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Abstract
This paper reports on the measurement of a flow-velocity profile using a laser Doppler velocimetry (LDV) system having a focus tunable lens (FTL). In the system, the FTL is installed in the transmitting optics of the LDV; therefore, it can measure the flow velocity profile by changing the measurement position without any mechanical scanning system. To demonstrate the concept of the technique, the velocity profile measurement of Poiseuille flow was conducted, and the measured velocity profile showed good agreement with the theoretical value.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent times, optical measurement techniques have become quite common for flow measurements. Because these are non-intrusive techniques, it is possible to investigate the flow fields without disturbing them. Therefore, this technique has an advantage over the other conventional probe measurement techniques such as hot-wire anemometry [1] and Pitot tubes [2], which are directly inserted into the flow fields. Laser Doppler velocimetry (LDV) [3] is one of the well-established optical measurement techniques for gas and liquid flows. Moreover, LDV is widely used in the industrial and research fields because of its high spatial and temporal resolution. Conventional LDV systems are composed of transmitting and receiving optics. However, such systems usually tend to be large, and as has been pointed out, such large systems are affected by ambient environmental conditions [4]. Therefore, miniaturized systems of LDV are in high demand from a practical point of view. With the recent rapid growth of microelectromechanical systems (MEMS) and optical MEMS, very small systems of LDV and their applications have been developed as sensing devices to realize the flow measurement in narrow flow fields [5–7].
Generally, LDV is a pointwise-measurement technique. However, to determine flow velocity profiles, an additional component must be installed in the LDV system to change the position of the measurement probe, which is formed by overlapping the laser beams. The typical approach involves mechanically moving mirrors and prisms, and as a result, such optical elements are included in the transmitting optics. Some studies [8–10] show examples of this typical approach. Furthermore, other studies [4,11,12] demonstrate a method that can change the measurement position by tuning the wavelength of laser beams with gratings. Moreover, techniques that can determine the velocity profile inside a measurement probe with high spatial resolution are demonstrated in other reports [13–22]. These reports used a technique called laser Doppler profile sensor [13] in which the measurement probe is formed by multiple laser beams having different wavelength. Although the spatial resolution of measurement is dependent on the measurement set up, the sensor achieved high-axial resolution of less than 1 µm for measuring microchannel [14]. Additionally, Haufe et al. conducted the flow velocity and the acoustic particle velocity measurements using the laser Doppler profile sensor with the axial resolution of 10 µm [15]. We can see the pattern that a lot of photodetectors are employed in receiving optics to realize the two-dimensional mapping of a velocity field [23].
Although such proposed techniques can determine the velocity profile in the measurement, such optical systems tend to become large, which is one of the disadvantages in terms of miniaturizing the entire systems of LDV [4,5]. Moreover, such techniques seem to be impractical and difficult to use because they need very complicated optics, with a very strict alignment. Therefore, other techniques that are suitable for LDV to scan measurement probes in a flow field should be considered.
The use of the focus tunable lens (FTL) has become one of the solutions to measure the velocity profile using LDV without any mechanical scanning system. The FTL can electronically adjust the focal length by tuning the refractive index or shape of the lens. As a result, the focal point can be changed along the optical axis (z-axis). The detailed principles of the FTL are described in Ref. [24]. In recent studies, because the FTL can easily and quickly change the focal point, the applications built in microscopy systems have been proposed to demonstrate the three-dimensional imaging of the biological phenomena and the three-dimensional flow configuration measurement in microchannels [25–30]. In addition, the FTL has the potential to be an alternative to the conventional z-direction scanner, which is used in a confocal laser scanning microscopy [31,32]. In these studies, FTL is only integrated in the optical path, and thus, it is not necessary to include other complicated optics. Therefore, the FTL is also suitable for conventional LDV systems as a simple z-axis scanner. Some commercially available FTLs are very small [24]. Hence, very small velocity profile sensors can be realized by combining them with MEMS-based LDV sensors.
On the basis of these backgrounds, we applied the FTL to a commercially available conventional LDV system to measure the velocity profile on the z-axis without any mechanical scanning system. Hereafter, this system is called focus tunable LDV (FT-LDV).
The aim of this study is to demonstrate the concept of the FT-LDV measurement. For the demonstration, the laminar flow in the parallel plate channel was measured with FT-LDV. The measured velocity profile was compared with the theoretical value. When the probe location of the LDV was traversed using the FTL, the fringe space in the measurement probe was also changed. Because of the principle of LDV [33], the fringe space is necessary to calculate the velocity from the Doppler frequency. Therefore, the relationship between the focal length of the FTL and the fringe space of the LDV was also calibrated before the flow velocity measurement. Furthermore, the influence of fringe space and Doppler frequency variation on the velocity profile was evaluated by performing the uncertainty analysis. To the best of our knowledge, this is the first study about the velocity profile measurement by the LDV with the FTL. The results showed that the FT-LDV was practically applicable for flow field measurements. The remainder of this manuscript is organized as follows. Section 2 shows details of the experimental setup and calibration procedure of the FT-LDV. Section 3 presents the results of the calibration and velocity measurement. Section 3 also discusses the results of the uncertainty analysis. Section 4 summarizes conclusions drawn from this study.
2. Experimental systems
2.1 FT-LDV setup and calibration procedure of fringe spacing
Figure 1 shows the experimental setup for the demonstration of the FT-LDV. The parallel plate channel is also shown in Fig. 1, with the coordinate system of the measurement. We utilized the LDV system, which can detect the backscattering signal from tracer particles in the flow channel (Smart LDV III 8743-S, Kanomax). This system can also determine the one-component velocity. The measurement head shown in Fig. 1 includes the transmitting and receiving optics. The laser beam of this system has a wavelength λ of 660 nm and a power of 60 mW, and two beams were emitted from the measurement head, as shown in Fig. 1.
Fig. 1. Schematics of the experimental setup. The width of the parallel plate channel is W = 120 mm (for the y-direction). The FT-LDV system was controlled by a PC.
Figure 2 shows a schematic about the location of transmitting lens (TL) and FTL. The distance between two laser beams Δstl at the TL was approximately 38 mm. This measurement head had the TL, the focal length of which was ftl = 150 mm. In addition, the FTL (EL-16-40-TC-VIS-20D-C, Optotune) was placed in front of the measurement head. The distance between the TL and FTL was set at approximately d1 = 92 mm. This FTL is a type of shape-changing polymer lens [24] and has an aperture of 16 mm. To change the focal length of the FTL fFTL, the curvature and thickness of the polymer was controlled electronically. Furthermore, to control the value of fFTL, a software package (Lens Driver Controller, Optotune) was utilized. The laser beams were focused through the FTL, and the measurement probe that have the fringe space of ΔxFT was formed, as shown in Fig. 2. The value of the focal length fFLT is changed by controlling the value of diopter dpt ( = 1000 mm/fFTL) using the software package. The range of dpt of the FTL is from −10 to +10. Therefore, the range of fFTL becomes fFTL = +100 mm ∼ +∞ in positive and -∞ ∼ −100 mm in negative value. When the value of dpt is input to the software, the value of fFTL changes instantaneously.
Fig. 2. Schematics about the location of the transmitting lens (TL) and focus tunable lens (FTL). The laser beams pass two lenses and cross with the angle of θcr. In the measurement probe where two beams cross, the fringe space ΔxFT is formed.
When the tracer particles passed through the measurement point, the signal of the Doppler frequency f was acquired using a photomultiplier. Then, the flow velocity for x-axis direction ux was determined as follows:
Here, ΔxFT was determined by the crossing angle of two laser beams and the wavelength λ of the laser. Generally, when these laser beams cross with the angle of θcr, ΔxFT is defined as follows:From Eqs. (3) and (4), if the accurate value of the BFL is measured, then ΔxFT can be easily determined. To accurately measure the BFL, instruments that can adjust and determine the distance from the lens and the point where the two-laser beams cross precisely should be employed. However, this method seemed to be impractical. Therefore, the calibration procedure that estimates the value of ΔxFT by acquiring the Doppler frequency f was utilized. In this study, we employed the calibration procedure used by Shirai et al. [18] and Akiguchi et al. [23]. The fringe space ΔxFT was determined based on Eq. (1) by obtaining the Doppler frequencies f when the objects passed in the measurement probe for the tangential direction with a known velocity as shown in Fig. 3. As a passing object, we employed a tungsten wire with a diameter of 10 µm and attached it onto the rotating disk. The disk was rotated using DC motor. The distance r between the center of the rotating disk and the measurement location was 14 mm. The average rotating speed N of the disk was 641.1 rpm, which was measured by a tachometer. As a result, the average passing velocity of the wire at the measurement location ( = 2πrN/60) was 0.94 m/s. The rotating disk was attached to a slide stage. When fFTL was changed, the location of the disk was also changed manually in the z-axis direction. The position of the stage was adjusted with 0.1 mm in resolution. Thus, 20 samples of the Doppler frequencies were obtained and averaged at each value of fFTL. During the calibration procedure, the signal of the Doppler burst was monitored using an oscilloscope. Then, the position where the signal becomes the strongest was estimated by adjusting the position of the rotating disk. At that position, we judged that the tungsten wire passes through inside of the measurement volume.
2.2 Flow channel and velocity measurement
In this study, the laminar flow velocity component ux for the streamwise direction (x-axis direction) in the parallel plate channel was acquired. The parallel plate channel was H = 10 mm in height and W = 120 mm in width. The velocity measurement was conducted at the test section, which is far from the inlet of the channel (more than 1 m), where the flow was fully developed. Owing to the height of the stage, which clamped the measurement head, the measurement position for the spanwise direction (y-axis direction) was set at y = 11 mm (here, the center of the test section was set as y = 0 mm). As shown in Fig. 1, the lower side of the test section was made of a glass plate for the optical access. In addition, the upper side of the channel was made of an aluminum plate. The glass plate had a thickness of t = 5 mm and refractive index of nglass = 1.457. Furthermore, the distance between the FTL and the lower position of the flow channel (i.e., the position of z = 0 mm) was 41.2 mm, and the flow channel was a blowing type. Air was employed as the working fluid. The Reynolds number Re was 430, based on the hydraulic diameter of the channel Dh ( = 2WH/(W + H)) of 18.5 mm and the cross-sectional averaged velocity Uref of 0.35 m/s. Here, Uref was determined from the flow rate Q, which was controlled by the blower. As the tracer particle, dioctyl sebacate (DOS) droplets were employed and seeded upstream at the inlet of the channel. These DOS droplets were generated using an atomizer and compressor. In the atomizer, an impactor or separator which can remove DOS droplets other than a specific size was not attached. Therefore, the diameter of the droplets was not controlled. The velocity measurement was conducted by changing the location of the measurement point for the z-direction in the flow channel by controlling the value of fFTL. The sampling rate of the burst signal was set to 2 MHz. Additionally, the resolution of velocity become approximately 10−3 m/s. As indicated above, the measurement was conducted through the glass plate. As a result, the refractive index nglass does not have any influence on the value of θcr. Nonetheless, the value of the measurement location for the z-direction in the flow channel was affected by nglass. Therefore, the measurement location should be offset by considering the value of t(1–1/nglass).
The volume of the measurement probe changes when the fFTL changes, and the length of the probe for axial direction increases as fFTL (i.e., the value of z/H) increases. In other words, the spatial resolution of the FT-LDV changes along the z-axis direction (in our experimental conditions, the averaged typical length of the measurement probe was approximately 0.7 mm). The length was estimated to be increased up to approximately 35% from z/H = 0 to z/H = 1. This means that the length becomes approximately 0.6 mm at z/H = 0 and 0.8 mm at z/H = 1. To evaluate the length of the measurement probe more strictly, it is necessary to consider the influences of the distortion of the spot size of the laser beam and astigmatism on the length [33]. Therefore, care must be taken while applying this technique to the objected flow field, which needs high spatial resolution.
3. Results and discussion
3.1 Calibration curve and measurement position
The calibration curve between fFTL and ΔxFT is shown in Fig. 4(a). The curve was obtained by following the procedure described above. Here, the value of fFTL was changed from approximately 180 mm to 1515 mm (the value of dpt was changed from 5.56 to 0.66). The error bars corresponded to the standard deviation of the fringe space at each fFTL. We applied Eq. (5) to the points shown in Fig. 4(a) to estimate the calibration curve by least square fitting. The calibration curve is shown in Fig. 4(a) as a broken line. The constants in Eq. (5), A = 431 and B = 54.3 mm, were obtained from the fitting. Using these values, Eq. (4) was calculated, and the BFL was obtained. The standard estimated error (SEE) of ΔxFT which was regressively obtained from the result shown in Fig. 3(a) was found to be 4.64 × 10−2 µm. Considering both the range of ΔxFT obtained from calibration curve and sampling frequency, the maximum limit of the velocity measurement would become 1.5 ∼ 2.4 m/s. This indicates that the proposed system can determine the velocity profile in the flow channel and this is described in Section 3.2. In addition, the measurement position for the z-direction in the channel was acquired. These values are depicted in Fig. 4(b). Here, the value of z was normalized by H. From the relationship between the value of fFTL and z/H, the measurement location in the channel was also determined. Here, the value of z/H was calculated by considering the influence of nglass.
Fig. 4. (a) Calibration curve between fFTL and ΔxFT. The dashed line corresponds to the fitting curve indicated as Eq. (5). (b) Relationship between fFTL and BFL and between fFTL and z/H.
The measurement position z/H by the FT-LDV depends on the value of BFL, and the BFL depends on the calibration curve of ΔxFT indicated in Fig. 4(a), because of the value of the constants A and B. Therefore, the determination error of A and B, i.e., the error of ΔxFT, affected the measurement position. The uncertainty of the calibration curve on the measurement position was preliminarily estimated, and we then confirmed that the order of the uncertainty of the measurement position U(z/H) becomes less than O(U(z/H)) ≈ 10−6. This fact indicates that the error of the measurement position is negligible. Therefore, the calibration procedure described above is suitable for the FT-LDV system. Note that when the distance between the TL and FTL (= d1) is kept constant, there is no need to conduct the calibration procedure again. However, when the focal length of TL (= ftl) or d1, or both are changed, it becomes necessary to conduct the calibration procedure again.
3.2 Velocity profile measurement
The measurement position was changed from the glass wall location by controlling the value of fFTL. At z/H = 0.083, the system initially obtained the Doppler frequency f from the scattering of the tracer particles. From this point, we started the measurement of the streamwise velocity ux. The interval of the adjacent measurement location was approximately 0.4–0.5 mm.
By following the procedure indicated above, we acquired the streamwise velocity ux profile in the z-direction. At each z/H, more than 300 data of the instantaneous velocity ux were obtained and subsequently averaged. The temporary average value of ux, i.e., ux ave, is depicted in Fig. 5. The value of ux ave was normalized using the value of Uref. Error bars in the diagram correspond to the standard deviation of ux, σux, which is normalized by Uref at each z/H. The average value of σux ave/Uref is 0.06. The standard deviation of the Doppler frequency f (σf) was also acquired, and the average value of σf became 10.6 kHz.
Fig. 5. Streamwise velocity distribution ux ave/Uref in a parallel plate channel in the depth direction obtained by the FT-LDV. The theoretical calculations are also shown.
Afterward, the profile of ux ave was compared with that of the theoretical calculation, which is depicted in Fig. 5. The theoretical ux of Poiseuille flow and the flow rate Q in a rectangular channel are described as follows, respectively [34]:
From Fig. 5, we can confirm that the distribution of the measured ux ave shows a good agreement with the theoretical calculation. The maximum difference between the measured and theoretical values was 3.5% of the reading. The concept of the FT-LDV was successfully demonstrated.
In Fig. 5, in the region where the velocity was unobtainable, i.e., at z/H > 0.8, the Doppler frequency was not obtained at all. As described in Section 2.2, the wall that was on the opposite side of the observation window was made of aluminum plate. Therefore, the noise, produced by the scattering and reflection from the aluminum wall, could be a reason the Doppler signal was not obtained. In reality, by visually using the oscilloscope, the signal was monitored throughout the experiment, and when the measurement probe moved close to the aluminum plate, the noise increased, and the Doppler burst did not appear. In the region where z/H is less than 0.08, the Doppler frequency was also not obtained. In the near-wall region, the signal-to-noise ratio declined because the density of tracer particles decreased, and the wall scattering increased [14,18]. Because of the aforementioned reasons, we did not obtain the velocity at the location where the Doppler frequency was not presented. Therefore, the velocity profile was not plotted at z/H < 0.08 and z/H > 0.8. To decrease the effect of scattering and reflection of wall and increase the signal-to-noise ratio, a material that can mitigate the scattering and reflection should be used as a channel wall, and the laser power should be increased.
In the velocity profile shown in Fig. 5, we can see the error bars of various sizes can be observed. When the flow measurement was conducted, the diameter of tracer particles was not controlled. Owing to the non-uniformity of the diameter of particles, there is a possibility that velocity dispersion occurred at each measurement position. When the particles exist in the shear flow, large particles which show the low ability to follow the flow motion are generally move to the area where the flow velocity becomes large due to the Saffman lift force. Additionally, when the large particles which are larger than the fringe space pass through the measurement probe, there is a possibility that high-frequency noise is superimposed on the burst signal. Thus, the measured velocity is possibly estimated larger than actual flow velocity [33]. Moreover, as indicated in Section 2.2, the size of measurement probe for axial direction becomes large as the probe moves closing to z/H = 1. This means that the spatial resolution of measurement declines, resulting in larger error bars. These factors may have affected the velocity dispersion around z/H = 0.6, as a result, the error bars possibly became wider. To reduce the size of error bars, it is necessary to control the particle diameter and make it uniform.
3.3 Measurement uncertainty
The absolute difference between the measured and theoretical values of the velocity (Δux) was calculated from the data shown in Fig. 5. The values of Δux were normalized by ux ave in Fig. 6. The maximum value of Δux/ux ave was less than 0.05.
Fig. 6. Comparison between the value of uncertainty U(ux) analyzed by utilizing Eq. (8) and Δux that corresponds to the differences between the measured velocity and theoretical ones indicated in Fig. 5.
The uncertainties of the measured velocity U(ux) which is normalized ux ave at each measurement position (U(ux)/ux ave), are also shown in Fig. 6. The uncertainty was estimated as follows. In the FT-LDV system, the fringe spacing ΔxFT changes when the value of fFTL changes. Therefore, the error of ΔxFT is a factor of the uncertainty and the error of the Doppler frequency f [14]. In this study, to investigate the influences of measuring f and ΔxFT on the measured ux, the velocity measurement uncertainty caused by the FT-LDV was analyzed by considering the error propagation [36].
According to the definition of the velocity determination indicated in Eq. (1), the uncertainty of velocity U(ux) is expressed as follows:
4. Conclusion
In this study, an FT-LDV system composed of a conventional LDV system and a focus tunable lens was proposed and demonstrated as the velocity profile sensor. The calibration procedure associated with the focal length of FTL (fFTL) and fringe space of ΔxFT was proposed. The FT-LDV successfully measured the streamwise velocity profile of the laminar flow in the parallel plate channel without mechanically traversing for the axial direction. The obtained velocity profile showed good agreement with the theoretical calculation of Poiseuille flow. Moreover, the measurement uncertainty U(ux) of the FT-LDV was analyzed. The value of the standard estimated error of fringe space determined from the calibration result was found to be 4.64 × 10−2 µm, and the averaged value of the uncertainty of the measured velocity by the proposed FT-LDV technique became U(ux)/ux ave = 0.05. The proposed system has a potential to be a valid and promising technique as a small-scale velocity profile sensor. In the future, the application of very small FT-LDV systems will be utilized in the science and industrial field.
Disclosures
The authors declare no conflicts of interest.
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