND ENGINEERING (8EACWE) Proceedings, 11-14.
  57. Li, C.Y., Chen, Z., Tse, T.K.T., Weerasuriya, A.U., Zhang, X., Fu, Y. and Lin, X. (2023), "The linear-time-invariance notion of the Koopman analysis. Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake", J. Fluid Mech., 959, A15. https://doi.org/10.1017/jfm.2023.36.
  58. Li, C.Y., Chen, Z., Zhang, X., Tse, T.K.T. and Lin, C. (2023), "Koopman analysis by the dynamic mode decomposition in wind engineering", J. Wind Eng. Ind. Aerod., 232, 105295. https://doi.org/10.1016/J.JWEIA.2022.105295.
  59. Li, C.Y., Tse, T.K.T. and Hu, G. (2020a), "Reconstruction of flow field around a square prism using dynamic mode decomposition", The 4th Hong Kong Wind Engineering Society Workshop, 61.
  60. Li, C.Y., Tse, T.K.T. and Hu, G. (2020b), "Dynamic mode decomposition on pressure flow field analysis: Flow field reconstruction, accuracy, and practical significance", J. Wind Eng. Ind. Aerod., 205, 104278. https://doi.org/10.1016/j.jweia.2020.104278.
  61. Lilly, D.K. (1967), "The representation of small-scale turbulence in numerical simulation experiments", Proc. IBM Scientijc Comput. Symp. Environ. Sci., 195-210. https://doi.org/10.5065/D62R3PMM.
  62. Luo, S.C., Tong, X.H. and Khoo, B.C. (2007), "Transition phenomena in the wake of a square cylinder", J. Fluids Struct., 23(2), 227-248. https://doi.org/10.1016/j.jfluidstructs.2006.08.012.
  63. Luo, X. and Kareem, A. (2021), "Dynamic mode decomposition of random pressure fields over bluff bodies", J. Eng. Mech., 147(4), 04021007. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001904.
  64. Lusch, B., Kutz, J.N. and Brunton, S.L. (2018), "Deep learning for universal linear embeddings of nonlinear dynamics", Nature Commun., 9(1), 4950. https://doi.org/10.1038/s41467-018-07210-0.
  65. Mauroy, A. and Mezic, I. (2012), "On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics", Chaos: Interdiscipl. J. Nonlinear Sci., 22(3), 033112. https://doi.org/10.1063/1.4736859.
  66. Mauroy, A. and Mezic, I. (2013), "A spectral operator-theoretic framework for global stability", Proceedings of the IEEE Conference on Decision and Control, 5234-5239. https://doi.org/10.1109/CDC.2013.6760712.
  67. Mauroy, A. and Mezic, I. (2016), "Global stability analysis using the Eigenfunctions of the Koopman Operator", IEEE Transact, Automatic Control, 61(11), 3356-3369. https://doi.org/10.1109/TAC.2016.2518918.
  68. Menter, F.R. (2012), Best Practice: Scale-Resolving Simulations in ANSYS CFD. In ANSYS Germany GmbH.
  69. Mezic, I. (2005), "Spectral properties of dynamical systems, model reduction and decompositions", Nonlinear Dyn., 41(1-3), 309-325. https://doi.org/10.1007/s11071-005-2824-x.
  70. Mezic, I. (2013), "Analysis of fluid flows via spectral properties of the Koopman Operator", Annu. Rev. Fluid Mech., 45(1), 357-378. https://doi.org/10.1146/annurev-fluid-011212-140652.
  71. Muld, T.W., Efraimsson, G. and Henningson, D.S. (2012a), "Flow structures around a high-speed train extracted using Proper Orthogonal Decomposition and Dynamic Mode Decomposition", Comput. Fluids, 57, 87-97. https://doi.org/10.1016/j.compfluid.2011.12.012.
  72. Muld, T.W., Efraimsson, G. and Henningson, D.S. (2012b), "Mode decomposition on surface-mounted cube", Flow, Turbulence Combustion, 88(3), 279-310. https://doi.org/10.1007/s10494-011-9355-y.
  73. Paidoussis, M.P., Price, S.J. and de Langre, E. (2010), "Fluid-structure interactions: Cross-flow-induced instabilities", Cambridge University Press. https://doi.org/10.1017/CBO9780511760792
  74. Pan, S., Arnold-Medabalimi, N. and Duraisamy, K. (2021), "Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces", J. Fluid Mech., 917, A18. https://doi.org/10.1017/jfm.2021.271.
  75. Pope, S.B. (2000), Turbulent Flows. Cambridge University Press. https://doi.org/10.1017/CBO9780511840531.
  76. Raissi, M., Perdikaris, P. and Karniadakis, G.E. (2019), "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations", J. Comput. Phys., 378, 686-707. https://doi.org/10.1016/j.jcp.2018.10.045.
  77. Richecoeur, F., Hakim, L., Renaud, A. and Zimmer, L. (2012), "DMD algorithms for experimental data processing in combustion", Proceeding of the 2012 Summer Program, Center of Turbulence Research, 459-468.
  78. Robichaux, J., Balachandar, S. and Vanka, S.P. (1999), "Three-dimensional Floquet instability of the wake of square cylinder", Phys. Fluids, 11(2-3), 560-578. https://doi.org/10.1063/1.869930.
  79. Rowley, C.W. and Dawson, S.T.M.M. (2017), "Model reduction for flow analysis and control", Annu. Rev. Fluid Mech., 49(1), 387-417. https://doi.org/10.1146/annurev-fluid-010816-060042.
  80. Rowley, C.W., Mezic, I., Bagheri, S., Schlatter, P. and Henningson, D.S. (2009), "Spectral analysis of nonlinear flows", J. Fluid Mech., 641, 115-127. https://doi.org/10.1017/S0022112009992059.
  81. Roy, S., Hua, J.C., Barnhill, W., Gunaratne, G.H. and Gord, J.R. (2015), "Deconvolution of reacting-flow dynamics using proper orthogonal and dynamic mode decompositions", Phys. Rev. E -Statistic. Nonlinear, Soft Matter Phys., 91(1), 013001. https://doi.org/10.1103/PhysRevE.91.013001.
  82. Rudy, S.H., Brunton, S.L., Proctor, J.L. and Kutz, J.N. (2017), "Data-driven discovery of partial differential equations", Sci. Adv., 3(4). https://doi.org/10.1126/sciadv.1602614.
  83. Sarkar, S., Ganguly, S., Dalal, A., Saha, P. and Chakraborty, S. (2013), "Mixed convective flow stability of nanofluids past a square cylinder by dynamic mode decomposition", Int. J. Heat Fluid Flow, 44, 624-634. https://doi.org/https://doi.org/10.1016/j.ijheatfluidflow.2013.09.004.
  84. Sarmast, S., Dadfar, R., Mikkelsen, R.F., Schlatter, P., Ivanell, S., Sorensen, J.N. and Henningson, D.S. (2014), "Mutual inductance instability of the tip vortices behind a wind turbine", J. Fluid Mech., 755, 705-731. https://doi.org/10.1017/jfm.2014.326.
  85. Sarpkaya, T. (1979), "Vortex-induced oscillations: A selective review", J. Appl. Mech., 46(2), 241-258. https://doi.org/10.1115/1.3424537.
  86. Sayadi, T., Schmid, P.J., Nichols, J.W. and Moin, P. (2014), "Reduced-order representation of near-wall structures in the late transitional boundary layer", J. Fluid Mech., 748, 278-301. https://doi.org/10.1017/jfm.2014.184.
  87. Schmid, P.J. (2010), "Dynamic mode decomposition of numerical and experimental data", J. Fluid Mech., 656, 5-28. https://doi.org/10.1017/S0022112010001217.
  88. Smagorinsky, J. (1963), "General circulation experiments with the primitive equations", Month. Weather Rev., 91(3), 99-164. https://doi.org/10.1175/15200493(1963)091<0099:gcewtp>2.3.co;2.
  89. Statnikov, V., Bolgar, I., Scharnowski, S., Meinke, M., Kahler, C. J. and Schroder, W. (2016), "Analysis of characteristic wake flow modes on a generic transonic backward-facing step configuration", Europ. J. Mech., B/Fluids, 59, 124-134. https://doi.org/10.1016/j.euromechflu.2016.05.008.
  90. Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V. and Ukeiley, L.S. (2017), "Modal analysis of fluid flows: An overview", AIAA J., 55(12), 4013-4041. https://doi.org/10.2514/1.J056060.
  91. Tissot, G., Cordier, L., Benard, N. and Noack, B.R. (2014), "Model reduction using Dynamic Mode Decomposition", In Comptes Rendus - Mecanique, 342(6-7), 410-416. https://doi.org/10.1016/j.crme.2013.12.011.
  92. Tominaga, Y., Mochida, A., Yoshie, R., Kataoka, H., Nozu, T., Yoshikawa, M. and Shirasawa, T. (2008), "AIJ guidelines for practical applications of CFD to pedestrian wind environment around buildings", J. Wind Eng. Ind. Aerod., 96(10-11), 1749-1761. https://doi.org/10.1016/j.jweia.2008.02.058.
  93. Towne, A., Schmidt, O.T. and Colonius, T. (2018), "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis", J. Fluid Mech., 847, 821-867. https://doi.org/10.1017/jfm.2018.283.
  94. Tse, T.K.T., Li, C.Y. and Hu, G. (2021), "High-fidelity flow field reconstruction and revelation of flow mechanisms around high-rise structures - A dynamic mode decomposition approach", Proceedings for The Infrastructure Construction Conference (ICC21).
  95. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L. and Kutz, J.N. (2014), "On dynamic mode decomposition: Theory and applications", J. Comput. Dyn., 1(2), 391-421. https://doi.org/10.3934/jcd.2014.1.391.
  96. Unal, M.F. and Rockwell, D. (1988), "On vortex formation from a cylinder. Part 1. The initial instability", J. Fluid Mech., 190, 491-512. https://doi.org/10.1017/S0022112088001429.
  97. Wan, Z.-H., Zhou, L., Wang, B.-F. and Sun, D.-J. (2015), "Dynamic mode decomposition of forced spatially developed transitional jets", Europ. J. Mech. - B/Fluids, 51, 16-26. https://doi.org/10.1016/j.euromechflu.2014.12.001.
  98. Wang, B. and Yu, M. (2016), "Analysis of wake structures behind an oscillating square cylinder using dynamic mode decomposition", The 46th AIAA Fluid Dynamics Conference. https://doi.org/10.2514/6.2016-3779.
  99. Wang, S., Liu, H., Wang, Y., Qiao, Y., Wang, L., Bai, J., Tse, T.K. T., Li, C.Y. and Fu, Y. (2022), "Experimental study on the seismic performance of shear walls with different coal gangue replacement rates", Appl. Sci., 12(20), 10622. https://doi.org/10.3390/app122010622.
  100. Weerasuriya, A.U., Zhang, X., Tse, T.K.T., Liu, C.H. and Kwok, K.C.S. (2022), "RANS simulation of near-field dispersion of reactive air pollutants", Build. Environ., 207, 108553. https://doi.org/10.1016/J.BUILDENV.2021.108553.
  101. White, F. (2006), Viscous Fluid Flow, McGraw Hill.
  102. Williams, M.O., Kevrekidis, I.G. and Rowley, C.W. (2015), "A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition", J. Nonlinear Sci., 25(6), 1307-1346. https://doi.org/10.1007/s00332-015-9258-5.
  103. Wu, J., Sheridan, J., Hourigan, K. and Soria, J. (1996), "Shear layer vortices and longitudinal vortices in the near wake of a circular cylinder", Experiment. Thermal Fluid Sci., 12(2), 169-174. https://doi.org/10.1016/0894-1777(95)00087-9.
  104. Zhang, X., Weerasuriya, A.U., Wang, J., Li, C.Y., Chen, Z., Tse, K. T. and Hang, J. (2022), "Cross-ventilation of a generic building with various configurations of external and internal openings", Build. Environ., 207, 108447. https://doi.org/10.1016/j.buildenv.2021.108447.
  105. Zhang, X., Weerasuriya, A.U., Zhang, X., Tse, K.T., Lu, B., Li, C. Y. and Liu, C.-H. (2020), "Pedestrian wind comfort near a super-tall building with various configurations in an urban-like setting", Build. Simul, 13(6), 1385-1408. https://doi.org/10.1007/s12273-020-0658-6.
  106. Zhou, L., Tse, K.T., Hu, G. and Li, C.Y. (2021a), "Mode interpretation of interference effects between tall buildings in tandem and side-by-side arrangement with POD and ICA", Eng. Struct., 243, 112616. https://doi.org/10.1016/j.engstruct.2021.112616.
  107. Zhou, L., Tse, K.T., Hu, G. and Li, Y. (2021b), "Higher order dynamic mode decomposition of wind pressures on square buildings", J. Wind Eng. Ind. Aerod., 211(June 2020), 104545. https://doi.org/10.1016/j.jweia.2021.104545.