Correction to: Psi-PIV: a novel framework to study unsteady microfluidic flows

The original version of this article contained a mistake.


Introduction
In micro-scale application, Particle image velocimetry (PIV) is generally referred to as micro-PIV (Santiago et al. [1998]; Meinhart et al. [1999]). The length scales of microfluidic device in the stream-wise and span-wise directions of the flow are considerably larger compared to the channel height in the wall normal direction. Hence, the Hele-Shaw cell conditions are valid for such devices.
The Poiseuille flow profile in the gap between two plates of a Hele-Shaw cell causes wall-normal variation of the flow velocity where fluid in the centre-line travels faster than the fluid near the walls. Large field-of-view results in large depth-of-field in the measurement volume. PIV yields low signal-to-noise ratio (SNR) due to the variation of the velocity along the depth-of-field and low seeding density. To reduce the effect of velocity gradient on the SNR, an enormous amount of tracer particles is required which is not feasible.
19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics LISBON | PORTUGAL JULY 16 -19, 2018 Meinhart et al. [2000] showed that the ensemble correlation of the image pairs substantially improves the SNR. Taking ensemble correlation average essentially increases the effective image density to determine the velocity field. However, this method leads to significant loss in instantaneous information. Ehyaei and Kiger [2014] from synthetic image simulation showed that a likelihood of 94% valid vectors can be reached if an aggregate of maximum displacement ( x max ) and neighbouring displacements ( x max -1 , x max -2) are taken as valid vectors.
Nonetheless, this does not give a reliable sub-pixel estimation of the particle displacement. It was shown by Kloosterman et al. [2011] that the measured PIV velocity can under-estimate upto 25% of the maximum Poiseuille flow velocity. Streamline image velocimetry by Keinan et al. [2013] uses long exposure images to capture path lines of the particle in fully developed laminar flow and determine velocity field from the fact that the volumetric flow rate in stream tube is conserved in low Reynolds number. The drawback of this method is its inability to measure velocity field in unsteady laminar flow cases.
This work focuses on determining the velocity field by reducing the image density and taking the advantage of the potential flow theory. Thus, making the current work relevant for measuring flow fields in steady and unsteady laminar flow. Quantitative analysis by means of PIV for large field-of-view is done in this investigation.

Theoretical description
The correlation peak for Poiseuille flow under low number of frames and/or less image density splits up into multiple peaks, although the particles in the interrogation area follows the flow direction. Fig. 1a shows the top-view of the correlation map for 2700 averaged images, where the smeared peak is at an angle of 63 w.r.t the x-axis. The corresponding normalized correlation function is shown in Fig. 1c. For the same interrogation window, the top view of the correlation map for 40 averaged image is shown in Fig. 1b. In this case, multiple peaks are present but it can be seen that the strong peaks are in the flow direction. The corresponding normalized correlation function along the 63 direction is shown in Fig. 1d which clearly shows multiple peaks.
This section describes a PIV algorithm which reduces the effective seeding density and yield similar if not higher SNR compared to conventional PIV. We call this algorithm '' -PIV''. Fig. 2 shows a flowchart describing the processing steps taken by -PIV. Consider a case where conventional micro-PIV requires an ensemble average of correlation plane of around 1000 frames to get the correct velocity field due to low seeding density and variation of the in-plane motion over the channel height. In case of -PIV, first the direction of the flow is determined from the correlation plane for all the interrogation areas (shown as dashed arrows in Fig. 2). The cross-19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics LISBON | PORTUGAL JULY 16 -19, 2018 correlation gives the highest peak in the direction in which the fluid is moving even with a fewer number of frames and with lower seeding density. Next, virtual particles are advected using a Runge-Kutta 4th order scheme through the interrogation areas. The path taken by the advection of the virtual particles are considered as streamlines. For simplicity, the flow is considered to have a uniform flow at the inlet, each streamline can be given a constant value at the inlet. In this way, the stream function from the (instantaneous) streamline pattern can be assigned.
The stream function ( ) is calculated for the centre location of each interrogation window by interpolating the nearest streamline values. Finally, using Eq. 1 the velocity components in x-and y-direction are calculated.
(1) 19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics LISBON | PORTUGAL JULY 16 -19, 2018 Validation Synthetic images were generated in order to validate the -PIV algorithm. The images were made to mimic the uniform flow in span-wise direction under Hele-Shaw conditions. Thus, by randomly distributing the particles across the channel height leads to variability in velocity field for the uniform flow. Secondary effects such as Brownian motion and background noise from the image sensor are not taken into account for simplicity. The tracer particles in the image domain are reinserted at random positions at the in-flow boundary in order to keep the image density constant across all frames. All images are generated with image size of 1024 1024 pixels with mean particle image diameter of 3 pixels. The centre-line displacement for the Poiseuille flow profile is 8 pixels.
The processing of the synthetic images was done using single-pass FFT cross-correlation. The interrogation windows of 32 32 pixels with no overlap was chosen.
The valid detection probability shows the probability of determining the displacement peak correctly compared to the tallest random peak. Keane and Adrian [1990] showed that probability of determining true displacement peaks improves as the average number of particle pair in the interrogation area increases. In conventional PIV, the general guideline for having valid detection probability greater than 95%, N I F I F O of 8-10 particles for uniform flow test case is desired (Keane and Adrian [1992]; Adrian and Westerweel [2010]). In this case, the flow is in the channel between the two plates and volume illumination is used. Hence, there is no out-of-plane motion i.e F O =1. performance. This is due to the particle size biasing effect where the maximum peak is located at 1 pixel less than the true displacement value (Ehyaei and Kiger [2014]). In case of -PIV, the valid detection probability reaches 1 substantially quicker due to the fact that less image density is required to determine the flow direction.

Experiment Setup
The PIV measurement was carried out in a Hele-Shaw cell consisting of two plexi-glass plates with length and breadth of 300mm and 100mm respectively. The

Result
The PIV processing is done by single-pass cross-correlation with interrogation windows of 32 32pixel with 0% overlap. Fig. 5a shows the velocity field result from correlation average of 217 frames. The velocity direction is from bottom to top. The velocity near the boundary of the cylinder decreases because Hele-Shaw condition is no longer valid near the wall. The Hele-Shaw condition only applies when the in-plane dimension (L) is much greater than the channel height (h) i.e. L h. Hence, the velocity decreases near the wall where distance is O(h). Fig. 5b shows the velocity field with correlation average of 9 frames. The velocity vectors are not well resolved because seeding density is too low for correlation averaging to give a realistic outcome. Fig. 5c shows the velocity field for -PIV with 9 frames. The result is in close agreement with the PIV results from 217 averaged images (Fig. 5a). The increase in the velocity near the cylinder wall is better captured by -PIV compared to PIV with the same number of frames.
A quantitative comparative study of the velocity field from analytically, PIV and -PIV are shown in Fig. 6. The velocity field is made non-dimensional w.r.t. free stream.
The analytic solution is calculated using the inviscid potential flow theory, hence there is a slip velocity on the edge of the cylinder.
PIV result of 217 frames shows that the velocity field is in close agreement with the analytic solution except near the cylinder's edge because Hele-Shaw condition is violated. Hence the data point closest to the edge of the cylinder would not be considered for comparison. PIV result from 9 frames is clearly under-predicted due to low image density. -PIV result for 9 frames noticeably matches the PIV result of 217 frames. For -PIV, maximum difference in measured velocity with respect to reference velocity is 0.2 throughout the velocity distributions whereas for PIV with 9 frames the maximum difference is 0.8 throughout the velocity distribution w.r.t reference velocity. This shows the improvement in velocity field by -PIV measurement alongside lower frame requirement.

Discussion
The comparative study between PIV and -PIV (Fig. 3) shows that -PIV furnishes valid detection probability of more than 0.95 with effective image density of around 20 particle image pairs in an interrogation area. This means that either the seeding density needs to be high enough to take time resolved data or the ensemble average of few frames needs to be taken if the seeding density is low. To measure the angle precisely, the time separation between two frames should be