Abstract
Fluctuating surface pressures on a bluff body exposed to a boundary layer flow generally are characterized as a spatiotemporally varying random field. In this paper, a dynamic mode decomposition (DMD) was applied to extract dominant features embedded in these random pressure fields. Utilizing an unsupervised machine learning algorithm, spatial modes and their temporal variations were grouped into different clusters at scales, e.g., macro, meso, and micro. A proper orthogonal decomposition (POD) of the experimental data was carried out to observe commonalities and distinctive perspectives each decomposition offers. A comprehensive examination of the DMD/POD for their convergence criteria, data sufficiency, and modal components analysis was conducted. The physical interpretation of the spatiotemporal pressure field based on these decomposition schemes was discussed. At different scales, the DMD modes can capture the evolution of aerodynamic features, e.g., convection of vortices (or vortex tubes) and other structures. The distribution of energy among these three broad scales also reflects an energy cascade in pressure fluctuations akin to turbulence.
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available in a repository online (https://xihaier.github.io/) in accordance with funder data retention policies.
Acknowledgments
This work was supported in part by the National Science Foundation (NSF) under Grant No. 1562244. This support is gratefully acknowledged. The authors also gratefully acknowledge the use of the Tokyo Polytechnic University aerodynamic database.
References
Arbabi, H., and I. Mezic. 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.
Arthur, D., and S. Vassilvitskii. 2007. “k-means++: The advantages of careful seeding.” In Proc., 18th Annual ACM-SIAM Symp. on Discrete Algorithms, 1027–1035. Philadelphia: Society for Industrial and Applied Mathematics.
Aubry, N., P. Holmes, J. L. Lumley, and E. Stone. 1988. “The dynamics of coherent structures in the wall region of a turbulent boundary layer.” J. Fluid Mech. 192: 115–173. https://doi.org/10.1017/S0022112088001818.
Baker, C. 2000. “Aspects of the use of proper orthogonal decomposition of surface pressure fields.” Wind Struct. 3 (2): 97–115. https://doi.org/10.12989/was.2000.3.2.097.
Berkooz, G., P. Holmes, and J. L. Lumley. 1993. “The proper orthogonal decomposition in the analysis of turbulent flows.” Annu. Rev. Fluid Mech. 25 (1): 539–575. https://doi.org/10.1146/annurev.fl.25.010193.002543.
Bistrian, D. A., and I. M. Navon. 2015. “An improved algorithm for the shallow water equations model reduction: Dynamic mode decomposition vs pod.” Int. J. Numer. Methods Fluids 78 (9): 552–580. https://doi.org/10.1002/fld.4029.
Brunton, S. L., B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz. 2017. “Chaos as an intermittently forced linear system.” Nat. Commun. 8 (1): 1–9.
Budišić, M., R. Mohr, and I. Mezić. 2012. “Applied Koopmanism.” Chaos: Interdiscip. J. Nonlinear Sci. 22 (4): 047510. https://doi.org/10.1063/1.4772195.
Carassale, L. 2012. “Analysis of aerodynamic pressure measurements by dynamic coherent structures.” Probab. Eng. Mech. 28 (Apr): 66–74. https://doi.org/10.1016/j.probengmech.2011.08.010.
Carassale, L., and M. M. Brunenghi. 2011. “Statistical analysis of wind-induced pressure fields: A methodological perspective.” J. Wind Eng. Ind. Aerodyn. 99 (6–7): 700–710. https://doi.org/10.1016/j.jweia.2011.03.011.
Carassale, L., G. Piccardo, and G. Solari. 2001. “Double modal transformation and wind engineering applications.” J. Eng. Mech. 127 (5): 432–439. https://doi.org/10.1061/%28ASCE%290733-9399%282001%29127%3A5%28432%29.
Carassale, L., and G. Solari. 2002. “Wind modes for structural dynamics: a continuous approach.” Probab. Eng. Mech. 17 (2): 157–166. https://doi.org/10.1016/S0266-8920(01)00036-4.
Cermak, J. 1976. “Aerodynamics of buildings.” Annu. Rev. Fluid Mech. 8 (1): 75–106. https://doi.org/10.1146/annurev.fl.08.010176.000451.
Chen, K. K., J. H. Tu, and C. W. Rowley. 2012. “Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses.” J. Nonlinear Sci. 22 (6): 887–915. https://doi.org/10.1007/s00332-012-9130-9.
Chen, X. 2013. “Estimation of stochastic crosswind response of wind-excited tall buildings with nonlinear aerodynamic damping.” Eng. Struct. 56 (Nov): 766–778. https://doi.org/10.1016/j.engstruct.2013.05.044.
Chen, X., and A. Kareem. 2004. “Equivalent static wind loads on buildings: New model.” J. Struct. Eng. 130 (10): 1425–1435. https://doi.org/10.1061/(ASCE)0733-9445(2004)130:10(1425).
Chen, X., and A. Kareem. 2005. “Proper orthogonal decomposition-based modeling, analysis, and simulation of dynamic wind load effects on structures.” J. Eng. Mech. 131 (4): 325–339. https://doi.org/10.1061/%28ASCE%290733-9399%282005%29131%3A4%28325%29.
Cui, W., and L. Caracoglia. 2018. “A fully-coupled generalized model for multi-directional wind loads on tall buildings: A development of the quasi-steady theory.” J. Fluids Struct. 78 (Apr): 52–68. https://doi.org/10.1016/j.jfluidstructs.2017.12.008.
Dawson, S. T., M. S. Hemati, M. O. Williams, and C. W. Rowley. 2016. “Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition.” Exp. Fluids 57 (3): 42. https://doi.org/10.1007/s00348-016-2127-7.
Ding, F., A. Kareem, and J. Wan. 2019. “Aerodynamic tailoring of structures using computational fluid dynamics.” Struct. Eng. Int. 29 (1): 26–39. https://doi.org/10.1080/10168664.2018.1522936.
Frisch, U. 1995. Turbulence: The legacy of A. N. Kolmogorov. Cambridge, UK: Cambridge University Press.
Giannakis, D. 2019. “Data-driven spectral decomposition and forecasting of ergodic dynamical systems.” Appl. Comput. Harmon. Anal. 47 (2): 338–396. https://doi.org/10.1016/j.acha.2017.09.001.
Grassberger, P., and I. Procaccia. 2004. “Measuring the strangeness of strange attractors.” In The theory of chaotic attractors, 170–189. New York: Springer.
Holmes, P., J. L. Lumley, G. Berkooz, and C. W. Rowley. 2012. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge, UK: Cambridge University Press.
Huang, G., and X. Chen. 2007. “Wind load effects and equivalent static wind loads of tall buildings based on synchronous pressure measurements.” Eng. Struct. 29 (10): 2641–2653. https://doi.org/10.1016/j.engstruct.2007.01.011.
Kareem, A., and J. E. Cermak. 1984. “Pressure fluctuations on a square building model in boundary-layer flows.” J. Wind Eng. Ind. Aerodyn. 16 (1): 17–41. https://doi.org/10.1016/0167-6105(84)90047-3.
Kennel, M. B., R. Brown, and H. D. I. Abarbanel. 1992. “Determining embedding dimension for phase-space reconstruction using a geometrical construction.” Phys. Rev. A 45 (6): 3403. https://doi.org/10.1103/PhysRevA.45.3403.
Kim, B., K. Tse, and Y. Tamura. 2018. “POD analysis for aerodynamic characteristics of tall linked buildings.” J. Wind Eng. Ind. Aerodyn. 181 (Oct): 126–140. https://doi.org/10.1016/j.jweia.2018.09.001.
Klus, S., P. Koltai, and C. Schütte. 2015. “On the numerical approximation of the Perron-Frobenius and Koopman operator.” Preprint, submitted December 18, 2015. https://arxiv.org/abs/1512.05997.
Koopman, B. O. 1931. “Hamiltonian systems and transformation in Hilbert space.” Proc. Natl. Acad. Sci. USA 17 (5): 315. https://doi.org/10.1073/pnas.17.5.315.
Kutz, J. N., S. L. Brunton, B. W. Brunton, and J. L. Proctor. 2016. Dynamic mode decomposition: Data-driven modeling of complex systems. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Le Clainche, S., and J. M. Vega. 2017. “Higher order dynamic mode decomposition.” SIAM J. Appl. Dyn. Syst. 16 (2): 882–925. https://doi.org/10.1137/15M1054924.
Lee, B. 1975. “The effect of turbulence on the surface pressure field of a square prism.” J. Fluid Mech. 69 (2): 263–282. https://doi.org/10.1017/S0022112075001437.
Leonard, A. 1975. “Energy cascade in large-eddy simulations of turbulent fluid flows.” In Vol. 18 of Advances in geophysics, 237–248. Amsterdam, Netherlands: Elsevier.
Lin, N., C. Letchford, Y. Tamura, B. Liang, and O. Nakamura. 2005. “Characteristics of wind forces acting on tall buildings.” J. Wind Eng. Ind. Aerodyn. 93 (3): 217–242. https://doi.org/10.1016/j.jweia.2004.12.001.
Luo, X., and A. Kareem. 2020. “Bayesian deep learning with hierarchical prior: Predictions from limited and noisy data.” Struct. Saf. 84 (May): 101918. https://doi.org/10.1016/j.strusafe.2019.101918.
Lusch, B., J. N. Kutz, and S. L. Brunton. 2018. “Deep learning for universal linear embeddings of nonlinear dynamics.” Nat. Commun. 9 (1): 1–10.
Majda, A. J., and Y. Lee. 2014. “Conceptual dynamical models for turbulence.” Proc. Nat. Acad. Sci. 111 (18): 6548–6553. https://doi.org/10.1073/pnas.1404914111.
Mezić, 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.
Muld, T. W., G. Efraimsson, and D. S. Henningson. 2012a. “Flow structures around a high-speed train extracted using Proper Orthogonal Decomposition and Dynamic Mode Decomposition.” Comput. Fluids 57 (Mar): 87–97. https://doi.org/10.1016/j.compfluid.2011.12.012.
Muld, T. W., G. Efraimsson, and D. S. Henningson. 2012b. “Mode decomposition on surface-mounted cube.” Flow Turbul. Combust. 88 (3): 279–310. https://doi.org/10.1007/s10494-011-9355-y.
Murphy, K. P. 2012. Machine learning: A probabilistic perspective. London: MIT Press.
Rowley, C. W., I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson. 2009. “Spectral analysis of nonlinear flows.” J. Fluid Mech. 641 (1): 115–127. https://doi.org/10.1017/S0022112009992059.
Schmid, P. J. 2010. “Dynamic mode decomposition of numerical and experimental data.” J. Fluid Mech. 656: 5–28. https://doi.org/10.1017/S0022112010001217.
Sirovich, L. 1987. “Turbulence and the dynamics of coherent structures. I. Coherent structures.” Q. Appl. Math. 45 (3): 561–571. https://doi.org/10.1090/qam/910462.
Solari, G. 2016. “Thunderstorm response spectrum technique: Theory and applications.” Eng. Struct. 108 (Feb): 28–46. https://doi.org/10.1016/j.engstruct.2015.11.012.
Solari, G., L. Carassale, and F. Tubino. 2007. “Proper orthogonal decomposition in wind engineering—Part 1: A state-of-the-art and some prospects.” Wind Struct. 10 (2): 153–176. https://doi.org/10.12989/was.2007.10.2.153.
Taira, K., S. L. Brunton, S. T. M. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, and L. S. Ukeiley. 2017. “Modal analysis of fluid flows: An overview.” AIAA J. 55 (12): 4013–4041. https://doi.org/10.2514/1.J056060.
Takens, F. 1981. “Detecting strange attractors in turbulence.” In Dynamical systems and turbulence, Warwick 1980, 366–381. New York: Springer.
Tamura, Y., S. Suganuma, H. Kikuchi, and K. Hibi. 1999. “Proper orthogonal decomposition of random wind pressure field.” J. Fluids Struct. 13 (7–8): 1069–1095. https://doi.org/10.1006/jfls.1999.0242.
Tanaka, H., Y. Tamura, K. Ohtake, M. Nakai, and Y. C. Kim. 2012. “Experimental investigation of aerodynamic forces and wind pressures acting on tall buildings with various unconventional configurations.” J. Wind Eng. Ind. Aerodyn. 107 (Aug): 179–191. https://doi.org/10.1016/j.jweia.2012.04.014.
Tennekes, H., and J. L. Lumley. 1972. A first course in turbulence. London: MIT Press.
Towne, A., O. T. Schmidt, and T. Colonius. 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.
TPU (Tokyo Polytechnic University). 2007. “Aerodynamic database: Wind pressure database based on wind tunnel experiment for high-rise building.” Accessed September 30, 2020. http://wind.arch.t-kougei.ac.jp/system/eng/contents/code/tpu.
Tu, J. H., C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz. 2013. “On dynamic mode decomposition: Theory and applications.” Preprint, submitted November 29, 2020. https://arxiv.org/abs/1312.0041.
Vickery, B. 1966. “Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream.” J. Fluid Mech. 25 (3): 481–494. https://doi.org/10.1017/S002211206600020X.
Williams, M. O., I. G. Kevrekidis, and C. W. Rowley. 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.
Yalla, S. K., and A. Kareem. 2001. “Beat phenomenon in combined structure-liquid damper systems.” Eng. Struct. 23 (6): 622–630. https://doi.org/10.1016/S0141-0296(00)00085-7.
Yin, C., X. Luo, and A. Kareem. 2020. “Probabilistic evolution of stochastic dynamical systems: A meso-scale perspective.” Preprint, submitted April 11, 2020. https://arxiv.org/abs/2004.06803.
Zhang, Q., Y. Liu, and S. Wang. 2014. “The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition.” J. Fluids Struct. 49 (Aug): 53–72. https://doi.org/10.1016/j.jfluidstructs.2014.04.002.
Zhao, Y., M. Zhao, X. Li, Z. Liu, and J. Du. 2019. “A modified proper orthogonal decomposition method for flow dynamic analysis.” Comput. Fluids 182 (Mar): 28–36. https://doi.org/10.1016/j.compfluid.2019.01.020.
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Received: Jul 12, 2020
Accepted: Nov 11, 2020
Published online: Jan 20, 2021
Published in print: Apr 1, 2021
Discussion open until: Jun 20, 2021
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