## Abstract

A single-camera color particle image velocimetry (PIV) system that can acquire the PIV data of three separated layers has been redesigned to make it more suitable for wind tunnel applications. We target smoke images that have particle-per-pixel values higher than unity. The system consists of a high-power color-coding illuminator and a digital color high-speed video camera. RGB values in the recorded image include severe color contaminations due to five optical and digital sequences. To quantify this, a snapshot calibration method is proposed to describe the contamination matrix equation. Taking the inverse matrix allows the in-plane PIV in each color layer to be accurately implemented. We also derive the mathematical limits in the operation of colored smoke PIV, which are explained by the matrix properties. The feasibility of the proposed method was demonstrated by application to a turbulent wake behind a delta wing at a quasi-stall angle of attack.

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## Acknowledgements

This work was financially supported by the *f*3-engineering Center, Hokkaido University. The authors express thanks to Norio Yonezawa, Yuhei Sugano, Kento Yamada, Daisuke Noto, Hokkaido University, and Yoshihiko Oishi, Muroran Institute of Technology, for their technical support.

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## Appendix

### Appendix

The linearity of the color sensing between the three-color LCDP and RGB color camera is mathematically explained below. Because the color intensities for the LCDP are set as *R*_{0}, *G*_{0}, and *B*_{0}, the total light spectrum emitted by the LDCP is expressed as

where *l* is the wavelength of light. In addition, *r*(l), *g*(l), and *b*(l) are normalized spectra that depend on the molecular filters adapted in the LCDP. A color camera receives the light of Eq. (33) with different spectral sensitivity, *r’*(l), *g’*(l), and *b’*(l), so that the camera obtains RGB components as

Substituting Eq. (33) into Eq. (34) yields

where the three integrals in Eq. (37) are given as constant values and rewritten as

This proves that the brightness *R*_{1} recorded by the camera is always given by a linear sum of *R*_{0}, *B*_{0}, and *G*_{0}. The linearity is conserved, regardless of the combination of the two spectra. In the same way, *B*_{1} and *G*_{1} are also given linearly with respect to *R*_{0}, *B*_{0}, and *G*_{0} with different proportional constants, as follows.

Summarizing these relations, the linear algebraic description is expressed as follows:

Note that Eq. (1) in Sect. 2.2 is given by normalizing the matrix constant with its diagonal component and by considering nonzero background brightness.

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Murai, Y., Yumoto, T., Park, H.J. *et al.* Color-coded smoke PIV for wind tunnel experiments improved by eliminating optical and digital color contamination.
*Exp Fluids* **62**, 231 (2021). https://doi.org/10.1007/s00348-021-03315-3

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DOI: https://doi.org/10.1007/s00348-021-03315-3