Abstract
Proper orthogonal decomposition (POD) is used widely in experimental fluid dynamics for reducing noise in a measured flow field. The efficacy of POD is governed by the selection of modes used for the velocity field reconstruction. Currently, the determination of which or how many modes to keep is a user-defined subjective choice, where an arbitrary amount of energy to retain in the reconstruction, such as 99% cumulative energy, is chosen. Here, we present a novel, fully autonomous, and objective mode-selection method, which we term the entropy-line fit (ELF) method. The ELF method computes the Shannon entropy of the spatial discrete cosine transform of the eigenmodes, and using a two-line fit of the entropy mode spectrum, distinguishes between the modes carrying meaningful signal and those containing noise. We compare the ELF and existing methods using the analytical Hama flow field, synthetic PIV velocity fields derived from DNS turbulent channel flow data, and experimental particle image velocimetry vortex ring data. Overall, the ELF method improves the effectiveness of POD at removing noise from experimentally measured flow fields and subsequently the accuracy of post-processing calculations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abutaleb AS (1989) Automatic thresholding of gray-level pictures using two-dimensional entropy. Comput Vision, Graph Image Process 47:22–32. doi:10.1016/0734-189X(89)90051-0
Adrian RJ, Christensen KT, Liu Z-C (2000) Analysis and interpretation of instantaneous turbulent velocity fields. Exp Fluids 29:275–290. doi:10.1007/s003489900087
Ahmed N, Natarajan T, Rao KR (1974) Discrete Cosine Transform. IEEE Trans Comput 90–93.
Algazi VR, Sakrison DJ (1969) A note on the optimality of the Karhunen-Loeve expansion. IEEE Trans Inf Theory 15:319–321. doi:10.1016/0167-8655(83)90025-9
Aubry N (1991) On the hidden beauty of the proper orthogonal decomposition. Theor Comput Fluid Dyn 2:339–352. doi:10.1007/BF00271473
Aubry N, Guyonnet R, Lima R (1991) Spatiotemporal analysis of complex signals: theory and applications. J Stat Phys 64:683–739
Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25:539–575. doi:10.1146/annurev.fl.25.010193.002543
Brindise MC, Chiastra C, Burzotta F, et al (2016) hemodynamics of stent implantation procedures in coronary bifurcations: an in vitro study. Ann Biomed Eng 1–33. doi:10.1007/s10439-016-1699-y
Bulusu K, Plesniak M (2015) Shannon entropy-based wavelet transform method for autonomous coherent structure identification in fluid flow field data. Entropy 17:6617–6642. doi:10.3390/e17106617
Charonko JJ, Vlachos PP (2013) Estimation of uncertainty bounds for individual particle image velocimetry measurements from cross-correlation peak ratio. Meas Sci Technol 24:065301. doi:10.1088/0957-0233/24/6/065301
Charonko J, Karri S, Schmieg J et al (2010a) In vitro comparison of the effect of stent configuration on wall shear stress using time-resolved particle image velocimetry. Ann Biomed Eng 38:889–902. doi:10.1007/s10439-010-9915-7
Charonko JJ, King CV, Smith BL, Vlachos PP (2010b) Assessment of pressure field calculations from particle image velocimetry measurements. Meas Sci Technol 21:105401. doi:10.1088/0957-0233/21/10/105401
Charonko JJ, Kumar R, Stewart K et al (2013) Vortices formed on the mitral valve tips aid normal left ventricular filling. Ann Biomed Eng 41:1049–1061. doi:10.1007/s10439-013-0755-0
Chen WH, Pratt WK (1984) Scene Adaptive Coder. IEEE Trans Commun 32:225–232. doi:10.1109/TCOM.1984.1096066
Didden N (1979) On the formation of vortex rings: rolling up and production of circulation. J Appl Math Phys 30:101–116
Eckstein A, Vlachos PP (2009a) Digital particle image velocimetry (DPIV) robust phase correlation. Meas Sci Technol 20:055401. doi:10.1088/0957-0233/20/5/055401
Eckstein A, Vlachos PP (2009b) Assessment of advanced windowing techniques for digital particle image velocimetry (DPIV). Meas Sci Technol 20:075402. doi:10.1088/0957-0233/20/7/075402
Eckstein AC, Charonko J, Vlachos P (2008) Phase correlation processing for DPIV measurements. Exp Fluids 45:485–500. doi:10.1007/s00348-008-0492-6
Epps BP, Techet AH (2010) An error threshold criterion for singular value decomposition modes extracted from PIV data. Exp Fluids 48:355–367. doi:10.1007/s00348-009-0740-4
Etebari A, Vlachos PP (2005) Improvements on the accuracy of derivative estimation from DPIV velocity measurements. Exp Fluids 39:1040–1050. doi:10.1007/s00348-005-0037-1
Gharib M, Rambod E, Shariff K (1998) A universal time scale for vortex ring formation. J Fluid Mech 360:121–140. doi:10.1017/S0022112097008410
Graham J, Lee M, Malaya N, et al (2013) Turbulent channel flow data set. 1–8.
Hama FR (1962) Streaklines in a perturbed shear flow. Phys Fluids 5:644. doi:10.1063/1.1706678
Holmes P, Lumley JL, Berkooz G, Rowley CW (1996) Turbulence, coherent structures, dynamical systems and symmetry
Jørgensen BH (1999) Application of POD to PIV images of flow over a wall mounted fence. IUTAM Symp Simul Identif Organ Struct Flows 52:397–407. doi:10.1007/978-94-011-4601-2_35
Karri S, Charonko J, Vlachos PP (2009) Robust wall gradient estimation using radial basis functions and proper orthogonal decomposition (POD) for particle image velocimetry (PIV) measured fields. Meas Sci Technol 20:045401. doi:10.1088/0957-0233/20/4/045401
Kostas J, Soria J, Chong MS (2005) A comparison between snapshot POD analysis of PIV velocity and vorticity data. Exp Fluids 38:146–160. doi:10.1007/s00348-004-0873-4
Krumm J (2001) Savitzky-Golay filters for 2D images.
Li Y, Perlman E, Wan M, et al (2008) A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J Turbul 9:N31. doi:10.1080/14685240802376389
Müller SHR, Böhm B, Gleißner M et al (2010) Flow field measurements in an optically accessible, direct-injection spray-guided internal combustion engine using high-speed PIV. Exp Fluids 48:281–290. doi:10.1007/s00348-009-0742-2
Oxlade AR, Valente PC, Ganapathisubramani B, Morrison JF (2012) Denoising of time-resolved PIV for accurate measurement of turbulence spectra and reduced error in derivatives. Exp Fluids 53:1561–1575. doi:10.1007/s00348-012-1375-4
Perlman E, Burns R, Li Y, Meneveau C (2007) Data exploration of turbulence simulations using a database cluster. Proc 2007 ACM/IEEE Conf Supercomput (SC’07). doi:10.1145/1362622.1362654
Raiola M, Discetti S (2016) Personal Communication
Raiola M, Discetti S, Ianiro A (2015) On PIV random error minimization with optimal POD-based low-order reconstruction. Exp Fluids. doi:10.1007/s00348-015-1940-8
Savitzky A, Golay MJE (1964) Smoothing and differentiation of data by simplified least squares procedures. Anal Chem 36:1627–1639. doi:10.1021/ac60214a047
Scarano F (2002) Iterative image deformation methods in PIV. Meas Sci Technol 13:R1–R19. doi:10.1088/0957-0233/13/1/201
Schanz D, Gesemann S, Schröder A (2016) Shake-The-Box: Lagrangian particle tracking at high particle image densities. Exp Fluids. doi:10.1007/s00348-016-2157-1
Shariff K, Leonard A (1992) Vortex rings. Annu Rev Fluid Mech 24:235–279
Siegel SG, Seidel J, Fagley C, et al (2008) Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition
Sirovich L (1987) Turbulence and the Dynamics of Coherent Structures, I-III. Q Appl Math 45:561–571
Smith TR, Moehlis J, Holmes P (2005) Low-dimensional modelling of turbulence using the proper orthogonal decomposition: a tutorial. Nonlinear Dyn 41:275–307. doi:10.1007/s11071-005-2823-y
Soon IY, Koh SN, Yeo CK (1998) Noisy speech enhancement using discrete cosine transform. Speech Commun 24:249–257. doi:Doi:10.1016/S0167-6393(98)00019–3
Stewart KC, Vlachos PP (2012) Vortex rings in radially confined domains. Exp Fluids 53:1033–1044. doi:10.1007/s00348-012-1343-z
Tropea C, Yarin AL, Foss JF (2007) Handbook of experimental fluid mechanics
Walpot RJE, van der Geld CWM, Kuerten JGM (2007) Determination of the coefficients of Langevin models for inhomogeneous turbulent flows by three-dimensional particle tracking velocimetry and direct numerical simulation. Phys Fluids. doi:10.1063/1.2717688
Westerweel J, Scarano F (2005) Universal outlier detection for PIV data. Exp Fluids 39:1096–1100. doi:10.1007/s00348-005-0016-6
Xue Z, Charonko JJ, Vlachos PP (2014) Particle image velocimetry correlation signal-to-noise ratio metrics and measurement uncertainty quantification. Meas Sci Technol 25:115301. doi:10.1088/0957-0233/25/11/115301
Yu H, Kanov K, Perlman E, et al (2012) Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database. J Turbul 13:N12. doi:10.1080/14685248.2012.674643
Zhou J, Adrian RJ, Balachandal S, Kendall TM (1999) Mechanisms for generating coherent packets of hairpin vortices in channel flow. J Fluid Mech 387:S002211209900467X. doi:10.1017/S002211209900467X
Acknowledgements
The authors would like to thank Marco Raiola and Stefano Discetti for providing the synthetic PIV data and their help ensuring their method was properly implemented for this work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brindise, M.C., Vlachos, P.P. Proper orthogonal decomposition truncation method for data denoising and order reduction. Exp Fluids 58, 28 (2017). https://doi.org/10.1007/s00348-017-2320-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00348-017-2320-3