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Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments

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Abstract

We theoretically analyze the effect of density/refractive index gradients on the measurement precision of background-oriented schlieren (BOS) experiments by deriving the Cramer–Rao lower bound (CRLB) for the 2D centroid estimation process. A model is derived for the diffraction limited image of a dot viewed through a medium containing density gradients that includes the effect of the experimental parameters such as the magnification and f-number. It is shown using the model that nonlinearities in the density gradient field lead to blurring of the dot image. This blurring amplifies the effect of image noise on the centroid estimation process, leading to an increase in the CRLB and a decrease in the measurement precision. The ratio of position uncertainties of a dot in the reference and gradient images is shown to be a function of the ratio of the dot diameters and dot intensities. We termed this parameter the amplification ratio (\(A_{\text{F}}\)), and a methodology for reporting position uncertainties in tracking-based BOS measurements is proposed. The theoretical predictions of the dot position estimation variance from the CRLB are compared to ray tracing simulations, and agreement is obtained. The uncertainty amplification is also demonstrated on experimental BOS images of flow induced by a spark discharge, where it is seen that regions of high amplification ratio correspond to regions of density gradients. This analysis elucidates the dependence of the position uncertainty on density and refractive index gradient-induced distortion parameters, provides a methodology for accounting its effect on uncertainty quantification and provides a framework for optimizing experiment design.

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Abbreviations

\(\alpha\) :

Image exposure

\(\beta\) :

Blurring coefficient

\(\gamma\) :

Gray value per unit exposure

\(\delta\) :

Dirac delta function

\(\Delta \theta_{0}\) :

Angle of the ray cone

\(\epsilon\) :

Angular deflection

\(\zeta\) :

Tangential coordinate of the light ray trajectory

\(\eta\) :

Standard deviation of the Gaussian intensity profile on the image plane

\(\theta\) :

Light ray angle

\(\lambda\) :

Wavelength of light

\(\rho\) :

Density

\(\sigma\) :

Standard deviation

\(\tau\) :

Point spread function

\(\varvec{a}\) :

Model parameter vector

\({\text{AR}}\) :

Amplification ratio

\(d_{r}\) :

Pixel pitch

\(\varvec{E}\) :

Expectation operator

\(f\) :

Focal length of the camera lens

\(f_{\# }\) :

F-number

\(g\) :

Image gray level or signal

\(I\) :

Image intensity

\(I_{0,r}\) :

Peak image intensity for a single light ray

\(J_{ij}\) :

Fisher information matrix

\(K\) :

Gladstone–Dale constant

\(k,l\) :

Pixel indices

\(m\) :

Model

\(M\) :

Magnification

\(\hat{n}\) :

Thermal noise

\(n_{0}\) :

Ambient refractive index

\(\varvec{N}\) :

Normal distribution

\(N_{\text{R}}\) :

Number of light rays

\(p\) :

Probability density function (PDF)

\(r\) :

Light ray index

\(t\) :

Spatial coordinate on the density field

\(x,y,z\) :

Coordinates in the object space

\(X,Y\) :

Spatial coordinates on the camera sensor/image space

\(z_{\text{D}}\) :

Distance from dot pattern to camera lens

\(z_{\text{D}}\) :

Distance from dot pattern to density gradient field

\(\Delta z\) :

Thickness of the density gradient field

References

  • Adrian RJ, Yao CS (1985) Pulsed laser technique application to liquid and gaseous flows and the scattering power of seed materials. Appl Opt 24(1):44–52

    Google Scholar 

  • Anderson JD (2004) Fundamentals of aerodynamics, 4th edn. McGraw-Hill, New York

    Google Scholar 

  • Bhattacharya S, Vlachos PP (2019) Uncertainty quantification in volumetric PTV. In: 13th international symposium on particle image velocimetry

  • Brady MR, Raben SG, Vlachos PP (2009) Methods for Digital Particle Image Sizing (DPIS): comparisons and improvements. Flow Meas Instrum 20(6):207–219

    Google Scholar 

  • Charonko JJ, Vlachos PP (2013) Estimation of uncertainty bounds for individual particle image velocimetry measurements from cross-correlation peak ratio. Meas Sci Technol 24(6):065301

    Google Scholar 

  • Clem MM, Brown CA, Fagan AF (2013) Background Oriented Schlieren implementation in a jet-surface interaction test. In: 51st AIAA Aerosp Sci Meet Incl New Horizons Forum Aerosp Expo

  • Dalziel SB, Hughes GO, Sutherland BR (2000) Whole-field density measurements by ‘synthetic schlieren’. Exp Fluids 28(4):322–335

    Google Scholar 

  • Elsinga GE, Orlicz GC (2015) Particle imaging through planar shock waves and associated velocimetry errors. Exp Fluids 56(6):129

    Google Scholar 

  • Elsinga GE, Van Oudheusden BW, Scarano F (2005a) Evaluation of aero-optical distortion effects in PIV. Exp Fluids 39(2):246–256

    Google Scholar 

  • Elsinga GE, van Oudheusden BW, Scarano F (2005b) Modeling of particle imaging through shock waves. Opt Model Perform Predict 5867:58670M–58670M11

    Google Scholar 

  • Elsinga GE, van Oudheusden BW, Scarano F (2005c) The effect of particle image blur on the correlation map and velocity measurement in PIV. In: Proceedings of SPIE, vol 5880, p 588010

  • Goldhahn E, Seume J (2007) The background oriented schlieren technique: sensitivity, accuracy, resolution and application to a three-dimensional density field. Exp Fluids 43(2–3):241–249

    Google Scholar 

  • Meier G (2002) Computerized background-oriented schlieren. Exp Fluids 33(1):181–187

    Google Scholar 

  • Nicolas F et al (2016) A direct approach for instantaneous 3D density field reconstruction from background-oriented schlieren (BOS) measurements. Exp Fluids 57(1):1–21

    MathSciNet  Google Scholar 

  • Nobach H, Tropea C (2007) Fundamentals of data processing. Springer Handbook of Experimental Fluid Mechanics. Springer, Berlin, pp 1399–1417

    Google Scholar 

  • Raffel M (2015) Background-oriented schlieren (BOS) techniques. Exp Fluids 56(3):1–17

    Google Scholar 

  • Rajendran LK, Bane SPM, Vlachos PP (2019a) Dot tracking methodology for background-oriented schlieren (BOS). Exp Fluids 60(11):162

    Google Scholar 

  • Rajendran LK, Zhang J, Bhattacharya S, Bane S, Vlachos P (2019b) Uncertainty quantification in density estimation from background oriented schlieren (BOS) measurements. Meas Sci Technol 31(5):054002

    Google Scholar 

  • Rajendran LK, Bane SPM, Vlachos PP (2019c) PIV/BOS synthetic image generation in variable density environments for error analysis and experiment design. Meas Sci Technol 30(8):085302

    Google Scholar 

  • Richard H, Raffel M (2001) Principle and applications of the background oriented schlieren (BOS) method. Meas Sci Technol 12(12):1576–1585

    Google Scholar 

  • Singh B, Rajendran LK, Gupta P, Scalo C, Vlachos PP, Bane SPM (2019a) Experimental and numerical study of flow induced by nanosecond repetitively pulsed discharges. In: AIAA Scitech 2019 Forum

  • Singh B, Rajendran LK, Vlachos PP, Bane SPM (2019b) Two regime cooling in flow induced by a spark discharge. Phys Rev Fluids 5(1):014501

    Google Scholar 

  • Singh B, Rajendran LK, Vlachos P, Bane SPM (2020) Study of cooling and the effect of energy deposited in a single nanosecond spark plasma discharge using simultaneous 50 kHz PIV and BOS. In: AIAA Scitech 2020 Forum

  • Sourgen F, Leopold F, Klatt D (2012) Reconstruction of the density field using the Colored Background Oriented Schlieren Technique (CBOS). Opt Lasers Eng 50(1):29–38

    Google Scholar 

  • van den Bos A (2007) Parameter estimation for scientists and engineers. Wiley, New York

    MATH  Google Scholar 

  • Wernet MP, Pline A (1993) Particle displacement tracking technique and Cramer–Rao lower bound error in centroid estimates from CCD imagery. Exp Fluids 15(4–5):295–307

    Google Scholar 

  • Westerweel J (2000) Theoretical analysis of the measurement precision in particle image velocimetry. Exp Fluids 29(7):S003–S012

    Google Scholar 

  • Willert M, Gharib CE (1991) Digital particle image velocimetry experiments. Exp Fluids 10:181–193

    Google Scholar 

  • Xue Z, Charonko JJ, Vlachos PP (2014) Particle image velocimetry correlation signal-to-noise ratio metrics and measurement uncertainty quantification. Meas Sci Technol 25(11):115301

    Google Scholar 

  • 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

    Google Scholar 

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Acknowledgements

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award Number DE-SC0018156. Bhavini Singh is acknowledged for help with the spark discharge experiment. Jiacheng Zhang, Javad Eshraghi and Adib Ahmadzadegan are acknowledged for reviewing the manuscript drafts. This work was also supported by NSF 1706474.

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Authors and Affiliations

  1. School of Aeronautics and Astronautics, Purdue University, West Lafayette, USA

    Lalit K. Rajendran & Sally P. M. Bane

  2. School of Mechanical Engineering, Purdue University, West Lafayette, USA

    Pavlos P. Vlachos

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  1. Lalit K. Rajendran
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  2. Sally P. M. Bane
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  3. Pavlos P. Vlachos
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Correspondence to Pavlos P. Vlachos.

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Rajendran, L.K., Bane, S.P.M. & Vlachos, P.P. Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments. Exp Fluids 61, 139 (2020). https://doi.org/10.1007/s00348-020-02978-8

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