Color-coded three-dimensional micro particle tracking velocimetry and application to micro backward-facing step flows

Abstract

In this work, the authors proposed a microscopic particle tracking system based on the previous work (Tien et al. in Exp Fluids 44(6):1015–1026, 2008). A three-pinhole plate, color-coded by color filters of different wavelengths, is utilized to create a triple exposure pattern on the image sensor plane for each particle, and each color channel of the color camera acts as an independent image sensor. This modification increases the particle image density of the original monochrome system by three times and eliminates the ambiguities caused by overlap of the triangle exposure patterns. A novel lighting method and a color separation algorithm are proposed to overcome the measurement errors due to crosstalk between color filters. A complete post-processing procedure, including a cascade correlation peak-finding algorithm to resolve overlap particles, a calibration-based method to calculate the depth location based on epipolar line search method, and a vision-based particle tracking algorithm is developed to identify, locate and track the Lagrangian motions of the tracer particles and reconstruct the flow field. A 10X infinity-corrected microscope and back-lighted by three individual high power color LEDs aligning to each of the pinhole is used to image the flow. The volume of imaging is 600 × 600 × 600 μm³. The experimental uncertainties of the system verified with experiments show that the location uncertainties are less than 0.10 and 0.08 μm for the in-plane and less than 0.82 μm for the out-of-plane components, respectively. The displacement uncertainties are 0.62 and 0.63 μm for the in-plane and 0.77 μm for the out-of-plane components, respectively. This technique is applied to measure a flow over a backward-facing micro-channel flow. The channel/step height is 600/250 μm. A steady flow with low particle density and an accelerating flow with high particle density are measured and compared to validate the flow field resolved from a two-frame tracking method. The Reynolds number in the current work varies from 0.033 to 0.825. A total of 20,592 vectors are reconstructed by time-averaged tracking of 156 image pairs from the steady flow case, and roughly 400 vectors per image pair are reconstructed by two-frame tracking from the accelerating flow case.

Appendix

The radial basis function (RBF) interpolation is a method for approaching a function with given data points. The RBF is a real-valued function depending only on the distance from a certain point,

$$ \varphi \left( r \right) = \varphi \left( {\left|| {x - x_{i} } \right||} \right), $$

(15)

where $r = \left( {\left|| {x - x_{i} } \right||} \right)$ is the Euclidean distance in the current work.

For N given points, the target function can be approximated by the sum of N radial basis function,

$$ f\left( x \right) = c_{0} + c_{1} x + \mathop \sum \limits_{i = 1}^{N} \lambda_{i} \varphi \left( {\left| {x - x_{i} } \right|} \right), $$

(16)

where the coefficients c ₀ , c ₁, and λ _i are chosen to match the function values at the known data points (interpolation nodes). Because the approximation function is linear to the coefficients, these coefficients can be estimated using matrix methods of linear least squares. Several basis functions can be used are listed below:

Gaussian:

$$ \varphi \left( r \right) = \exp \left( { - \frac{{r^{2} }}{{2\sigma^{2} }}} \right), $$

(17)

Multiquadrics:

$$ \varphi \left( r \right) = \exp \sqrt {\left( {1 + \frac{{r^{2} }}{{\sigma^{2} }}} \right)} , $$

(18)

Linear:

$$ \varphi \left( r \right) = r, $$

(19)

Cubic:

$$ \varphi \left( r \right) = r^{3} , $$

(20)

Thinplate:

$$ \varphi \left( r \right) = \ln (r + 1) $$

(21)

once coefficients c ₀ , c ₁,and λ _i are found, this expression can be used to estimate value of the function at any point.