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The article introduces a direct detection method for cusp thickness in wavy thin-film flow using an optical waveguide film. This method addresses the challenges posed by complex interfacial waves and varying flow conditions in industries such as nuclear reactors, steam turbines, and semiconductor manufacturing. By directly detecting cusps with superior resolution, the optical waveguide film method overcomes the limitations of existing techniques, ensuring accurate and reliable measurements even during disturbance wave flow. The method's high spatial and temporal resolution makes it a valuable tool for predicting critical phenomena like liquid film dryout and droplet entrainment, thereby enhancing the safety and efficiency of industrial processes.
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Abstract
The wave crest (cusp) of the disturbance wave in thin liquid film flow is an important factor contributing to heat/mass transfer, e.g., fuel rods in boiling water reactors, stator/rotor blades in steam turbines, and cleaning/drying wafer processes in semiconductor manufacturing. We developed a new technique for directly detecting the thickness of wave cusps using array-based sensing with an optical waveguide film (OWF). This technique, based on geometrical optics assumptions, simultaneously obtains information on liquid films’ thickness and their interfacial shape, i.e., whether or not the local interface is convex upward. We first performed a pseudo-film flow measurement using a metal specimen to confirm the basic principle. According to the results, a meaningful signal indicating the wave-cusp passage, along with a thickness signal, was detected simultaneously. The OWF signal processing for cusp thickness detection was newly established based on this fact. We then applied this technique to a wavy liquid film flow formed on a flat plate in the entry region. A series of experiments were performed over a wide range of air speeds (jG = 20–70 m/s). As a result, the cusp thicknesses of relatively large waves on the wavy interface were successfully extracted from the OWF output signal. Further, the major thickness variables (i.e., base film thickness, median film thickness, and cusp thickness) were compared with those of conventional thickness estimation methods, which showed reasonable agreement. This paper provides a framework for wavy thin-film flow measurements via OWF that is specialized for directly detecting local thickness profiles.
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1 Introduction
Liquid film flow that ranges in thickness from a few hundred microns occurs in a wide range of industries, e.g., fuel rods in boiling water reactors (Nishida et al. 1994; Kawahara et al. 2015; Robers et al. 2021), stator and rotor blades in steam turbines (Hammitt et al. 1981; Simon et al. 2016; Gribin et al. 2020), cleaning and drying wafer processes in semiconductor manufacturing (Staudegger et al. 2009; Kobayashi and Abe 2013), falling film absorbers (Killion and Garimella 2001; Mortazavi et al. 2015), falling film evaporators (Yung et al. 1980; Fernández-Seara and Pardiñas 2014), and ice accretion on aircraft wings (Fortin et al. 2006; Liu and Hu 2018). These thin liquid film flows are often accompanied by complex interfacial waves spatially and temporally multiscale depending on the gas and liquid flow conditions (Inoue and Maeda 2021). Such waves include those that behave in a solitary manner, others coalesce, and others link in axial and transverse directions, creating 2D/3D wave crests, referred to as “cusps,” throughout the liquid film’s interface. Once the superficial gas velocity is sufficiently larger than that of the liquid, the cusps on the liquid film interface are characterized by large waves referred to as “disturbance waves” and accompanying smaller waves referred to as “ripple waves” (Berna et al. 2014). These cusps play an important role in heat and mass transfer in liquid film flows. A schematic of the flow regime is illustrated in Fig. 1. The cusp of the disturbance wave carries considerable liquid mass; therefore, the disturbance wave is known as a primary source of phenomena such as liquid films’ dryout and droplet entrainment. However, predicting the disturbance waves’ emergence in the actual system is unfeasible since every flow condition, including channel geometry (e.g., spacers and rods of nuclear fuel assemblies), significantly affects wavy liquid film formation (Kawahara et al. 2015; Robers et al. 2021). Hence, a direct measurement of the cusp thickness is essential to ensure safe and efficient operation of the equipment.
Fig. 1
Schematic representation of the interfacial structure in disturbance wave flow
A great variety of measurement techniques are available to estimate the thickness of liquid films, including electrical methods based on resistance and impedance measurements (Fukano 1998; Damsohn and Prasser 2009; Zhao et al. 2013; Bonilla-Riaño et al. 2019; Rivera et al. 2022; Zhang et al. 2023), laser-induced fluorescence (LIF) (Alekseenko et al. 2008, 2013; Cheng et al. 2010; Cherdantsev et al. 2014; Zadrazil et al. 2014; Chang et al. 2019, 2023), an ultrasonic echo technique (UET) (Kamei and Serizawa 1998; Pedersen et al. 2000; Aoyama et al. 2014, 2016; Yang et al. 2015), laser focus displacement meter (LFD) (Hazuku et al. 2005, 2008; Han and Shikazono 2009; Leng et al. 2018; Lapp et al. 2022), total internal reflection method (TIRM) (Hurlburt and Newell 1996; Pautsch and Shedd 2006; Kabardin et al. 2011; Moreira et al. 2020, 2023; Grasso et al. 2024), X-ray tomography (Hu et al. 2005; Hubers et al. 2005; Heindel 2011; Skjæraasen and Kesana 2020; Robers et al. 2021), and a fiber-optic method (Alekseenko et al. 2003; Zaitsev et al. 2003; Oliveira et al. 2006; Mizushima 2021, 2023; Okui et al. 2022; Watanabe et al. 2023). Each of these techniques has excellent characteristics and has made a significant contribution to our understanding of liquid film flow. However, these methods often ignore or simplify the local shape of the liquid interface due to spatial, temporal, or thickness resolution limitations. Thus, the measured values are unintentionally averaged under higher gas velocities and more unsteady conditions than ever before, which may lead to a decline in measurement accuracy during disturbance wave flow. To resolve this issue, it is necessary to directly detect the cusp on the interface with superior resolution compared to existing methods.
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We developed a method for measuring liquid film thickness using an optical waveguide film (OWF) (Miyachi et al. 2022; Furuichi et al. 2023a,b). The OWF is a flexible polyimide film with several built-in optical waveguides. It is easily installed on a channel wall, such as inside a circular tube or another curved surface. This method is based on a simple but robust measurement principle that uses the light intensity reflected by the surface of a liquid film in a manner similar to that used in the fiber-optic method. The optical response of the OWF depends on the refractive index difference between the gas and liquid phases, and the refractive index of water has a very small temperature dependence (Schiebener et al. 1990). Therefore, the OWF method is expected to be applicable to the aforementioned industrial processes. Table 1 summarizes the measurement limitations of typical techniques for measuring film thickness, including the OWF method. Our OWF sensors can be designed in sizes ranging tens of micrometers, enabling high spatial resolution. The time resolution of the OWF method depends on the photoelectric conversion systems, thus making it possible to achieve sampling rates of over a few hundred kHz at a relatively low cost. Moreover, the OWF method is highly responsive to thin films (typically < 100 μm), whereas earlier methods tend to be less responsive to such thin films. Therefore, OWF has the potential to enable the detection of instantaneous thickness fluctuations more accurately than conventional methods. The effectiveness of liquid film measurements via the OWF in annular flow was verified experimentally and numerically (Furuichi et al. 2023a,b).
Table 1
Measurement limitations of the typical techniques used for the film thickness
Measurement methoda
Spatial resolutionb,c
Temporal resolutionb
Thickness resolutionb, c
Measurement rangeb
Contaminates liquid phase?b
Electrical method
~ 10−3 m
< 10 kHz
~ 10−6 m
50 μm–3 mm
Yes
LIF
~ 10−4 m
< 10 kHz
~ 10−7 m
5 μm–3 mm
Yes
UET
~ 10−3 m
< 10 kHz
~ 10−5 m
60 μm–2 mm
No
LFD
~ 10−6 m
< 2 kHz
~ 10−8 md
0.25 μm–2 mm
No
TIRM
~ 10−4 m
< 15 kHz
~ 10−5 m
18 μm–1.4 mm
No
X-ray tomography
~ 10−4 m
< 50 Hz
~ 10−6 m
64 μm–6.5 mm
No
OWF
~ 10−5 m
< 1 MHz
~ 10−6 m
5 μm–2 mm
No
aThe fiber-optic method is omitted because of the same principle as OWF. LIF: laser-induced fluorescence, UET: ultrasonic echo technique, LFD: laser focus displacement meter, TIRM: total internal reflection method, OWF: optical waveguide film
bThe information in these columns is based only on the literature referred to in this study
cThis information represents representative values
dThe value in the case in which there is no layer other than air between the sensor and the measurement object, and the object is a mirror surface
In this paper, we develop a new OWF method that is specialized for measuring cusp thickness and is applicable to complex disturbance wave flow. This method utilizes geometrical optics-based array sensing to simultaneously estimate the liquid film thickness and interfacial wave shape from the OWF output signal. First, to test the basic measurement principle in the experiments, we conducted OWF measurements of a pseudo-film flow using a metal specimen simulating interfacial waves. A meaningful signal indicating the wave-cusp passage (CP signal) was observed, and the signal processing for cusp detection was established using the characteristics of the CP signal. We then conducted OWF measurements of the liquid film flow driven by high-speed gas flow (jG = 20–70 m/s) at the entry region on a flat plate. There are few reports on shear-driven liquid films in the entry region despite the importance of engineering. Measurements in this region are preferable for confirming the proposed detection method, as the onset and development of the turbulent boundary layer are expected to cause more unsteady and wavy interface motions than those in fully developed regions. As a result, the cusp thicknesses of the wavy interface were successfully extracted from the original OWF signals. Further, the measured thickness data showed reasonable agreement with conventional thickness estimation methods. Based on these results, we demonstrated the effectiveness of the developed method.
2 Basis of liquid film measurements with OWF
2.1 OWF structure
Figure 2 is a structural schematic of the OWF used in this study. The OWF allows light to propagate by connecting the connector to the optics, as seen in Fig. 2a. A close-up view of the sensor group is also provided in the figure. The sensor array consists of three channels located at a pitch of 0.25 mm in a row. We defined channel numbers #1, #2, and #3. The cross-sectional view of a single channel (Fig. 2c) shows that each channel consists of a core layer, a cladding layer, and a polyimide layer, with a micromirror at the core end. Micromirrors were used as sensors in the channels. The OWF is 0.1 mm thick, and the core is square with a 42-μm side length. Since the refractive index of the core is slightly higher than that of the cladding, the light causes total reflection at the core–cladding interface and propagates in the core. At the core end, a micromirror reflects the propagating light vertically and passes it through the OWF’s surface.
A significant advantage of the OWF is its high degree of flexibility in production thanks to the photo-printing technique (Mori et al. 2010), allowing the sensor number, sensor position, and waveguide shape to obtain broad spatial distributions of the liquid film profiles. Additionally, the surface layer of the OWF is a polyimide, which can easily dope ionic substances to arrange its surface wettability if needed. The OWF specifications in this study, i.e., the sensor pitch and untreated surface, were determined to directly measure the cusp thickness of disturbance waves in an air–water two-phase system using the minimum necessary processes.
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2.2 Measurement principle of OWF
Figure 3 is a schematic of the basic liquid film thickness measurement using the OWF. The OWF is fixed to the channel wall; in the thickness measurement, the target is the liquid phase covering the sensor. The light reflected off the micromirror is directed into the liquid. The emitted light is partially reflected at the gas–liquid interface and then returns into the OWF again. The intensity of this reflected light depends on the liquid film’s thickness. The relationship between the liquid film’s thickness and the reflected intensity derived from the preliminary experiments is depicted in Fig. 4. In the case of a thick liquid film, the intensity of the light returning to the OWF decreases due to the divergence angle of the emission beam’s profile. Conversely, in the case of a thin liquid film, the reflected light intensity increases. This relationship is obtained beforehand as a calibration curve, and then the thickness can be estimated from the intensity level. The same approach is utilized in conventional fiber-optic measurements of liquid film flow (Alekseenko et al. 2003; Zaitsev et al. 2003; Oliveira et al. 2006; Mizushima 2021, 2023; Okui et al. 2022; Miyachi et al. 2022; Furuichi et al. 2023a, 2023b; Watanabe et al. 2023).
We introduce a new method for simultaneously detecting a film’s thickness and the shape of an interfacial wave by array sensing. Figure 5 illustrates the concept: channel #2 functions as an Active Ch, emitting light and receiving reflected light from the liquid film’s interface, whereas channels #1 and #3 each function as Passive Ch, receiving only the reflected light. These sensors are aligned in the direction of the liquid film flow. The figure in the frame is a cross-sectional view of the OWF sensor section, showing the peaks of the interfacial wave, i.e., the wave cusps, passing over the sensors in the order of times (I), (II), and (III). In this case, the internal reflection is detected in the order of channels #1, #2, and #3 due to the geometrical optics relationship. Channel #2 detects all wave extrema (i.e., the wave’s cusps and troughs) and measures the film thickness using the calibration curve (Fig. 4). Thanks to the small sensor size, the reflected light intensity of waves with an interfacial curvature of less than 5.0 mm-1 is consistent with that of a flat film surface (Furuichi et al. 2023b). Therefore, the uncertainty in the measured thickness caused by the shape topology of the interface is negligible for almost all waves arising on the liquid film (Chang et al. 2019). In contrast, channels #1 and #3 are turned on only when the cusps pass over the sensor array. Therefore, we can directly measure the cusp thickness by extracting the intensity of the channel #2 signal when the signals from channels #1 and #3 appear clearly. Hereafter, we refer to this meaningful peak in #1 and #3 as a cusp passage signal (CP signal).
Fig. 5
Principle of OWF array sensing for wave-cusp detection. (I)&(III) Wave shape detection: channels #1 and #3 detect the passage of the wave cusp. (II) Film thickness measurement: channel #2 measures the thickness of all wave extrema. The measured thickness, when channels #1 and #3 detect internal reflection (i.e., the appearance of the CP signal), corresponds to the cusp thickness
Figure 6 is a schematic of the optics in the measurement system using the OWF. A laser diode (cat. no. OBIS637LX, Coherent Corp., Saxonburg, PA, USA) was connected only to channel #2, and the laser beam was guided through an objective lens (cat. no. 101-40S6, Yashima Optical Co., Tokyo, Japan) into an optical circulator and OWF. The reflected light on the liquid film interface was received by the core of the OWF and traveled in the opposite direction in the optical circulator (model WMC1H1F, Thorlabs, Newton, NJ, USA). The reflected light was converted into an electrical signal by a photomultiplier tube (cat. no. R928, Hamamatsu Photonics, Hamamatsu, Japan) and recorded as the output voltage by a recorder (#8861-50, Hioki, Nagano, Japan) through an amplifier (cat. no. C7319, Hamamatsu Photonics). This method requires a series of measurement systems for each channel.
Fig. 6
Optics of the OWF measurement system. The light from the laser diode is introduced to channel #2. The reflected light from the liquid film interface is converted into an electrical signal by the photomultiplier tubes connected to each channel
In this experiment, we investigated the characteristics of the CP signal using the solid–liquid interface. Figure 7 is a schematic of the experimental apparatus. We used a test specimen made of stainless steel with a sinusoidal wave shape because it is difficult to experimentally simulate arbitrary wave shapes with a gas–liquid interface. The surface of the specimen was mirror-polished. The OWF was fixed to the bottom of an acrylic vessel filled with ultrapure water. Table 2 lists the experimental conditions for the test specimen. The specimen was installed such that the smallest distance δbase from the OWF surface to the specimen was 0.2 mm. The specimen parameters were as follows: amplitude a of 75 μm, wavelength λ of 1.08 mm, and curvature κ of 2.53 mm−1. We set these conditions based on a report on liquid film flow on a flat plate (Chang et al. 2019, 2023).
Fig. 7
Schematic of the experimental apparatus for the pseudo-film flow measurement
The specimen was moved just above the OWF sensors by applying an actuator (cat. no. HS-KT45 20A0500L, THK Co., Tokyo) in the x-axis direction. We recorded the output signal from the OWF with a recorder and visualized the specimen with a high-speed camera (model FASTCAM SA-X2, Photron, Tokyo) that was synchronized with the recorder. We binarized the captured images and calculated the distance between the specimen and OWF surface just above channel #2. The image captured by the shadowgraph method was 4.8 µm/pixel. The sampling frequency of the OWF was 1 MHz. The frame rate and shutter speed were set to 1,000 frames per second (fps) and 1/2,000 s, respectively.
3.2 True film flow measurement
We applied the OWF to liquid film flow measurements on a flat plate in the entry region to confirm the characteristics of the CP signal. Figure 8 is a schematic of the experiment. An airflow was generated by a blowout-type wind tunnel apparatus, and a liquid film flow was formed by the shear force generated between the airflow and liquid. The airflow velocity was controlled by the inverter frequency of the blower (model CTB-1003OB, Teral Inc., Fukuyama, Japan). The airflow passed through the settling chamber of the honeycomb structure and accelerated in the nozzle channel; it was then sent to the test section. Tap water driven by a water pump (model PMD-121B7B1, Sanso Electric Co., Himeji, Japan) from the water tank was sent to the test section. The volumetric flow rates of air and water were measured by flow meters (air: model TV1100, Oval Corp., Tokyo; water: model FD-SS02A, Keyence, Osaka, Japan). The test environment was at room temperature and atmospheric pressure.
Fig. 8
Overview of the experimental apparatus for the true film flow measurement. Airflow is introduced from the wind tunnel, which is accompanied by water in the test section, resulting in the formation of a wavy liquid film flow. At the downstream end of the test section, the flows are released into the atmosphere
A schematic of the test section is provided in Fig. 9. The square flow channel (50 mm wide, 30 mm high) is made of acrylic material. The tap water flowed from the channel base through a slit. The OWF was fixed on a stainless steel base plate installed downstream of the flow channel, with the distance x from the slit to channel #2 set to approximately 160 mm. The OWF sensors were installed in the center of the channel width, parallel to the flow direction.
We measured the horizontal liquid film flow by the OWF. The output signal from the OWF was recorded by a recorder, and the interfacial wave shape of the liquid film flow passing over the OWF sensors was visualized from a side view by a high-speed camera (FASTCAM SA-X2, Photron) synchronized with the recorder. The image captured by the shadowgraph method was 42 µm/pixel. The gas superficial velocity jG was changed every 10 m/s from 20 to 70 m/s, and the liquid volumetric flow rate Q was set at 300 mL/min. The liquid Reynolds number Ref = Q/(bνL) for each condition was approximately 113 on average. Here, νL is the liquid kinematic viscosity. b is the liquid film width, which was measured from the photos taken by an overhead view camera after the OWF measurements. These flow rate conditions are chosen so as to cover different flow regimes (i.e., ripple flow and disturbance wave flow) based on the flow pattern map reported by Fukano (1998). The ripple flow regime is characterized by wavy flow with an elongation breadthwise to a certain length that is not always identical across different instances, corresponding to jG = 20 and 30 m/s in this study. In contrast, the disturbance wave flow regime, as illustrated in Fig. 1, has more three-dimensional interface structures compared to ripple flow, corresponding to jG > 40 m/s. The OWF sampling frequency was 100 kHz, and the measurement time was 50 s. The frame rate and shutter speed of the high-speed camera were set to 100,000 fps and 1/200,000 s, respectively.
4 Results and discussion
4.1 Characteristics of the cusp passage signal
Figure 10 depicts the output signals delivered from the OWF and the time-series thickness above the Active Ch (channel #2) by visualization. The output signal of the Active Ch has clear peaks at the local maximum and minimum heights of the wavy interface. This sensing method is also sensitive to the interface inclination. Active Ch receives the most reflection at the locally flat part of the wave, i.e., the wave’s relative maximum and minimum. Therefore, the signal consists of many peaks. Note that the peak levels are inversely related to the thickness. The peak values are extracted from the Active Ch’s signal, and then the film thickness at the crests and troughs of the interfacial wave can be evaluated (Okui et al. 2022; Mizushima 2023; Furuichi et al. 2023a), using the calibration curve shown in Fig. 4.
Fig. 10
Time-series thickness obtained by visualization and output signal delivered from the OWF. Note that the visualization result is the time-series change in the distance between the test specimen and OWF surface directly above channel #2. Based on the analysis, the peaks at times (I′) to (III′) correspond to those at times (I) to (III) in Fig. 5
Focusing on the output signal from the Passive Ch, the CP signals of #1 and #3 clearly appear just before and after the crest’s passing, respectively. This should be an indicator that the wave cusp has passed over the array sensors of the OWF, as illustrated in Fig. 5. We parametrically investigated how the CP signal appears by the following analysis. Figure 11a shows the geometrical optics calculation model. This model approximates the laser beam as a single ray, ignoring scattering and multiple reflections. This assumption is based on the fact that the laser beam profile is dominated by the ray near the central axis. The distance l from channel #2 to the terminus of the reflected ray is,
$$l=\delta\; \text{tan}\left(2\theta \right)$$
(1)
where δ is the film thickness, and θ is the interface angle. Here, the output levels of #1, I#1, are the function of l,
where Δs is the sensor pitch (0 < l ≤ 2Δs, Δs = 0.250 mm in this study). I#1 distributes shown in Fig. 11b as a function of δ and θ (0 < δ < 1.2 mm and 0 < θ < 30°). The trends of δ and θ, calculated based on the experimental conditions in Table 2, are plotted as a red solid line. The other conditions (Chang et al. 2019, 2023) are also shown as black dashed lines for reference. The calculated range corresponds to the passage of the cusp’s ascending limb, as depicted in Fig. 11c. For details of the calculation, please see Appendix A. According to the current experimental conditions (red solid line), I#1 increases from the edge of the wave cusp (white triangle), reaches a maximum value, and then decreases to the wave cusp (white square). That is, the output signal of #1 is estimated to peak just before the cusp passes over channel #2. In contrast, the output signal of #3 should peak just after the cusp passes. These trends agree with the actual signal in Fig. 10. It should be noted that the above trends significantly depend on Δs. According to other results, the overall levels of I#1 become small at low liquid superficial velocity (Ref = 26) when Δs = 0.250 mm, and it might be difficult to find the CP signal. Nevertheless, our interest is to find the disturbance wave that frequently emerges at a relatively large liquid flow rate. For Ref > 100, the estimated signals of #1 are similar to that of the current experiment; therefore, we will undoubtedly find the cusp passage based on the appearance of the CP signal. We further emphasize that our experiment proved the cusp detection for thinner liquid film conditions than conventional literature, which makes it challenging to receive a clear CP signal.
Fig. 11
Parametric analysis of the CP signal based on the geometrical optics calculation
4.2.1 Confirmation of the cusp passage signal characteristics
As a typical example of the output signal, the OWF signal obtained at jG = 20 m/s is shown in Fig. 12. There are numerous peak-like signal changes in #2, indicating that channel #2 constantly measures thickness fluctuations. In contrast, the CP signals are infrequent, as the actual waves are not necessarily two-dimensional waves like in the experiment on pseudo-film flow measurement, and the cusps do not always pass in a straight line over the sensor array. We emphasize that the OWF measurements successfully obtained the output signals without temporal leakage. The measurement methods for the few-dozens-kHz band are unable to detect rapid thickness fluctuations, such as those shown in the inset.
Fig. 12
Typical output signal (jG = 20 m/s). jG: gas superficial velocity
A temporally magnified signal and visualization result are provided in Fig. 13, in which the CP signals clearly appear in both #1 and #3. The red arrows in the image represent the positions of channels #1 to #3. As shown in the figure’s panel (a), the signal peaks at times (I’) to (III’) appear in the order of #1, #2, and #3, and it can thus be estimated that the wave cusp passes over the sensors as shown in Fig. 5. This situation is in good agreement with that illustrated in Fig. 13b. This correspondence was observed in almost all cases where relatively large-amplitude waves existed over the sensor array. However, in Fig. 14, no CP signals exist in #1 and #3, although the cusp passage over the sensor arrays is visualized. According to Fig. 11b, the CP signal inevitably decreases in small-amplitude waves with a thin thickness and slight curvature. The visualized image exactly shows the detection of such a small wave. As we concluded from the model analysis, the CP signal indicates the passage of cusps with a relatively large amplitude, even in the actual wavy film flow.
Fig. 13
Example of the OWF signal and visualization result when the CP signal appeared (jG = 20 m/s)
In addition, the CP signal levels provide information about the cusp shape. The CP signal intensities in #1 and #3 in Fig. 10 are similar; however, they differ in Fig. 13a due to an asymmetric cusp shape. The difference in the CP signals arises from the difference in the thickness and angle of the local interface. The cusp shapes are symmetrical in Fig. 10; therefore, the positional relationship of channel #1, interface, and channel #2 is the mirror-image relationship of channel #2, interface, and channel #3. As a result, both CP signal levels are almost equal. In other words, when different CP signal levels are obtained, we can identify that the shape of the wave cusps passing over the sensor array is asymmetrical.
In Fig. 15, we organized the relationship between jG and “the occurrence rate α of the situation with different CP signal levels.” Here, α is evaluated by the following equation:
$$\alpha =\frac{n}{{N}_{\text{All}}}\times 100$$
(3)
where NAll is the total number of CP signals detected in the measurement time, and n is the number of patterns in which the CP signal level values differ between #1 and #3 in the measurement time. The colored area indicates the flow regimes, i.e., ripple flow and disturbance wave flow, based on the conventional classification (Fukano 1998). The transition from ripple flow to disturbance wave flow, marked by a significant increase in the asymmetry of the cusp shape, is a key point of interest due to the shear stress from a large jG (Berna et al. 2014). Figure 15 reveals a clear trend: as jG increases, α also increases, reaching a near-constant value after jG = 40 m/s. This transition is consistent with the flow regimes predicted by the existing literature, indicating that the cusp shape becomes increasingly asymmetric with increasing jG. It should be noted that all liquid film flow classifications were derived from researchers’ observations with transparent channels. However, by utilizing the CP signal, we can directly identify the passage of the wave cusps and their shape information from only the OWF signals, even in closed environments under unknown conditions.
Fig. 15
Relationship between jG and the occurrence rate of the situation with different CP signal levels
4.2.2 Detection algorithm for the wave-cusp passage and evaluation of the liquid film thickness
We establish a signal processing for wave-cusp extraction through array sensing of the OWF. Figure 16 is a flowchart of our new signal processing method by the OWF. Steps (1) to (3) are the same as those used in conventional signal processing methods (Furuichi et al. 2023a). In Step (1), all the acquired data of the output signal from the OWF are input. In step (2), the peaks of the output signals of all channels are detected. In step (3), all the peak values Vfp,#2 of channel #2 are converted into liquid film thickness δ according to the calibration curve. Note that δ is a mixture of the film thickness values of the cusps and troughs of interfacial waves. The appearance of the CP signal corresponds to the passage of the cusp with large amplitudes over the sensor array in Sect. 4.2.1. Therefore, we added step (4) to extract the cusp thickness. For details of the signal processing algorithm, please see Appendix B.
Fig. 16
Flowchart of the proposed signal processing method for the OWF
Figure 17 provides the probability density functions (PDF) of all liquid film thicknesses obtained in step (3). As jG increases, the profile of each PDF changes noticeably: the distribution becomes less symmetrical as jG approaches 30–40 m/s and appears to resemble a log-normal distribution. This jG range corresponds to the transition regime from rippled flow to disturbance wave flow, as shown in Fig. 15. These distributions and their trends are common patterns; however, we emphasize once again that the OWF successfully obtained them without temporal leakage, thanks to its 100 kHz resolution.
Fig. 17
Probability density functions of all liquid film thicknesses obtained in step (3)
Figure 18 shows the overlaid PDF before and after step (4). The black line in Fig. 18 represents the raw data of all film thicknesses (identical to Fig. 17), whereas the red line represents the extracted cusp thickness. The relatively large thickness, including the disturbance wave, is clearly extracted under every condition, and we can also clearly find their dominant mode values. Moreover, it is notable that the distributions deviate from a unimodal distribution, particularly when jG > 30 to 40 m/s. The extracted distribution consists of both the large waves and the accompanying ripple waves, as illustrated in Fig. 1, which is nothing but a characteristic of the disturbance wave flow. These results prove that the cusp thicknesses of the principal component in the wavy interface can be extracted from the original OWF signals using the newly established algorithm.
Fig. 18
Probability density functions of the extracted cusp thickness with new signal processing. The black line represents the raw data of all film thicknesses (identical to Fig. 17). The red line represents the cusp thickness extracted from the raw data
To demonstrate the effectiveness of our new method, we first evaluated the characteristic thicknesses measured by the OWF by comparing them with existing models for predicting liquid film thickness. The prediction models are typically categorized into two types: (i) those based on the local force balance between the gas and liquid shear forces across the interface (Inoue et al. 2022), and (ii) those based on the global force balance between the gas shear force and wall friction (Zhang et al. 2023). The shear stress τi of the turbulent gas flow acting on the interface is expressed as follows:
where fi and ρG are the interfacial friction factor [-] and gas density [kg/m3], respectively. Assuming that the liquid film flow is Couette flow, the local balance of shear stress across the interface is deduced:
where µL, uL, and δh are the liquid viscosity [Pa‧s], liquid film velocity [m/s], and height at the balance of forces [m], respectively. From Eqs. (4) and (5) and the flow rate equation Q = bδhuL, the thickness of model (i) is formulated as follows:
where Q and b are the liquid volume flow rate [m3/s] and liquid film width [m], respectively. fi is the only unknown factor for predicting the height (i.e., the liquid film thickness) in Eq. (6). In fact, there are very few models for the friction factor of liquid films on flat plates, whereas numerous models exist for circular pipes. Recently, Inoue et al. (2022) proposed appropriate scaling to alternate a circular model into a plate model with the following definition, such that L = x and U = Q/(bx). Here, L, U, and x are the characteristic length [m], characteristic velocity [m/s], and measurement point [m], respectively. Using this scaling, we apply the Wallis correlation given in Eq. (7) to fi,
This equation is widely used for gas–liquid two-phase flow in pipes. We will call the thickness prediction model (i) “Inoue model.”
The thickness prediction model (ii) is generally assumed for liquid film flow in the pipe. This model often introduces the dimensionless film thickness δ+ as follows:
Here, u* is the friction velocity, defined as u* = (τw/ρL)1/2, where τw and ρL are the wall shear stress [N/m2] and liquid density [kg/m3], respectively. Assuming a global balance between the wall shear stress and interface shear stress, u* is calculated as follows using Eq. (4): u* = (τi/ρL)1/2 = (0.5fi ρG jG2/ρL)1/2. fi is obtained from Eq. (7). Substituting the expression for the friction velocity into Eq. (8) and replacing δh with L, considering that the height at the balance of forces is a function of the characteristic length, the following simplified equation is obtained:
where the liquid Weber number, gas Weber number, and liquid Capillary number are defined as WeL = ρLU2L/σ, WeG = ρGjG2L/σ, and CaL = UµL/σ, respectively. Here, σ is the surface tension coefficient [N/m]. Zhang et al. (2023) approximated δ+ as the normalized liquid film thickness δh/L, and proposed the film thickness correlation given in Eq. (10):
We will apply this equation to liquid film flow on a flat plate through the scaling described above and call the thickness prediction model (ii) “Zhang model.”
Figure 19 schematically depicts the relationship between the major thickness levels analyzed in this study and the probability density function of liquid film thickness. The analyzed film thickness variables are base film thickness, median film thickness, and cusp film thickness. First, in order to confirm the accuracy of the conventional film thickness measurement method using OWF, we compare the thickness model with the base film thickness and median film thickness, which are characteristic values. Next, in order to demonstrate the effectiveness of the proposed method for measuring cusp thickness, we compare it with the maximum film thickness defined by the conventional 99% cumulative probability method.
Fig. 19
Major thickness levels analyzed in this study and their corresponding probability density functions of liquid film thicknesses. This schematic diagram depicts the typical relationship between interface structure and film thickness distribution in disturbance wave flow
Averaging is often employed to evaluate measured thickness against a prediction model; however, it is not appropriate to represent the liquid film thickness in our case. It is difficult to define a reasonable number of samples for averaging due to the assumed predominant non-steady state at the entry region during measurements. Therefore, we employed the mode and median values obtained from the film thickness distribution shown in Fig. 17 and compared them with the predicted thickness model, as shown in Fig. 20. According to these graphs, the mode values, referred to as the base film thickness, are in close agreement with the Inoue model, whereas the median values, referred to as the median film thickness, align closely with the Zhang model. This result states that the height at the balance of local forces in the model (i) corresponds to the base film thickness, whereas that of global forces in the model (ii) corresponds to the median film thickness.
Fig. 20
Comparison of the measured film thicknesses with those predicted by the analytical models. Notably, the mode and median of film thickness decrease as jG increases
Figure 21 summarizes the results of comparing the employed values of the measured film thickness with the corresponding models. The measurements are underestimated by up to approximately 25% compared to the models at jG = 20 and 30 m/s, possibly due to the non-uniform distribution along the film width. The measurement position in this study is located at the center of the channel width. Under low jG conditions, the thickness is distributed along the width of the channel due to the secondary flow induced by the rectangular channel, leading to a relatively thinner thickness at the center (Fukano et al. 1984). As jG increases, this distribution may become more uniform due to the predominance of the disturbance waves. Thus, the measurements were in good agreement with the models, with differences of less than 10% at jG > 40 m/s. In conclusion, we confirmed that the liquid film thicknesses measured by the OWF are in reasonable agreement with the thickness models.
Fig. 21
Comparison of the characteristic measured film thicknesses with predictions from the corresponding analytical model. This graph is a re-plotted version of Fig. 20, and compares the mode and median values of all measured film thicknesses with the predicted values of the Inoue model and Zhang model, respectively. Notably, the mode and median of film thickness decrease as jG increases
Further, we evaluate the cusp thicknesses extracted using the new algorithm. Figure 22 compares the dominant mode values of the extracted cusp thickness distribution in Fig. 18 with the maximum film thickness defined by Fukano and Furukawa (1998) and Hazuku et al. (2008). This maximum film thickness represents the wave thickness of the liquid film calculated from the 99% cumulative probability of the thickness distribution in Fig. 17. These independently obtained values result in very good agreement with each other. In this context, the cusp thickness in complex interface structures is certainly extracted only by identifying the CP signal, without the conventional maximum film thickness identification method requiring a large record length.
Fig. 22
Comparison of the dominant mode values of the extracted cusp thickness with the 99th percentile of all measured film thicknesses in Fig. 17
Throughout the experiment, the cusp thickness did not change significantly, in contrast to the base film thickness and median film thickness. This phenomenon may be an inherent characteristic of the airflow-induced wavy film flow in the short entrance region. However, other studies with experimental situations similar to ours (Leng et al. 2018; Chang et al. 2019, 2023) reported that cusp thickness tends to decrease with increasing jG. This discrepancy may stem from the differences in the resolution of the measurement method and aspect ratios of the flow channels. The characteristics of the liquid film flow in the entry region are critical for analyzing and predicting droplet impingement in the flow channel, as the wave cusps lead to droplet atomization. Nevertheless, there are currently no models of the wave cusp that account for entry regions, indicating the need for further detailed investigation in the future.
5 Conclusions
This report presents an experimental study measuring wavy liquid film flow driven by high-speed airflow using array-based sensing with an optical waveguide film (OWF). This study aims to demonstrate the effectiveness of this new technique for identifying wave cusp thickness from OWF signals. In the first experiment, we performed a pseudo-film flow measurement using a wavy test specimen to validate the basic measurement principle of the new method. We confirmed the appearance of a characteristic signal, referred to as the CP signal, derived from the cusp passage. A limitation in cusp detection at low liquid Reynolds number (Ref), restricted by the sensor pitch, was identified through geometrical optics analysis; however, this limitation is mitigated for Ref > 100. If necessary, this limitation can be easily addressed by modifying the OWF design. In the second experiment, we applied the new technique to a shear-driven liquid film flow on a flat plate in the entry region as part of a case study. A comparison between the OWF signal and visualization results of the interfacial wave shape confirmed that the CP signal from this sensor array is capable of detecting wave cusps with relatively large amplitudes. Moreover, the symmetry of the detected cusp shape was also estimated by the CP signal intensities, which showed a reasonable trend in agreement with the classification of conventional flow regimes.
Based on the characteristics of the CP signal, we developed an algorithm to detect the wave cusp thickness hidden in the raw OWF signal. To demonstrate the effectiveness of our new method, we first evaluated the accuracy of the film thickness measurements for wavy film flow using an OWF. Given the unsteady nature, we employed the mode value (i.e., base film thickness) and median value (i.e., median film thickness) of the measured thickness data as representatives. We compared them with those of conventional thickness models and found that they were in reasonable agreement. Further, the newly developed algorithm made it possible to extract the cusp thickness from the measured film thickness. The dominant mode value of the extracted cusp thickness was closely aligned with the maximum film thickness defined by the conventional 99% cumulative probability method. This result demonstrated that cusp thickness can be directly estimated by detecting the CP signal from the OWF measurement without a large record length. The characteristics of the liquid film flow in the entry region are crucial for analyzing and predicting droplet impingement in the flow channel; however, there are currently no models of the wave cusp that account for entry regions. Our new method could significantly contribute to understanding such uncharted wavy film flows.
Acknowledgements
This work was carried out under the collaborative research with Chubu Electric Power Co., Inc., Japan (Grant No. 2021-0057) and JSPS KAKENHI Grant Number JP20K14647 and JP23K17727.
Declarations
Conflict of interest
The authors declare no competing interests.
Ethical approval
Not applicable.
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Appendix A: Calculation procedures of the local thickness and interfacial angle for the pseudo-wavy film analysis
Figure 23 depicts the wavy liquid film interface model. As shown in Eq. (11), the wavy liquid film interface is modeled as a sinusoidal wave with amplitude a and wavelength λ.
Integrating the base film thickness δbase into equation (13), the distance (i.e., film thickness) δ between the wall surface and the interface is expressed as:
Assuming that a represents half of the wave height H (where H is defined as the difference between the maximum film thickness and base film thickness), the unknown parameters a, κ, and δbase in Eqs. (14) and (15) can be substituted with the experimental conditions in Table 2 or conventional measurement data (Chang et al. 2019, 2023).
Appendix B: Algorithm for calculating the cusp thicknesses from OWF signals
Algorithm 1 presents the pseudocode for the cusp thickness detection algorithm in the OWF. In this study, we used the findpeaks function in MATLAB (Math Works, Natick, MA, USA) to detect the peaks in all OWF signals.
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