Monte Carlo-based a posteriori uncertainty quantification for background-oriented schlieren measurements

Abdelhafidh Moumen, Veronique De Briey, Oussama Atoui, Delphine Laboureur, Johan Gallant, Patrick Hendrick

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Background-oriented schlieren (BOS) technique is a density-based optical measurement technique. BOS measurement is similar to particle image velocimetry (PIV) ones in terms of experiments design and the computation of the displacements. However, in the BOS technique, the reconstruction of the refractive index field involves further mathematical calculations, which depend on the flow geometry, such as Poisson solver, Abel inversion, algebraic reconstruction technique, and filtered back-projection. This lengthy combination of experimental measurements, cross-correlation evaluation, and mathematical computation complicates the uncertainty quantification of the reconstructed field. In this study, we present a detailed approach for an a posteriori estimation of uncertainty when using BOS measurements to reconstruct the refractive index/density field. The proposed framework is based on the Monte Carlo simulation (MCS) method and can consider all kinds of sources of error, ranging from experimental measurements to those arising from image processing. The key features of this methodology are its capacity to handle different mathematical reconstruction procedures and the ease with which it can integrate additional sources of error. We demonstrate this method first by using synthetic images and a Poisson solver with mixed boundary conditions in a 2D domain. The accuracy of the proposed approach is assessed by comparing analytical and MCS results. Then, the modular nature of the proposed framework is experimentally demonstrated using a combination of Abel inversion and inverse gradient techniques to reconstruct a 3D axisymmetric density field around a transonic projectile in free-flight. The results are compared with computational fluid dynamics (CFD) and show high levels of agreement with only limited discrepancies, which are attributed to the space-filtering effect within cross-correlation resulting from shock waves.
Originele taal-2English
Pagina's (van-tot)945-965
Aantal pagina's20
Tijdschrift Journal of Visualization
Nummer van het tijdschrift5
StatusPublished - 6 apr 2022

Bibliografische nota

Azijli I, Sciacchitano A, Ragni D, Palha A, Dwight RP (2016) A posteriori uncertainty quantification of PIV-based pressure data. Exp Fluids 57(5):72, DOI 10.1007/s00348-016-2159-z

Bhattacharya S, Charonko JJ, Vlachos PP (2018) Particle image velocimetry (PIV) uncertainty quantification using moment of correlation (MC) plane. Meas Sci Technol 29(11):115301, DOI 10.1088/1361-6501/aadfb4

Boomsma A, Bhattacharya S, Troolin D, Pothos S, Vlachos P (2016) A comparative experimental evaluation of uncertainty estimation methods for two-component PIV. Meas Sci Technol 27(9):094006, DOI 10.1088/0957-0233/27/9/094006

Coleman HW, Steele WG (2009) Experimentation, Validation, and Uncertainty Analysis for Engineers, 3rd edn. John Wiley & Sons, Hoboken, N.J, oCLC: 310400087

Fomin NA (1998) Speckle Photography for Fluid Mechanics Measurements. Experimental Fluid Mechanics, Springer, Berlin, oCLC: 845092443

Guillaume G, Beaulieu C, Braud P, David L (2018) Démarche d’estimation des incertitudes en PIV basée sur la méthode GUM. In: CFTL-16, CNRS,IRSN, Sep 2018, Dourdan, France, p 10

Hargather MJ, Settles GS (2010) Natural-background-oriented schlieren imaging. Exp Fluids 48(1):59–68, DOI 10.1007/s00348-009-0709-3

Hartmann U, Seume JR (2016) Combining ART and FBP for improved fidelity of tomographic BOS. Meas Sci Technol 27(9):097001, DOI 10.1088/0957-0233/27/9/097001

International Organization for Standardization (1989) ISO 2768-1, General tolerances Part 1: Tolerances for linear and angular dimensions without individual tolerance indications. Tech. rep., ISO

JCGM (2008) BIPM - Guide to the Expression of Uncertainty in Measurement (GUM). Tech. rep., JCGM

JCGM J (2008) 101: 2008 evaluation of measurement data–supplement 1 to the “guide to the expression of uncertainty in measurement”–propagation of distributions using a monte carlo method. International
Organisation for Standardisation, Geneva

Kolhe PS, Agrawal AK (2009) Abel inversion of reflectometric data: Comparison of accuracy and noise propagation of existing techniques. Appl Opt 48(20):3894, DOI 10.1364/AO.48.003894

Meier GEA (1998) New optical tools for fluid mechanics. Sadhana 23(5-6):557–567, DOI 10.1007/BF02744579

Moumen A, Grossen J, Ndindabahizi I, Gallant J, Hendrick P (2020) Visualization and analysis of muzzle flow fields using the Background-Oriented Schlieren technique. J Vis pp 1–15, DOI 10.1007/s12650-020-00639-w

Pan Z, Whitehead J, Thomson S, Truscott T (2016) Error propagation dynamics of PIV-based pressure field calculations: How well does the pressure Poisson solver perform inherently Meas Sci Technol 27(8):084012, DOI 10.1088/0957-0233/27/8/084012

Pretzier G (1991) A New Method for Numerical Abel-Inversion. Zeitschrift fur Naturforschung A 46(7):639–641, DOI 10.1515/zna-1991-0715

Raffel M (2015) Background-oriented schlieren (BOS) techniques. Exp Fluids 56(3):60, DOI 10.1007/s00348-015-1927-5

Raffel M, Willert CE, Scarano F, K¨ahler CJ, Wereley ST, Kompenhans J (2018) Particle Image Velocimetry: A Practical Guide, third edition edn. Springer, Cham

Rajendran LK, Zhang J, Bhattacharya S, Bane SPM, Vlachos PP (2019) Uncertainty Quantification in density estimation from Background Oriented Schlieren (BOS) measurements. Meas Sci Technol (1):25, DOI 10.1088/1361-6501/ab60c8

Sciacchitano A (2019) Uncertainty quantification in particle image velocimetry. Meas Sci Technol 30(9):092001, DOI 10.1088/1361-6501/ab1db8

Sciacchitano A, Wieneke B (2016) PIV uncertainty propagation. Meas Sci Technol 27(8):084006, DOI 10.1088/0957-0233/27/8/084006

Sciacchitano A, Wieneke B, Scarano F (2013) PIV uncertainty quantification by image matching. Meas Sci Technol 24(4):045302, DOI 10.1088/0957-0233/24/4/045302

Sciacchitano A, Neal DR, Smith BL, Warner SO, Vlachos PP, Wieneke B, Scarano F (2015) Collaborative framework for PIV uncertainty quantification: Comparative assessment of methods. Meas Sci Technol 26(7):074004, DOI 10.1088/0957-0233/26/7/074004

Stein M (1987) Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics 29(2):143–151, DOI 10.2307/1269769

Stone JA, Zimmerman JH (2011) Engineering metrology toolbox. URL http://emtoolbox nistgov/Wavelength/Edlen asp 20:24

MC-based uncertainty quantification for BOS 25 Thielicke W, Stamhuis E (2014) PIVlab – Towards User-friendly, Affordable and Accurate Digital Particle
Image Velocimetry in MATLAB. J Open Res Softw 2(1):e30 DOI

Timmins BH, Wilson BW, Smith BL, Vlachos PP (2012) A method for automatic estimation of instantaneous local uncertainty in particle image velocimetry measurements. Exp Fluids 53(4):1133–1147, DOI 10.1007/s00348-012-1341-1

Vinnichenko NA, Uvarov AV, Plaksina YY (2012) Accuracy of background oriented schlieren for different background patterns and means of refraction index reconstruction. In: ISFV-15, Minsk, Belarus, p 15

Xue Z, Charonko JJ, Vlachos PP (2015) Particle image pattern mutual information and uncertainty estimation for particle image velocimetry. Meas Sci Technol 26(7):074001, DOI 10.1088/0957-0233/26/7/074001


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