Proper Orthogonal Decomposition Analysis of Turbulent Energy Spectra within a Lagrangian Particle Tracking Framework

Articles in Press

Document Type : Research Paper

Author
Reza Hassanian
Faculty of Industrial Engineering, Mechanical Engineering and Computer Science, University of Iceland, Reykjavik, 102, Iceland
10.22055/jacm.2025.48733.5459
Abstract
This paper proposes an approach that applies Proper Orthogonal Decomposition (POD) analysis to experimental data obtained using the Lagrangian Particle Tracking (LPT) technique from the original experimental work [1]. Based on the experiment description, it was conducted under two flow conditions: (a) nearly Homogeneous Isotropic Turbulence (HIT) with a Taylor microscale Reynolds number in the range 100 < Reλ < 120, and (b) initially HIT flow subjected to strain deformation in the Y-direction with a mean strain rate of 8 s-1. The LPT-based POD analysis, using individual particle velocity data, yields several significant findings. Constructing the data matrix from the velocity fields of individual particles introduces a novel aspect to the analysis. POD is employed to uncover the effects of strain deformation on the energy spectra in contrast to the HIT flow. The energy spectra of flows with varying turbulence intensities and subjected to strain deformation illustrate how these factors influence the overall energy distribution. Furthermore, spectral analysis of turbulent structures shows that power-law behavior in the energy spectrum can be captured by POD analysis. Strain deformation alters the energy contributions of the POD modes, leading to a significantly increased concentration of energy in the first few dominant modes. The LPT-based POD exhibits significantly low computational cost per realization.
Keywords
Turbulent flow
Proper orthogonal decomposition
Energy spectra
Coherent structures
Computation
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Hassanian, R., Helgadottir, A., Bouhlali, L., Riedel, M., An experiment generates a specified mean strained rate turbulent flow: Dynamics of particles, Physics of Fluids, 35(1), 2023, 015124.
[2] Tennekes, H., Lumley, J. L., A First Course in Turbulence, MIT Press, Massachusetts, 1972.
[3] Davidson, P. A., Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, London, 2004.
[4] Berkooz, G, Holmes, P., Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annual Review of Fluid Mechanics, 25, 1993, 539.
[5] Batchelor, G. K., The Theory of Homogeneous Turbulence, Cambridge University Press, London, 1982.
[6] Pope, S. B., Turbulent Flows, Cambridge University Press, London, 2000.
[7] Jackson, D., Launder, B., Osborne Reynolds and the publication of his papers on turbulent flow, Annual Review of Fluid Mechanics, 39, 2007, 19. 
[8] Versteeg, V., An Introduction to Computational Fluid Dynamics, Pearson Education, London, 2007.
[9] Courant, R., Friedrichs, K., Lewy, H., Über die partiellen differenzengleichungen der mathematischen physik, Mathematische Annalen, 100, 1928, 32.
[10] Thom A., Taylor, G. I., The flow past circular cylinders at low speeds, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 141(845), 1993, 651.
[11] Hassanian, R., Myneni, H., Helgadottir, A., Riedel, M. Deciphering the dynamics of distorted turbulent flows: Lagrangian particle tracking and chaos prediction through transformer-based deep learning models, Physics of Fluids, 35(7), 2023, 075118.
[12] Hassanian, R., Aach, M., Lintermann, A., Helgadóttir, A., Riedel, M., Turbulent flow prediction-simulation: Strained flow with initial isotropic condition using a GRU model trained by an experimental Lagrangian framework, with emphasis on hyperparameter optimization, Fluids, 9(4), 2024, 84.
[13] Sathyanarayana, S., Bernardini, M., Modesti, D., Pirozzoli, S., Salvadore, F., High-speed turbulent flows towards the exascale: Streams-2 porting and performance, Journal of Parallel and Distributed Computing, 196, 2025, 104993.
[14] Hassanian, R., Helgadottir, A., Riedel, M., Deep learning forecasts a strained turbulent flow velocity field in temporal Lagrangian framework: Comparison of LSTM and GRU, Fluids, 7(11), 2022, 344.
[15] Hassanian, R., Helgadóttir, A., Gharibi, F., Beck, A., Riedel, M., Data-driven deep learning models in particle-laden turbulent flow, Physics of Fluids, 37(2), 2025, 023348.
[16] Hassanian, R., Riedel, M., Bouhlali, L., The capability of recurrent neural networks to predict turbulence flow via spatiotemporal features, In 10th IEEE Jubilee International Conference on Computational Cybernetics and Cyber-Medical Systems (ICCC), 2022, 000335–000338.
[17] Olesen, P. J., Hodžic, A., Andersen, S. J., Sørensen, N. N., Velte, S. M., Dissipation optimized proper orthogonal decomposition, Physics of Fluids, 35(1), 2023, 015131.
[18] Olesen, P. J., Soucasse, L., Podvin, B., Velte, C. M., Dissipation-based proper orthogonal decomposition of turbulent Rayleigh–Bénard convection flow, Physics of Fluids, 36(3), 2024, 035109.
[19] Hang, Y., Hodžic, A., Evrard, F., van Wachem, B., Velte, C. M., Phase proper orthogonal decomposition of non-stationary turbulent flow, Physics of Fluids, 35(4), 2023, 045109.
[20] Rovira, M., Engvall, K., Duwig, C., Proper orthogonal decomposition analysis of the large-scale dynamics of a round turbulent jet in counterflow, Physical Review Fluids, 6(1), 2021, 014701
[21] Sirovich, L., Turbulence and the dynamics of coherent structures. Part I: Coherent structures, Quarterly of Applied Mathematics, 45(3), 1987, 561.
[22] Podvina, B., Fraigneau, Y., A few thoughts on proper orthogonal decomposition in turbulence, Physics of Fluids, 29(2), 2017, 020709.
[23] Ouellette, N. T., Xu, H., Bodenschatz, E., A quantitative study of three-dimensional Lagrangian particle tracking algorithms, Experiments in Fluids, 40(2), 2006, 301.
[24] Kosambi, D. D., Statistics in Function Space, Springer India, New Delhi, 2016.
[25] Narasimha, R., Kosambi and proper orthogonal decomposition, Resonance, 16(6), 2011, 574.
[26] Borra, A., Flynn, Z., Goza, A., Saxton-Fox, T., Intrinsic phase-based proper orthogonal decomposition (iphab pod): a method for physically interpretable modes in near-periodic systems, Journal of Fluid Mechanics, 1002, 2025, A18.
[27] Dawson, S., The Proper Orthogonal Decomposition, Cambridge University Press, 2023.
[28] Chatterjee, A., An introduction to the proper orthogonal decomposition, Current Science, 78(7), 2000, 808. 
[29] Weiss, J., A tutorial on the proper orthogonal decomposition, In 2019 AIAA Aviation Forum, 17–21 June, Dallas, Texas, United States, American Institute of Aeronautics and Astronautics, 2019
[30] Hassanian, R., Riedel, M., Leading-edge erosion and floating particles: Stagnation point simulation in particle-laden turbulent flow via Lagrangian particle tracking, Machines, 11(5), 2023, 566.
[31] Sirovich, L., Ball, K. S., Keefe, L. R., Plane waves and structures in turbulent channel flow, Physics of Fluids, 2(12), 1990, 2217–2226.
[32] Moin, P., Moser, R. D., Characteristic-eddy decomposition of turbulence in a channel, Journal of Fluid Mechanics, 200, 1989, 471–509.
[33] Ball, K. S., Sirovich, L., Keefe, L. R., Dynamical eigenfunction decomposition of turbulent channel flow, International Journal for Numerical Methods in Fluids, 12, 1991, 585–604.
[34] Osman, H., Wang, W., Ikegaya, N., POD dimensional reduction for preserving turbulence statistics in urban environment, Building and Environment, 287, 2026, 113927.
[35] Schiødt, M., Hodžic, A., Evrard, F., Hausmann, M., Van Wachem, B., Velte, C. M., Characterizing Lagrangian particle dynamics in decaying homogeneous isotropic turbulence using proper orthogonal decomposition, Physics of Fluids, 34(6), 2022, 063303.
[36] Stewart, G. W., On the early history of the singular value decomposition, SIAM Review, 35, 1993, 551.
[37] SEN, M., Bhaganagar, K., Juttijudata, V., Application of proper orthogonal decomposition (POD) to investigate a turbulent boundary layer in a channel with rough walls, Journal of Turbulence, 8, 2007, https://doi.org/10.1080/14685240701615960.
[38] Lumely, J. L., Similarity and the turbulent energy spectrum, Physics of Fluids, 10, 1967a, 855-858.
[39] Hodžic, A., Meyer, K. E., George, W. K., Velte, C. M., Lumley Decomposition of the Turbulent Round Jet Far-field. Part 2 – Dynamics, arXiv preprint arXiv:1909.01307, 2019.
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Available Online from 05 December 2025