Optimization of multiplane μPIV for wall shear stress and wall topography characterization
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Abstract
Multiplane μPIV can be utilized to determine the wall shear stress and wall topology from the measured flow over a structured surface. A theoretical model was developed to predict the measurement error for the surface topography and shear stress, based on a theoretical analysis of the precision in PIV measurements. The main parameters that affect the accuracy of the measurement are identified. The effect of different parameter settings is studied by means of Monte Carlo simulations, and the results are compared with an experimental test case. The results are used to determine the recommended parameter settings for this measurement approach.
1 Introduction
This approach may be desirable or even necessary in applications where a direct measurement of topography or wall shear stress is problematic or not possible at all. This includes, for example, applications in microfluidics and in biological flows, where techniques such as scanning electron microscopy (SEM), atomic force microscopy (AFM), or force sensing probes cannot be applied due to the construction of the microchannels or the fragility of the biological samples. Optical methods, like interferometry, that rely on reflective surfaces are difficult to apply in the case of biological materials.
On the other hand, the measurement of the velocity field in these cases can be relatively simple using whole-field velocimetry techniques such as micro-particle-image velocimetry (μPIV) (Santiago et al. 1998). Stone et al. (2002) demonstrated that it is possible to determine the shape of the wall of a microfluidic device with a resolution approaching tens of nanometers using μPIV measurements of the fluid motion near a surface. Poelma et al. (2008) used μPIV to determine the local wall shear stress in vivo in a repeatable manner. From the flow measurements, the wall shear stress was derived either directly from the magnitude of the gradients or from fits of an analytical expression to the measured velocity profiles. The application of μPIV to in vivo experiments presents practical problems due to the peculiarity of the object of investigation, i.e., a living organism. However, under particular conditions, e.g., when it is possible to have appropriate optical access and to introduce seeding particles, the measurement accuracy that can be achieved is comparable to analogous in vitro experiments (Vennemann et al. 2007; Poelma et al. 2009).
Another possible application is the characterization of structures and surfaces in microfluidic devices. This approach was used by Joseph and Tabeling (2005) for the direct measurement of the apparent slip length in microchannels, and by Joseph et al. (2006) for the experimental characterization of liquid flow slippage over superhydrophobic surfaces made of carbon nanotube ‘forests’, incorporated in microchannels.
The topography and wall shear stress distribution are indirectly determined from the velocity measurement. The accuracy of the final result depends on a substantial number of interdependent parameters. In this paper, a model of the measurement method is described, and we use a Monte Carlo approach to optimize the performance of the system. The model is based on the theoretical analysis of the measurement precision in PIV developed by Keane and Adrian (1990, 1992) and Westerweel (2000, 2008). This paper intends to provide general guidelines for using this method or a similar approach.
The paper is structured as follows. First, the theoretical analysis is explained for the estimation of the error in the μPIV measurements (Sect. 2). This is used to perform a parametric optimization of the measurement method: first, the relevant parameters that affect the final result are identified, and subsequently, a Monte Carlo approach is used to analyze the effect of different parameter settings (Sect. 3). The conclusions of this study are discussed in the final section (Sect. 4).
2 Errors in μPIV for near-wall measurements
2.1 Theoretical analysis
Equation 9 shows that the relative error is a function of the distance z from the wall and that it depends on three dimensionless terms, i.e.: ΔX/d_{τ} and δ_{corr}/L_{0}, which are determined by the experimental parameters, and F(z), which depends on the flow velocity field. The mean displacement ΔX is determined by the exposure time delay Δt for the μPIV measurement.
2.2 Comparison with experimental results
μPIV measurements over a flat surface in a microchannel were performed to validate the expression in Eq. 9. A microchannel with a rectangular 0.127 × 2.5 mm^{2} cross-section was used, in which a steady flow was applied of 0.8 ml/min. The wall shear stress and topography measurements were taken over a glass coverslip with a nominal roughness of less than 1 nm.
For the μPIV measurements, an inverted microscope (Axiovert 200, Zeiss) was used with an objective lens (LD Achroplan) with an image magnification of M = 63, a numerical aperture of NA = 0.75, and working distance of WD = 1.57 mm. This configuration yields a depth of field of 1.2 μm (Inoué and Spring 1997) and a depth of correlation of 4.5 μm, which has to be multiplied by the factor k defined in Eq. 7. Since the immersion fluid of the lens is air (n_{0} = 1), and the working fluid is water (n_{w} = 1.33), the factor k is equal to 1.66. The objective lens was mounted on a piezo-electric positioning device (MIPOS500SG, Piezosystem Jena GmbH) with a precision of 8 nm.
Experimental settings
Image magnification | M | 63 |
NA of the objective lens | NA | 0.75 |
f-number of the objective lens | f^{#} | 0.67 |
Resolution piezo-electric position device | 8.0 (nm) | |
Refractive index of the immersion fluid | n_{0} | 1 |
Refractive index of the working fluid | n_{w} | 1.33 |
Fluorescent emission wavelength | λ_{emi} | 584 (nm) |
Particle diameter | d_{p} | 0.56 (μm) |
Particle-image diameter | d_{τ} | 10.9 (pixels) |
Nominal height of the channel | L_{0} | 127 (μm) |
Depth of correlation | δ_{corr} | 7.4 (μm) |
Mean pixel displacement (12 pixels) | ΔX/IW | 0.09 |
3 Parametric optimization
3.1 Relevant parameters in the optimization
- (a)
Δt is chosen as to keep the mean particle-image displacement ΔX constant in each plane;
- (b)
Δt is chosen as to keep the displacement error σ_{ΔX} constant, i.e., a = constant; c.f. Eq. 4
the number N of measurement planes;
the spacing Δz between subsequent planes;
the total height L of the measurement volume.
We set the distance of the first plane, which is the closest to the wall, to a fixed position at half the thickness of the measurement plane (i.e., correlation depth); see Fig. 5. Closer to the wall the measurement volume penetrates into the wall, and a bias occurs on the velocity measurement that will affect the estimation of the wall position and wall shear stress.
We do not take into account possible errors in the positioning of the plane, assuming that most of the traversing systems used to move the objective lens are accurate enough to neglect this source of error.
- (1)
the exposure time delay Δt;
- (2)
the height L of the measurement volume;
- (3)
the number N of measurement planes.
3.2 Monte Carlo simulations
We used a Monte Carlo method (Morgan 1984) to study how the three parameters defined in the previous paragraph influence the accuracy of the final result. Equation 8 is used to generate a velocity profile according to predefined settings of L and N. A random variation is applied to each velocity vector using the corresponding standard deviation obtained from Eq. 9 for a chosen Δt. From the generated velocity profile, the position of the wall h and the wall shear stress τ are extrapolated. The iteration of this procedure gives the mean values and the standard deviations of h and τ that can be achieved with the chosen parameter settings. We also evaluated the correlation of the estimates for h and τ.
Simulation 1 with Δt chosen in such a manner to keep the displacement ΔX constant in all planes and
Simulation 2 with Δt chosen to keep the variation a of the displacement in the interrogation volume constant in all measurement planes (i.e., this implies a constant error amplitude σ_{ΔX} in all planes; see Eq. 4).
3.2.1 Simulation 1
It can be noticed that h for large displacements and large L converges to a value less than zero, while one may expect h to converge to zero. Although we presently do not have an explanation for this effect, it is rather small, since the deviation from zero is one order of magnitude smaller than the random error.
3.2.2 Simulation 2
- (1)
A minimum height of the measurement volume is required to keep the bias error and the random error for the estimates of h and τ within acceptable limits. For the channel flow configuration we chose, this is around 0.25–0.3 times the channel height;
- (2)
Increasing the number of measurement planes does not significantly improve the quality of the estimates for h and τ;
- (3)
The two strategies investigated to set the exposure time delay Δt in each plane give similar results. When the exposure time delay Δt is set to have constant variation a of the particle-image displacement in all measurement planes, a value of a/d_{τ} ≈ 1–1.5 is advised. Although the two strategies show similar results, the second one is limited by the larger particle-image displacements obtained in the planes most distant from the channel wall as the value of a is increased.
3.3 Comparison with experimental results
To validate the results of our Monte Carlo simulations, measurement data obtained in the experiment presented in Sect. 2.2 are used to determine the mean value and standard deviation of the estimates for h and τ with different settings for the height L of the measurement volume and the number N of measurement planes. Since we take measurements in a channel with smooth and parallel walls, we expect that the actual values for h and τ are constant at all (x, y) locations.
4 Summary and conclusions
In this paper, we present how to optimize a measurement approach that uses μPIV measurements to determine the topography and wall shear stress distribution over a surface. Such measurements are relevant in studies of the response of endothelial cells to flow shear stress and in the assessment of structured surfaces in microfluidic devices. The topography and wall shear stress are determined from the velocity of the flow over the surface, measured in several planes parallel to the surface. Three relevant parameters for the accuracy of final result have been identified: the height L of the measurement volume, the number N of measurement planes, and the exposure time delay Δt (for each measurement plane) in the μPIV measurements. How the choice of these parameters modifies the final result was investigated by means of a Monte Carlo method. The theoretical result for the random error amplitude in the μPIV measurement as a result of velocity gradients in the interrogation volume (in particular in the out-of-plane direction) has been used to predict the random error amplitude in the velocity measurements in each measurement plane.
In general, the results show that a minimal height L of the measurement volume is required to maintain bias errors and random errors in the estimates for the surface height h and wall shear stress τ within acceptable limits, while the number N of measurement planes does not play a significant role in the accuracy of the final results. The minimum height needed depends on the experimental configuration (i.e., the magnification and numerical aperture of the objective microscope lens, and the characteristics of the PIV evaluation). This minimum height should be about 0.3 times the channel height (or other characteristic length of the velocity profile in other flow geometries) as studied in this work. With regard to the effect of the exposure time delay Δt for each measurement plane, two strategies were investigated: Δt set to keep the mean particle-image displacement constant and Δt set to keep the variation a of the particle-image displacement constant. In the first case, simulations show that a large value of ΔX is desirable (i.e., typically ΔX/d_{τ} > 1). In the second case, a large value of a is desirable as well, in this case larger than 1–1.5 times the particle-image diameter, although it has to be taken into account that large values of a may also produce very large particle-image displacements in the planes most distant from the wall. Results obtained in an actual experimental configuration confirm the findings of the Monte Carlo simulations.
Notes
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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