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Kriging regression of PIV data using a local error estimate

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Abstract

The objective of the method described in this work is to provide an improved reconstruction of an original flow field from experimental velocity data obtained with particle image velocimetry (PIV) technique, by incorporating the local accuracy of the PIV data. The postprocessing method we propose is Kriging regression using a local error estimate (Kriging LE). In Kriging LE, each velocity vector must be accompanied by an estimated measurement uncertainty. The performance of Kriging LE is first tested on synthetically generated PIV images of a two-dimensional flow of four counter-rotating vortices with various seeding and illumination conditions. Kriging LE is found to increase the accuracy of interpolation to a finer grid dramatically at severe reflection and low seeding conditions. We subsequently apply Kriging LE for spatial regression of stereo-PIV data to reconstruct the three-dimensional wake of a flapping-wing micro air vehicle. By qualitatively comparing the large-scale vortical structures, we show that Kriging LE performs better than cubic spline interpolation. By quantitatively comparing the interpolated vorticity to unused measurement data at intermediate planes, we show that Kriging LE outperforms conventional Kriging as well as cubic spline interpolation.

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Notes

  1. The error matrix is not a ‘nugget effect’, as it does not directly change the correlation matrix and does not represent a cascade of unresolved smaller flowscales [36].

  2. To avoid negative or infinitely large \(\epsilon\), we implement: \(\epsilon_{{\textsc{snr}},i}=\frac{c}{\max(\rm{SNR}_i-SNR_{\rm min},0.01)}\).

  3. A MLE optimization of the Kriging correlation ranges typically results in θ x  = θ y  = 0.26, and in the following, we use a fixed range of θ x  = θ y  = 0.25.

  4. One might conclude that the Low-Fi UM slightly underestimates the measurement uncertainty for high PMR. This conclusion could be attributed to the extremely simple Low-Fi UM; a slightly more sophisticated Low-Fi UM might incorporate the uncertainty associated with the subpixel fit as a lower uncertainty bound–which would roughly amount to \(\epsilon_{\rm subpix} \approx 0.1 {\hbox{m/s}}\) in the present case.

  5. Unlike in mpiv , PPR is defined in Davis.

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Acknowledgments

We are very grateful for the financial support by Technologiestichting STW, project number 10113 and 11023.

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Authors and Affiliations

  1. TU Delft, Kluyverweg 1, 2629 HS, Delft, The Netherlands

    Jouke H. S. de Baar, Mustafa Percin, Richard P. Dwight, Bas W. van Oudheusden & Hester Bijl

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  1. Jouke H. S. de Baar
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  2. Mustafa Percin
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  3. Richard P. Dwight
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  4. Bas W. van Oudheusden
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  5. Hester Bijl
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Correspondence to Jouke H. S. de Baar.

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Appendix: Detailed mpiv settings

Appendix: Detailed mpiv settings

For the cross-correlation of the synthetic images in Sect. 3, we use the following general settings: nx_window = 32; ny_window = 32; overlap_x = 0; overlap_y = 0; iu_max = 0; iv_max = 0; dt = 0.01; piv_type = ’cor’; i_recur = 2; i_plot = 0; By default mpiv assigns NaN to vectors with a SNR or PPR below 3.00, however, we want to control the SNR min top-level. Therefore, in the files piv_cor.m and piv_crs.m , we change these settings: r_SNR = 0.00; r_PPR = 0.00.

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de Baar, J.H.S., Percin, M., Dwight, R.P. et al. Kriging regression of PIV data using a local error estimate. Exp Fluids 55, 1650 (2014). https://doi.org/10.1007/s00348-013-1650-z

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