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

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Abstract

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.

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Abbreviations

A, B :

Uncertainty evaluation type

\(c_{50}\) :

Covariance factor (overlap = \(50\%\))

\(e_{ruler}\) :

Ruler fabrication error (mm)

\(e_{\varDelta y}\) :

Total error in \(\varDelta y\) computation (pixel)

F :

Refractive index reconstruction function

f :

Lens focal length (mm)

G :

Gladstone–Dale constant (\(m^3\)/kg)

h :

Ambient humidity (%)

k :

Coverage factor (%)

L :

Length read on the ruler image (mm)

M :

Background image magnification (mm/pixel)

\(\mathrm{Ma}\) :

Mach number

\(M_\mathrm{obj}\) :

Image magnification factor in the object plane (mm/pixel)

\(N_\mathrm{eff}\) :

Effective number of pixels (pixels)

\(N_\mathrm{s}\) :

Number of samples

\(N_\mathrm{T}\) :

Total number of matrix elements

n :

Refractive index

\(n_0\) :

Reference refractive index

\(n_n\) :

Normalized refractive index difference.

p :

Ambient pressure (kPa)

pix :

Number of pixels that corresponds to L (pixel)

r :

Radial coordinate (pixel)

s :

Grid size (pixel)

S :

Poisson equation source term

\(\sigma\) :

Distribution standard deviation

T :

Ambient temperature (C)

u :

Standard uncertainty

\(u_\mathrm{c}\) :

Combined uncertainty

\(U_\%\) :

Extended uncertainty

\(\chi _\bot\) :

Component \(\in (x,y)\) normal to the boundary

v :

True assigned displacement in y-direction (pixel)

\(Z_\mathrm{B}\) :

Distance object-background (m)

\(Z_\mathrm{T}\) :

Distance camera-background (m)

\(Z_\mathrm{W}\) :

Flow thickness (m)

\(\varDelta x,y\) :

Displacements in the image plane (pixel)

\(\varDelta x',y'\) :

Displacements in the background plane (pixel)

\(\delta\) :

Elemental error

\(\epsilon _{x,y}\) :

Light deflection angle in x,y-direction (rad)

\(\lambda\) :

Wavelength of light (m)

\(\rho\) :

Density (kg/\(m^3\))

\(\tau\) :

Background element particle image size (pixel)

\({\mathcal {V}}\) :

Extended vector

\(\varOmega\) :

Real but unknown quantity value

\(\omega\) :

Elemental measurement

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Authors and Affiliations

  1. Department of Weapon systems and Ballistics, Royal Military Academy, Avenue de la Renaissance 30, 1000, Brussels, Belgium

    Abdelhafidh Moumen, Véronique de Briey, Oussama Atoui & Johan Gallant

  2. Laboratory of Aero-Thermo-Mechanics, Université Libre de Bruxelles, Avenue F.D. Roosevelt 50 (CP 165/41), 1050, Brussels, Belgium

    Abdelhafidh Moumen & Patrick Hendrick

  3. Environmental and Applied Fluid Dynamics Department, von Karman Institute for fluid dynamics, Chaussée de Waterloo 72, 1640, Rhode-Saint-Genèse, Belgium

    Abdelhafidh Moumen & Delphine Laboureur

Authors
  1. Abdelhafidh Moumen
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  2. Véronique de Briey
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  3. Oussama Atoui
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  4. Delphine Laboureur
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  5. Johan Gallant
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  6. Patrick Hendrick
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Correspondence to Abdelhafidh Moumen.

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Moumen, A., de Briey, V., Atoui, O. et al. Monte Carlo-based a posteriori uncertainty quantification for background-oriented schlieren measurements. J Vis 25, 945–965 (2022). https://doi.org/10.1007/s12650-022-00838-7

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  • DOI: https://doi.org/10.1007/s12650-022-00838-7

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