Synchronized analysis of an unsteady laminar flow downstream of a circular cylinder centred between two parallel walls using PIV and mass transfer probes

Abstract

Experiments are carried out in the wake of a cylinder of d _c  = 10 mm diameter placed symmetrically between two parallel walls with a blockage ratio r = 1/3 and a Reynolds number varying between 75 ≤ Re ≤ 277. Particle image velocimetry is exerted to obtain the instantaneous velocity components in the cylinder wake. A snapshot proper orthogonal decomposition (POD) is also applied to these PIV results in order to extract the dominant modes through the implementation of an inhomogeneous filtering of these different snapshots, apart from an interpolation to estimate the wall shear rate at the lower wall downstream the cylinder. Mass transfer circular probes are placed at the lower wall downstream this obstacle so as to further determine the time evolution of the wall shear rate, by bringing the inverse method to bear on the convective-diffusion equation. Comparisons between the two synchronized techniques demonstrate that electrochemical method can give more accurate information about the coherent structures present in the flow and about the interaction of the von Kármán vortices with the walls of the tunnel as well. The comparison between the two measurement techniques in the flow regions concerns the spatiotemporal evolutions of the wall shear rate obtained from PIV measurements and the wall shear rate using mass transfer probes. Discrepancy between the PIV measurements and the electrochemical ones near the wall, where the secondary vortices P ₁′ are generated at wall, are caused by a PIV bias and a limitations of the singular mass transfer probes.

Appendix: Uncertainty analysis

1.1 Electrochemical uncertainty calculation

The total uncertainty t _u on the velocity calculation using PIV measurements is calculated as:

$$ t_{u} = \sqrt {P_{u}^{2} + B_{u}^{2} } $$

where B _u is the bias on the velocity and P _u is the precision on the measurements of a steady velocity, which can be calculated in absence of the cylinder in the same position and at the same Reynolds number as follows:

$$ P_{u} = t_{0.975} \cdot {\frac{{\hat{\sigma }_{u} }}{{\sqrt {N_{m} } }}} $$

where $t_{0.975} = t_{\text{student}} \left( {0.975;N_{m} - 1} \right)$ is the confidence coefficient for a confidence level of 95% and N _m  − 1 Freedom degree. t _0.975 ≈ 1.984 for N _m  = 100 measurement repetition. $\hat{\sigma }_{u}$ is the estimated standard deviation defined as:

$$ \hat{\sigma }_{u} = \sqrt {{\frac{1}{{N_{m} - 1}}}\sum\limits_{i = 1}^{{N_{m} }} {\left( {U_{i} \left( {x,y} \right) - \overline{{U\left( {x,y} \right)}} } \right)}^{2} } $$

and $\overline{{U\left( {x,y} \right)}} = {\frac{1}{{N_{m} }}}\sum\nolimits_{i = 1}^{{N_{m} }} {U_{i} \left( {x,y} \right)} .$

The following table resumes the experimental PIV conditions that are used for the calculation of the bias uncertainty.

Summary of measurement conditions

Target flow	2-D water flow
Measurement facility	Circulating water channel
Measurement area	81.73 × 60.59 mm2
Reference velocity	0.01 m/s
Calibration
Distance of reference points l r	81.73 mm
Distance of reference image L r	1,600 pixels
Magnification factor α	0.0511 mm/pixel
Flow visualization
Tracer particle	Spherical polyamide particle
Average diameter dp	0.05 mm
Standard deviation of diameter sp	0.005 mm
Average specific gravity	1.02
Light source	Double pulse Nd:YAG laser
Laser power	15 mJ
Thickness of laser light sheet	1.0 mm
Time interval Δt	25 ms
Image detection
Camera
Spatial resolution	1,600 × 1,086 pixels
Sampling frequency	30 Hz
Gray scale resolution	8 bit
Optical system
Distance from the target l t	1,100 mm
Length of focus	60 mm
F number of lens	f 2.8
Perspective angle θ	1°
Data processing
Pixel unit analysis	Adaptative cross-correlation method
Correlation area size	32 × 32 pixels

Flow speed U is calculated in PIV measurements using this equation:

$$ U = \alpha \cdot {\frac{\Updelta X}{\Updelta t}} + \delta u $$

where ΔX is the particle displacement, Δt is the time interval of successive images, α is the magnification factor, and δu are the errors caused by particles trajectories and by the dimensional effects.

The uncertainty of the velocity can be computed as

$$ \delta U = \sqrt {\left( {{\frac{\partial U}{{\partial \left( {\Updelta X} \right)}}}} \right)^{2} \left( {\delta \left( {\Updelta X} \right)} \right)^{2} + \left( {{\frac{\partial U}{\partial M}}} \right)^{2} \left( {\delta M} \right)^{2} + \left( {{\frac{\partial U}{{\partial \left( {\Updelta t} \right)}}}} \right)^{2} \left( {\delta \left( {\Updelta t} \right)} \right)^{2} + \left( {\delta u} \right)^{2} } $$

where

$$ {\frac{\partial U}{{\partial \left( {\Updelta X} \right)}}} = {\frac{\alpha }{\Updelta t}} $$

$$ {\frac{\partial U}{\partial \left( \alpha \right)}} = - {\frac{\Updelta X}{\Updelta t}} = - {\frac{U}{\alpha }} $$

$$ {\frac{\partial U}{{\partial \left( {\Updelta t} \right)}}} = - \alpha {\frac{\Updelta X}{{\left( {\Updelta t} \right)^{2} }}} = - {\frac{U}{\Updelta t}} $$

In order to compute the uncertainty of the velocity, uncertainties in magnification, displacement, and time between frames should be computed. The procedure adopted for uncertainty calculation is well discussed in (ITTC 2008).

 

Parameter	Category	Error sources	u (x i ) (unit)	c i (unit)	c i u i	u c
α (mm/pixel)	Calibration	Reference image	0.7 (pixel)	3.193 × 10−5	2.24 × 10−5	3.89 × 10−4
		Physical distance	0.02 (mm)	6.25 × 10−4	1.25 × 10−5
		Image distortion by lens	8 (pixel)	3.193 × 10−5	2.55 × 10−4
		Image distortion by CCD	0.0056 (pixel)	5.11 × 10−2	2.86 × 10−4
		Parallel board	0.035 (rad)	1.8 × 10−3	6.3 × 10−5
∆X (pixel)	Acquisition	Laser power fluctuation	10−5 (mm)	19.58	1.96 × 10−4	0.202
		Image distortion by CCD	5.6 × 10−3 (pixel)	1	5.6 × 10−3
		Normal view angle	0.035 (rad)	5.11 × 10−2	1.79 × 10−3
	Reduction	Mis-matching error	0.2 (pixel)	1.0	0.2
		Sub-pixel analysis	0.03 (pixel)	1.0	0.03
∆t (s)	Acquisition	Delay generator	2 × 10−9 (s)	1.0	2 × 10−9	1.02 × 10−8
		Pulse time	10−8 (s)	1.0	10−8
δu (mm/s)	Experiment	Particle trajectory	10−3 (mm/s)	1.0	10−3	7.3 × 10−3
		3-D effect	7.3 × 10−3 (mm/s)	1.0	7.3 × 10−3

Standard uncertainty u(x _i ), u _c is the combined uncertainty, $c_{i} = {\frac{\partial f}{{\partial x_{i} }}}$ and $u_{c} = \sqrt {\sum\nolimits_{i} {\left( {c_{i} \cdot u_{i} } \right)^{2} } }$

Example with a reference velocity U ≈ 10 mm/s

Parameter	Error sources	u c (x i ) (unit)	c i (unit)	c i u i
α	Magnification factor	3.89 × 10−4 (mm/pixel)	195.8 (pixel/s)	7.62 × 10−2 (mm/s)
∆X	Image displacement	0.202 (pixel)	2.554 (mm/pixel/s)	0.52 (mm/s)
∆t	Image interval	1.02 × 10−8 (s)	500 (mm/s2)	5.1 × 10−6 (mm/s)
δu	Experiment	7.3 × 10−3 (mm/s)	1.0	7.3 × 10−3 (mm/s)
		Combined uncertainty B u		0.522 (mm/s)

Example with a reference velocity U ≈ 18.5 mm/s

Parameter	Error sources	u(x i ) (unit)	c i (unit)	c i u i
α	Magnification factor	3.89 × 10−4 (mm/pixel)	362.04 (pixel/s)	7.62 × 10−2 (mm/s)
∆X	Image displacement	0.202 (pixel)	2.554 (mm/pixel/s)	0.52 (mm/s)
∆t	Image interval	1.02 × 10−8 (s)	925 (mm/s2)	5.1 × 10−6 (mm/s)
δu	Experiment	1.49 × 10−2 (mm/s)	1.0	1.49 × 10−2 (mm/s)
		Combined uncertainty B u		0.535 (mm/s)

The precision P _u is defined in the flow section for different Reynolds number in absence of the cylinder. The results on the relative precision uncertainty are presented in the following figure.

After calculating the precision uncertainty, the total uncertainty is calculated which is maximum near the wall about ${\frac{{t_{u} }}{{\bar{U}}}} = 12.4\%$ for Re = 159 and ${\frac{{t_{u} }}{{\bar{U}}}} = 9.4\%$ for Re = 277.

After calculating the total uncertainty near the wall, the uncertainty on the wall shear rate estimated using PIV method is calculated as follows:

$$ {\frac{{{\text{d}}\bar{S}_{\text{PIV}} }}{{\bar{S}_{\text{PIV}} }}} = {\frac{{{\text{d}}\overline{U} \left( {x,\Updelta y} \right)}}{{\overline{U} \left( {x,\Updelta y} \right)}}} + {\frac{{{\text{d}}\left( {\Updelta y} \right)}}{{\left( {\Updelta y} \right)}}} = {\frac{{t_{u} \left( {x,\Updelta y} \right)}}{{\overline{U} \left( {x,\Updelta y} \right)}}} + {\frac{{{\text{d}}\left( {\Updelta y} \right)}}{{\left( {\Updelta y} \right)}}} $$

since $S_{\text{PIV}} = \overline{U} \left( {x,\Updelta y} \right)/\Updelta y.$

The total relative uncertainty ${\frac{{\Updelta \bar{S}_{\text{PIV}} }}{{\bar{S}_{\text{PIV}} }}}$ is calculated as follows:

$$ {\frac{{\Updelta \bar{S}_{\text{PIV}} }}{{\bar{S}_{\text{PIV}} }}} = \sqrt {\left( {{\frac{{t_{u} \left( {x,\Updelta y} \right)}}{{\overline{U} \left( {x,\Updelta y} \right)}}}} \right)^{2} + \left( {{\frac{{\Updelta \left( {\Updelta y} \right)}}{{\left( {\Updelta y} \right)}}}} \right)^{2} } . $$

Finally and since ${\frac{{\Updelta \left( {\Updelta y} \right)}}{{\left( {\Updelta y} \right)}}}$ is about 5%, the relative uncertainty on wall shear rate calculated using PIV measurements is about 13.4% for Re = 159 and 10.34% for Re = 277.

1.2 Electrochemical uncertainty calculation

The electrical current delivered by a d _s diameter circular mass transfer probe in permanent regime can be expressed at high Peclet number as follows:

$$ I = 0,807nFC_{0} A_{S} \frac{D}{l}Pe^{{\frac{1}{3}}} = 0.6772nFC_{0} D^{{\frac{2}{3}}} \cdot d_{s}^{{\frac{5}{3}}} \cdot \bar{S}^{{\frac{1}{3}}} . $$

By differentiation, we obtain the following equation:

$$ {\frac{{{\text{d}}I}}{I}} = \frac{2}{3}{\frac{{{\text{d}}D}}{D}} + \frac{5}{3}{\frac{{{\text{d}}\left( {d_{s} } \right)}}{{d_{s} }}} + \frac{1}{3}{\frac{{{\text{d}}\bar{S}}}{{\bar{S}}}} + {\frac{{{\text{d}}C_{0} }}{{C_{0} }}}. $$

Finally, ${\frac{{\Updelta \bar{S}}}{{\bar{S}}}} = \sqrt {\left( {3 \cdot {\frac{\Updelta I}{I}}} \right)^{2} + \left( {2 \cdot {\frac{\Updelta D}{D}}} \right)^{2} + \left( {5 \cdot {\frac{{\Updelta \left( {d_{s} } \right)}}{{d_{s} }}}} \right)^{2} + \left( {3 \cdot {\frac{{\Updelta C_{0} }}{{C_{0} }}}} \right)^{2} }.$ In fact, the bias uncertainty on the wall shear is the due to an uncertainty on the current measurements and conversion binary data by using a 12-bit A/D converter that introduces an uncertainty of 0.1–0.2%; an uncertainty on the molecular diffusion coefficient which is about 1%; an uncertainty on the concentration estimation which is about 2% (Levich 1962); and an uncertainty on the active diameter of the mass transfer probe is about 2% (Sobolik et al. 1998). The total effective uncertainty is about ${\frac{{\Updelta \bar{S}}}{{\bar{S}}}} \approx 12\% ,$ and it is essentially due to the uncertainty on active probe diameter or surface. For this reason, it is interesting in many cases to make in situ probes calibration at high Peclet numbers and moderate fluctuations.