Experimental Validation of Falling Liquid Film Models: Velocity Assumption and Velocity Field Comparison

polymers

Article Experimental Validation of Falling Liquid Film Models: Velocity Assumption and Velocity Field Comparison

Ruiqi Wang, Riqiang Duan * and Haijun Jia

Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100085, China; [email protected] (R.W.); [email protected] (H.J.) * Correspondence: [email protected]

Abstract: This publication focuses on the experimental validation of film models by comparing constructed and experimental velocity fields based on model and elementary experimental data. The film experiment covers Kapitza numbers Ka = 278.8 and Ka = 4538.6, a Reynolds number range of 1.6–52, and disturbance frequencies of 0, 2, 5, and 7 Hz. Compared to previous publications, the applied methodology has boundary identification procedures that are more refined and provide additional adaptive particle image velocimetry (PIV) method access to synthetic particle images. The experimental method was validated with a comparison with experimental particle image velocimetry and planar laser induced fluorescence (PIV/PLIF) results, Nusselt’s theoretical prediction, and experimental particle tracking velocimetry (PTV) results of flat steady cases, and a good continuity equation reproduction of transient cases proves the method’s fidelity. The velocity fields are reconstructed based on different film flow model velocity profile assumptions such as experimental film thickness, flow rates, and their derivatives, providing a validation method of film model by

comparison between reconstructed velocity experimental data and experimental velocity data. The comparison results show that the ﬁrst-order weighted residual model (WRM) and regularized model

Citation: Wang, R.; Duan, R.; Jia, H. (RM) are very similar, although they may fail to predict the velocity field in rapidly changing zones Experimental Validation of Falling such as the front of the main hump and the first capillary wave troughs. Liquid Film Models: Velocity Assumption and Velocity Field Keywords: falling liquid films; PIV/PLIF; SPLIF; PTV; WRM Comparison. Polymers 2021, 13, 1205. https://doi.org/10.3390/ polym13081205 1. Introduction Academic Editor: Alexis Darras Falling liquid films are examples of open-flow systems that bring a laminar steady state to a disordered state in both time and space as the film develops [1,2]. Film flows Received: 6 March 2021 have a long wave nature, the typical length of which is much larger than the thickness Accepted: 6 April 2021 Published: 8 April 2021 of the film. These features enable modelling of the flow with reduced equations from the Navier–Stokes equations; however, the models need solid experimental data to be

Publisher’s Note: MDPI stays neutral validated and verified [1,3,4]. with regard to jurisdictional claims in To validate these different models, knowledge of the velocity field with the correlating published maps and institutional affil- film topology under various surface waves would be of great value [5]. High-quality iations. velocity field data with correlating film interfaces also have the following benefits: (i) velocity field data make it possible to calculate the transfer processes in flowing films [6]; (ii) they provide essential testing of the accuracy of models in the nonlinear regime by evaluating the accuracy of their velocity field predictions under waves [1]; and (iii) can directly validate basic assumptions applied in different models. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. Experimentally, the hydrodynamic characteristics of falling liquid films have been This article is an open access article investigated vigorously in the past, especially film thickness, which indicates the interface distributed under the terms and wave status of film flows. Various experiments have attempted to measure velocities in conditions of the Creative Commons falling liquid films since the 1950s, and the applied techniques include hot wire anemome- Attribution (CC BY) license (https:// try [7], stereoscopic photography [8], laser Doppler anemometry [9–11], high-speed video creativecommons.org/licenses/by/ photography [12,13], particle image velocimetry (PIV) [14–18], and particle tracking ve- 4.0/). locimetry (PTV) [15]. However, most of the previous investigations of the velocity field

Polymers 2021, 13, 1205. https://doi.org/10.3390/polym13081205 https://www.mdpi.com/journal/polymers Polymers 2021, 13, x FOR PEER REVIEW 2 of 20

Polymers 2021, 13, 1205 video photography [12,13], particle image velocimetry (PIV) [14–18], and particle tracking2 of 19 velocimetry (PTV) [15]. However, most of the previous investigations of the velocity field in wavy film flows, either time-averaged or instantaneous data, were compared to the parabolic profile, which is a theoretical model given by Nusselt for steady and flat film inflows. wavy Even film for flows, the eitherexperiments time-averaged stated in or the instantaneous context above, data, only were some compared experimentally to the parabolicmeasured profile, spatial which-temporal is a theoretical velocity fiel modeld data given within by transientNusselt for wavy steady falling and filmsflat film are flows.available. Even The for thelow experiments spatial or temporal stated in resolution the context also above, limits only the some further experimentally application of mea- ex- suredperimental spatial-temporal velocity field velocitys. field data within transient wavy falling films are available. The lowThis spatial paper or focuses temporal on resolutionthe high spatial also limits resolution the further and temporal application resolution of experimental of experi- velocitymental fields.velocity field measurements in transient film flows, which will be beneficial for furtherThis improving paper focuses the velocity on the highfield spatialmeasurement resolution capacity and temporalin a falling resolution film on an of in exper-clined imentalplate, allowing velocity for field quantitative measurements velocity in transient field research film flows, in films which with will submillimete be beneficialr thick- for further improving the velocity field measurement capacity in a falling film on an inclined ness, mostly corresponding to large Kapitza number liquids such as water. plate, allowing for quantitative velocity field research in films with submillimeter thickness, The remainder of this article is structured as follows: in Section 2, the experimental mostly corresponding to large Kapitza number liquids such as water. facility, including the film test section, flow loop, PIV/PLIF system and film experiment The remainder of this article is structured as follows: in Section2, the experimental fa- cases, is briefly described. In Section 3, the experimental methodology is illustrated, and cility, including the film test section, flow loop, PIV/PLIF system and film experiment cases, additional methodological validations are given, including Nusselt’s validation with is briefly described. In Section3, the experimental methodology is illustrated, and addi- PIV/PTV and an adaptive PIV method validation. In Section 4, data analysis is performed, tional methodological validations are given, including Nusselt’s validation with PIV/PTV mainly focusing on a transient velocity field comparison. In Section 5, conclusions are and an adaptive PIV method validation. In Section4, data analysis is performed, mainly drawn. focusing on a transient velocity field comparison. In Section5, conclusions are drawn.

2.2. ExperimentalExperimental Facility Facility 2.1.2.1. ExperimentalExperimental Film Film Test Test Section Section and and Flow Flow Loop Loop AA short short description description of of the the film film test test section section and and flow flow loop loop is is provided provided in inFigures Figures1 1and and2 2.. MoreMore details details can can be be found found in in the the work work of of Wang Wang [ 19[19].]. The The principal principal component component of of the the film film testtest section section is is a a 670 670× ×220 220× × 0.7 mm3 sodasoda–lime–lime glass plateplate thatthat servesserves asas thethefilm film substrate substrate andand a a polymethyl polymethyl methacrylate methacrylate (PMMA) (PMMA) box. box. The The measurements measurements in in this this article article were were taken taken 350350 mm mm downstream downstream of of the the slit slit and and center center in in the the spanwise spanwise direction. direction.

Figure 1. Schematic diagram of the experimental flow loop and film test section. The italic letters x, y, and z denote the Figure 1. Schematic diagram of the experimental flow loop and film test section. The italic letters x, y, and z denote the streamwise,streamwise, cross-stream, cross-stream and, and spanwise spanwise directions directions of of the the film film description description coordinate coordinate system, system, respectively. respectively.

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35°

Figure 2. ForFigure the film 2. test For section, the film high-speed test section, camera high-speed and laser camera sheet, and the laser high-speed sheet, the camera high- isspeed set up camera beside is the set test section, as illustratedup in thebeside figure. the test The section, left part as and illustrated right part in isthe the figure. same The test sectionleft part of and a high right speed part is camera the same and test laser sheet, just being given insection different of a views, high speed in which camera the leftand is laser from sheet, the film just cross-stream being given direction, in different and views, the right in which is from the the left streamwise is from the film cross-stream direction, and the right is from the streamwise direction. The high direction. The high speed camera’s pitch angle against the film spanwise direction is set to 35 degrees. speed camera’s pitch angle against the film spanwise direction is set to 35 degrees. Fluid in the flow loop is pumped with a peristaltic pump, which could provide an Fluid in accuratethe flow butloop pulsed is pumped small with flow a rate.peristaltic There pump, is a flow which distribution could provide part an in the film test accurate but pulsedsection small buffering flow andrate. dispensing There is a flow the fluiddistribution into a pulsation-free part in the film flow. test Thesection flow loop has an buffering andadditional dispensing recirculating the fluid into loop, a pulsation electrical-free heater, flow. and The ultrasonic flow loop plate, has whichan addi- could maintain tional recirculatingconstant loop, fluid elec temperaturetrical heater, and ultrasonic particle concentrations. plate, which could Additionally, maintain con- disturbances are stant fluid temperatureprovided with and compressedparticle concentrations. air regulated Additionally, by an electromagnetic disturbances valve, are and pro- the disturbance vided with compressedfrequency isair controlled regulated by by a digitalan electromagnetic square wave. valve, and the disturbance frequency is controlled by a digital square wave. 2.2. PIV/PLIF System 2.2. PIV/PLIF SystemTo acquire a 2D planar x–y cross section of the liquid film flow on an inclined plate, To acquirewhich a 2D has planar quasi-2D x–y cross characteristics, section of the with liquid macroscale film flow spanwise on an inclined dimensions plate, (0.2 m) and a which has quasimicroscale-2D characteristics, cross-section with (approximately macroscale spanwise 0.5 mm), highdimensions optical magnification(0.2 m) and a is required to microscale crossproperly-section resolve (approximately the streamwise 0.5 mm), velocity high profileoptical on magnification the millimeter is required scale of the film cross to properly resolvesection. the Moreover, streamwise the velocity image profile acquisition on the accessible millimete methodr scale of would the film be cross limited below the section. Moreover,glass plate.the image The cameraacquisition and lensaccessible are set method in an off-axis would setup, be limited and the below imaged the plane should glass plate. Thebe camera determined and lens by the are thin set laserin an sheetoff-axis to imagesetup, theand velocity the imaged field plane perpendicular should to the glass be determinedplate. by the A thin high laser recording sheet rateto image is required the velocity to properly field perpendicular resolve the film to hydrodynamicsthe glass due to plate. A high itsrecording transient rate nature. is required to properly resolve the film hydrodynamics due to its transient nature.To tune the normal 2D PIV system applicable for the stated velocity field and topology To tune themeasurement, normal 2D PIV the system adopted applicable experimental for the measurement stated velocity technique field and is topol- PIV/PLIF, which ogy measurement,combines the adopted PIV with experimental PLIF by using measurement a single camera technique to simultaneously is PIV/PLIF, acquirewhich the velocity combines PIVand with topology. PLIF by using a single camera to simultaneously acquire the velocity and topology. An adequate number of particles is the primary concern for the fidelity of velocity An adequatemeasurements, number of becauseparticles theis the measurement primary concern technique for the determines fidelity of thevelocity velocity of tracer measurements,particles because instead the measurement of the velocity technique of the determines fluid [20]. Thethe velocity particles of used tracer are par- 2 µm titanium ticles instead dioxideof the velocity particles. of the The fluid working [20]. fluidsThe particles in the experiment used are 2 μm are alsotitanium doped dioxide with florescent dye particles. The(rhodamine working fluids 6G). in The the additional experiment fluorophore are also doped could with radiate florescent florescence dye (rho- photons of longer damine 6G). wavelengths The additional after fluorophore excitation. For could PIV/PLIF radiate image florescence recording, photons no filter of longer is added in front of wavelengths theafter lens, excitation. and both For the PIV/PLIF particle-scattered image recording, light and no florescence filter is added are recorded. in front Theof latter could the lens, and bothprovide the particle the film-scattered topology light to distinguish and florescence the liquid are recorded. film from The the airlatter or could the glass plate. To satisfy the requirements of high optical magnification, a large depth of field and an provide the film topology to distinguish the liquid film from the air or the glass plate. adequate working distance, the chosen lens was an ultra-macro lens (Anhui Changgeng To satisfy the requirements of high optical magnification, a large depth of field and Optics Technology, LAOWA ULTRA MACRO lens, Hefei, Anhui, China). This lens can pro- an adequate working distance, the chosen lens was an ultra-macro lens (Anhui Chang- vide 2.5–5-fold optical magnification with a working distance of approximately 40–45 mm. geng Optics Technology, LAOWA ULTRA MACRO lens, Hefei, Anhui, China). This lens Unlike usual microscopy, an ultra-macro lens can provide an independent small aperture, can provide 2.5–5-fold optical magnification with a working distance of approximately 40–45 mm. Unlike usual microscopy, an ultra-macro lens can provide an independent

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small aperture, and the lens work has a large depth of field, which ensures that the whole film regionand can the be lensimaged work even has if a the large focused depth plane of field, does which not coincide ensures thatwith thethe wholedesired film region film cross-cansection. be imaged A detailed even image if the of focused the acquisition plane does setup not can coincide be found with in Figure the desired 2. Thefilm cross- lens and camerasection. we Are detailed mounted image on oftwo the-axis acquisition goniometers setup that can we bere found mounted in Figure on three2. The- lens and dimensionalcamera positioning were mounted stages. on two-axis goniometers that were mounted on three-dimensional The imagepositioning plane stages.position ambiguity was caused by the large depth of field and the off-axis setup ofThe the imagelens, which plane wa positions compensated ambiguity for was by causedcustomized by the thin large laser depth sheet of illu- field and the mination. Theoff-axis laser setup source of wa thes a lens, continuous which waswave compensated (CW) Nd:YAG for 532 by nm customized laser from thin Bei- laser sheet jing Ranbondillumination. Technology The Co., laser Ltd. source, Beijing, was China. a continuous It had a wavemaximum (CW) power Nd:YAG of 6 532 W. nmThe laser from beam laserBeijing was shaped Ranbond via a Technology plano-concave Co., cylindrical Ltd., Beijing, lens China. (f = −2 It5.4 had mm) a maximum diverging powerthe of 6 W. laser into aThe planar beam sheet laser, and was a shaped plano- viaconvex a plano-concave cylindrical lens cylindrical (f = 100 lens mm) (f =reducing−25.4 mm) the diverging beam waistthe diameter. laser into As a planarshown sheet,in Figure and 3 a, plano-convexthe laser optics cylindrical were acco lensmmodated (f = 100 with mm) an reducing the optical cagebeam system waist using diameter. four rigid As shownsteel rods in Figure to ensure3, the a common laser optics optical were axis accommodated and better with an mobility. Inoptical the experiments, cage system the using cage four system rigid wa steels mounted rods to ensure on a three a common-dimensional optical man- axis and better mobility. In the experiments, the cage system was mounted on a three-dimensional manual ual translation stage for precise positioning and focusing. translation stage for precise positioning and focusing.

(a) (b)

Figure 3. PhotographsFigure 3. Photographs of PIV/PLIF of measurement PIV/PLIF measurement experiments. experiments. (a,b) were photos (a,b) were corresponding photos corresponding to to schematic diagramsschematic in diagrams Figure 2. in Figure2.

Finally, theFinally, illuminated the illuminated film cross filmsection cross (x section–y plane (x in–y Figureplane in1) Figurewas recorded1) was recorded with with a a high-speedhigh-speed camera (Vision camera Resea (Visionrch, Research, model Phantom model Phantom V711, Wayne, V711, NJ, Wayne, USA). NJ, USA). When usedWhen in structured used in structuredplanar laser planar induced laser fluorescence induced fluorescence (SPLIF) mode, (SPLIF) a long mode,- a long- pass filter passwas filtermounted was in mounted front of inthe front lens, of and the a lens, structured and a structured glass plate glass was plateadded was to added to shadow theshadow light sheet the light into sheetparallel into structured parallel structured illumination. illumination. The long-pass The filter long-pass (LOPF filter- (LOPF- 25C-561 from25C-561 OptoSigma, from OptoSigma, Tokyo, Japan) Tokyo, provides Japan) a provides high transmittance a high transmittance range wavelength range wavelength of 571.1–900of nm571.1–900 with over nm 93%with transmittance. over 93% transmittance. The structured The plate structured was a customized plate was atar- customized get (glass substratetarget (glass with substrate vacuum with-sputtered vacuum-sputtered chrome, laser chrome, carved laser50 mm carved × 50 50mm mm and× 550 mm and lines per millimete5 lines perr in millimeter a Ronchi rulings in a Ronchi pattern) rulings to form pattern) structured to form illumination. structured illumination. The developedThe developed single-camera single-camera PIV/PLIF implementation, PIV/PLIF implementation, the first of theits kind first ofto itsthe kind to the best of thebest authors’ of the knowledge, authors’ knowledge, avoids the avoids alignment the alignmentand registration and registration issues associated issues associated with dual-withcamera dual-camera PIV/PLIF combinations PIV/PLIF combinations [21]. In addition, [21]. In the addition, CW laser the is CW continuous laser is continuous and stable,and while stable, the dual while-pulsed the dual-pulsed laser has an laser alignment has an alignmentissue: dual issue:-pulse dual-pulse PIV lasers PIVdo lasers do not perfectlynot overlap perfectly [22], overlap and one [22 frame], and onemight frame not be might in the not same be in focus the same as the focus other as frame, the other frame, causing a causingrelatively a low relatively depth lowof field depth compared of field comparedto a normal to lens. a normal Moreover, lens. Moreover,with the with the ultra-macroultra-macro lens option, lens there option, would there not wouldbe enough not be room enough below room the belowfilm glass the filmplate glass for plate for two high-speedtwo high-speed cameras together cameras with together the lens with properly the lens positioned. properly positioned. Ultimately, Ultimately, the high the high frame rate,frame which rate, could which be used could to beresolve used tothe resolve transient the wavy transient film wavyflow field, film flowmotivated field, motivated our developmentour development of a single- ofcamera a single-camera PIV/PLIF solution. PIV/PLIF solution.

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2.3. Film Conditions In the work presented in this article, the working ﬂuids include deionized water and aqueous glycerol water solution (glycerol concentration 45% by volume), and the experimental conditions cover the range of Re = 16.34~52.54 for water and Re = 1.61~7.43 for the aqueous glycerol water solution. The detailed properties of the working ﬂuids are given in Table1 below.

Table 1. Working ﬂuid properties.

Working Fluid ρ kg/m3 µ kg/(m·s) σ N/m Deionized water 0.998 × 103 10.3 × 10−4 70.7 × 10−3 Glycerol water 1.120 × 103 81.0 × 10−4 65.4 × 10−3 solution

With all the properties, the dimensionless groups are determined with the following equations [2]. Reynolds number: Q/W Re = (1) ϑl

where Q represents the volume flow rate, W is the film width, and ϑl is the kinematic viscosity of the working fluid. In the following equation, q = Q/W is used to symbolize the unit width flow rate. Kapitza number: σ Ka = l (2) 4/3 1/3 ρlϑl (sin(β)g)

where σl represents the surface tension of the working liquid, ρl is the density of the working ﬂuid, and g represents the gravitational acceleration.

3. Experimental Methodology In this publication, the experimental methodology is essentially the same as that presented previously [19]. However, the interface identiﬁcation and methodology validation are improved. The elaboration here mainly focuses on the changes, and the details of the remaining methodology can be found in Wang [19]. Figure4 provides a schematic illustration and an example of the whole set image process.

3.1. Image Calibration and Dewarping With nonuniform refraction index media in the optical path and a high-magniﬁcation off-axis image acquisition system, the recorded images are distorted due to nonuniform magniﬁcation in different directions, partial defocusing, and imaged plane tilting from the nominal focused plane. Image correction and calibration processes are essential before further investigation and analysis. The corresponding procedures of image distortion correction were accomplished with Dantecs software DynamicStudio by means of multicamera calibration methods with a manual calibration target and pinhole image mode. Polymers 2021, 13, x FOR PEER REVIEW 6 of 20 Polymers 2021, 13, 1205 6 of 19

Filtered image

(a) (b)

FigureFigure 4 4.. SchematicSchematic illustration illustration of of PIV/PLIF PIV/PLIF process. process. (a) ( aPIV/PLIF) PIV/PLIF process. process. (b) PIV/PLIF (b) PIV/PLIF process process example. example. Each Eachof the of pro- the ceduresprocedures in (a in) corresponds (a) corresponds to an to image an image outcome outcome in (b in) in (b )the in theorder order of top of topto bottom to bottom,, and and then then left leftto right to right [19]. [19 ].

3.2. InterfaceSPLIF, a Identificationtechnique evolved with PLIF from PLIF, was performed to decompose the acquired image composition. In SPLIF, the planar light sheet is shadowed with a structured plate The imaged film region was uneven and thin, and complicated image composition to form parallel linear illumination, and explicit discrepancies in the fluorescence intensity has been reported by many researchers. The working fluids used in the experiment were or orientation emerge after reflection or refraction when light passes through the inter- mixed with both the florescent dye and the titanium dioxide particles recorded in the faces [23], which makes it feasible to discern the interfaces and different domains in the images; therefore, it was necessary to identify the composition of the acquired image prior image. Figure 5 displays a film SPLIF image, from which it could be inferred that there is to further processing. a refraction zone above the film–air interface. The bottom of the film domain is estimated SPLIF, a technique evolved from PLIF, was performed to decompose the acquired to be a projection of the laser plane, the thickness of which is not negligible when the image composition. In SPLIF, the planar light sheet is shadowed with a structured plate to camera is tilted. In addition, the laser sheet thickness is a crucial factor deciding the parti- form parallel linear illumination, and explicit discrepancies in the fluorescence intensity cle density in the imaged plane and influences the exact velocity profile by the overlay or orientation emerge after reflection or refraction when light passes through the inter- effect in finite volume. In experiments, a thin laser and high particle concentration were faces [23], which makes it feasible to discern the interfaces and different domains in the adoptedimage. Figure to guarantee5 displays sufficient a film SPLIF particle image, density from with which an explicit it could laser be inferred sheet position. that there is a refraction zone above the film–air interface. The bottom of the film domain is estimated to be a projection of the laser plane, the thickness of which is not negligible when the camera is tilted. In addition, the laser sheet thickness is a crucial factor deciding the particle density in the imaged plane and influences the exact velocity profile by the overlay effect in finite volume. In experiments, a thin laser and high particle concentration were adopted to guarantee sufficient particle density with an explicit laser sheet position.

Figure 5. Raw PIV image (above) and SPLIF image (below). The two images here do not rigorously represent the same state of a capillary wave.

With the image constitutions decomposed, it is possible to identify the interfaces ac- cording to the greyscale image. The particle images and fluorescence mixing nature some- times make the results of gradient-based interface identification noisy. Hence, an interface

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Filtered image

(a) (b) Figure 4. Schematic illustration of PIV/PLIF process. (a) PIV/PLIF process. (b) PIV/PLIF process example. Each of the procedures in (a) corresponds to an image outcome in (b) in the order of top to bottom, and then left to right [19].

SPLIF, a technique evolved from PLIF, was performed to decompose the acquired image composition. In SPLIF, the planar light sheet is shadowed with a structured plate to form parallel linear illumination, and explicit discrepancies in the fluorescence intensity or orientation emerge after reflection or refraction when light passes through the interfaces [23], which makes it feasible to discern the interfaces and different domains in the image. Figure 5 displays a film SPLIF image, from which it could be inferred that there is a refraction zone above the film–air interface. The bottom of the film domain is estimated to be a projection of the laser plane, the thickness of which is not negligible when the camera is tilted. In addition, the laser sheet thickness is a crucial factor deciding the parti- Polymers 2021, 13, 1205 cle density in the imaged plane and influences the exact velocity profile by the overlay7 of 19 effect in finite volume. In experiments, a thin laser and high particle concentration were adopted to guarantee sufficient particle density with an explicit laser sheet position.

FigureFigure 5.5.Raw Raw PIVPIV imageimage (above(above)) and and SPLIF SPLIF image image ( below(below).). The The two two images images here here do do not not rigorously rigor- representously represent the same the state same of state a capillary of a capillary wave. wave. Polymers 2021, 13, x FOR PEER REVIEW 7 of 20 WithWith thethe imageimage constitutions deco decomposed,mposed, it it is is possible possible to to identify identify the the interfaces interfaces ac- accordingcording to tothe the greyscale greyscale image image.. The The particle particle images images and fluorescence and fluorescence mixing mixing nature nature some- sometimestimes make make the results the results of gradient of gradient-based-based interface interface identification identification noisy. Hence, noisy. an Hence, interface an interfaceidentification identification method depending method depending on the fluorescence on the fluorescence background backgroundcontribution in contribution image pixel invalues image wa pixels applied values in image was applied pre-processing. in image This pre-processing. interface identification This interface method identification was still ac- methodcomplished was with still accomplishedthe convenience with of the the MATLAB convenience link of in the DynamicStudio MATLAB link soft inware. DynamicStu- Figure 6 dioshows software. an example Figure of6 shows interface an exampleidentification, of interface where identification, the white pixels where represent the white pixels pixels fall- representing into the pixels greyscale falling range into theand greyscale the red lines range are and the thefinal red interface lines are results the final obtained interface after resultsfitting.obtained after fitting.

FigureFigure 6.6. InterfaceInterface identiﬁcationidentification example.example.

3.3.3.3. PIVPIV MethodMethod TheThe adaptiveadaptive PIVPIV schemescheme parametersparameters werewere asas follows:follows: maximummaximum interrogationinterrogationareas areas (IA)(IA) size:size: 6464 × 32;32; minimum minimum IA IA size: size: 32 32 × ×16;16; and and grid grid step step size: size: 8 × 8 4.× The4. Thestep step size sizewas waschosen chosen to better to better adapt adapt the velocity the velocity field field grid grid to the to film the filmboundary boundary interface. interface. With With the cho- the chosensen step step size, size, pixel pixel element element dimension dimension and and optical optical magnification, magnification, the spatialspatial resolutionresolution µ couldcould bebe inferredinferred toto bebe approximatelyapproximately 25 25 µmm per per vector vector in in the the cross-stream cross-stream direction. direction. DuringDuring thethe adaptiveadaptive adjustmentadjustment iterativelyiteratively appliedapplied toto eacheach IA,IA, thethe particleparticle densitydensity limitslimits thethe IAIA size,size, andand thethe velocityvelocity gradientgradient limitslimits thethe IAIA shapeshape[ 24[24––2626].]. Specifically,Specifically, thethe particle density setting limits the lowest particle detection in each IA and the desired particle density setting limits the lowest particle detection in each IA and the desired num- number of particles in each IA, which were set to four and eight, respectively, which are ber of particles in each IA, which were set to four and eight, respectively, which are rela- relatively low to adapt to the high optical magnification and small investigated zone. For tively low to adapt to the high optical magnification and small investigated zone. For the the velocity gradient, the absolute magnitude of each of the four gradients was limited to velocity gradient, the absolute magnitude of each of the four gradients was limited to 0.1, 0.1, and the combined effect of all four gradients was limited to 0.2, which were the default and the combined effect of all four gradients was limited to 0.2, which were the default parameters presented by DynamicStudio. parameters presented by DynamicStudio. Finally, the results were tested with a universal outlier detection algorithm [27], peak Finally, the results were tested with a universal outlier detection algorithm [27], peak height validation, and moving average validation (5 × 3 area with acceptance factor 0.1 height validation, and moving average validation (5 × 3 area with acceptance factor 0.1 and iteration 3). and iteration 3). 3.4. Adaptive PIV Test via Synthetic Particle Images 3.4. Adaptive PIV Test via Synthetic Particle Images The quality of the adaptive PIV measurement method was additionally evaluated usingThe synthetic quality particle of the adaptive images with PIV knownmeasurement properties method [28,29 was]. Theadditionally considered evaluated image effectsusing synthetic in synthetic particle particle images images with include known the properties particle image[28,29]. diameter, The considered particle image density, ef- andfects motion in synthetic blur. Theparticle synthetic images images include have the dimensions particle image of 512 diameter,× 256 pixels.particle From density, bottom and tomotion top, pixels blur. 1–64The synthetic of 256 are images left blank have as dimensions the boundary, of and512 × pixels 256 pixels. 65–256 From are filled bottom with to synthetictop, pixels particle 1–64 of images 256 are with left a blank parabolic as the velocity boundary, profile. and pixels 65–256 are filled with synthetic particle images with a parabolic velocity profile. Under optimal conditions (particle image diameter ≈ 3 pixels, particle density ≈ 0.02 particles/pixel, no noise, no particle pair loss), the bias error of selected adaptive PIV algorithms is smaller than 0.1 pixels, and the random error is below 0.1 pixels, which might be even more precise and accurate if the lowest position is ignored. As shown in Figure 7, the largest random error and bias error are usually at the edge of the boundary; hence, in the calculation of wall shear stress, the second and third vectors are used instead of the first velocity.

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Under optimal conditions (particle image diameter ≈ 3 pixels, particle density ≈ 0.02 particles/pixel, no noise, no particle pair loss), the bias error of selected adaptive PIV algorithms is smaller than 0.1 pixels, and the random error is below 0.1 pixels, which might be even more precise and accurate if the lowest position is ignored. As shown in Figure7, Polymers 2021, 13, x FOR PEER REVIEW 8 of 20 the largest random error and bias error are usually at the edge of the boundary; hence, in the calculation of wall shear stress, the second and third vectors are used instead of the ﬁrst velocity.

(a) (b)

Figure 7. AdaptiveFigure 7. Adaptive PIV test PIV with test synthetic with synthetic particle particle images: images: (a (a,,bb)) synthetic particle particle images; images (c,d;) ( testc,d) results test results of bias errorof bias error and randomand error, random and error, the and color the colorscheme scheme associated associated with with the datadata points points corresponds corresponds to the to vertical the vertical pixel position. pixel Textsposition. in Texts in (c,d) suggested(c,d) suggested the applied the applied adaptive adaptive PIV PIV parameters: parameters: thethe initial initial 4-digit 4-digit number number were were the minimum the minimum IA size, andIA size the last, and 4 the last 4 representedrepresented the grid the step grid size step. size.

3.5. Experimental3.5. Experimental Methods Methods Validation Validation 3.5.1. Streamwise Velocity Profile Validation via PIV/PLIF, PTV and Nusselt’s Theory 3.5.1. StreamwiseThe PIV/PLIF Velocity measurement Profile wasValidation validated via by comparingPIV/PLIF, the PTV experimental and Nusselt’s PIV/PLIF Theory Theresults PIV/PLIF with the experimentalmeasurement PTV wa resultss validated and theoretical by comparing results under the steady experimental smooth film PIV/PLIF resultsflow with conditions. the experimental For the theoretical PTV aspects, results when and Re theoretical < Rec [30] (5/4cot( resultsβ ),under approximately steady smooth 3.4 for β = 20◦) film flow falling on an inclined plate is developed, and laminar flow film flow conditions. For the theoretical aspects, when Re < Rec [30] (5/4cot(β), approxi- submits to Nusselt’s flat film solution [30]. Then, the theoretical velocity profile could matelybe 3.4 reconstructed for β = 20° with) film flow flow rate falling and fluid on an properties. inclined Forplate the is PTV developed, parts, it is and expected laminar flow submitsthat to the Nusselt’s PTV results flat are film more solution accurate [30]. and haveThen, higher the theoretical resolution than velocity the PIV profile results could be reconstructedbecause PTV with operates flow rate on individual and fluid particles properties. instead For of the a collection PTV parts, of particles it is expected [22,31]. that the PTV resultsIn the PTV are application more accurate for Re =and 1.61, have f = 0 higher Hz and Reresolution = 2.22, f = than 0 Hz the glycerol PIV film results flow because cases, and determination of the mean velocity profile is achieved by fitting the data with PTV operates on individual particles instead of a collection of particles [22,31]. In the PTV application for Re = 1.61, f = 0 Hz and Re = 2.22, f = 0 Hz glycerol film flow cases, and determination of the mean velocity profile is achieved by fitting the data with a second- order polynomial function in MATLAB with the curve fitting toolbox using default settings. The boundaries were recovered from the polynomial, where the zero streamwise velocity suggested that the cross-stream position is the liquid–solid interface and the maximum streamwise velocity corresponds to the liquid–gas interface. Figure 8c,d illustrates comparisons among the PIV/PLIF experimental results, PTV experimental results and corresponding analytical velocity profiles based on Nusselt’s equation for the Re = 1.61, f = 0 Hz and Re = 2.22, f = 0 Hz glycerol water film flow cases. The comparison showed good agreement among the overall datasets, and a general conclusion could be drawn that the PIV/PLIF film flow experimental results match well with Nusselt’s theoretical prediction for both the film thickness and the velocity profile. However, the PIV/PLIF velocity in the near-wall region (0–0.1 mm) is biased for the average effect where a strong velocity gradient near the wall exists [31,32], and the profile

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Polymers 2021, 13, x FOR PEER REVIEW a second-order polynomial function in MATLAB with the curve fitting toolbox using 9 of 20 default settings. The boundaries were recovered from the polynomial, where the zero streamwise velocity suggested that the cross-stream position is the liquid–solid interface and the frommaximum the PTV streamwise results avoids velocity the corresponds wall effect to thein PIV. liquid–gas Interestingly, interface. above the near-wall region, theFigure PIV/PLIF8c,d illustrates results comparisonsand PTV results among coalesce the PIV/PLIFd perfectly. experimental For the results, zone PTVnear the liq- uidexperimental–air interface, results the experimental and corresponding velocity analytical was larger velocity than profiles Nusselt’s based velocity. on Nusselt’s Compared equation for the Re = 1.61, f = 0 Hz and Re = 2.22, f = 0 Hz glycerol water film flow to the analytical film thickness, PIV/PLIF and PTV slightly overestimated the film thick- cases. The comparison showed good agreement among the overall datasets, and a general ness.conclusion could be drawn that the PIV/PLIF film flow experimental results match well with Nusselt’s theoretical prediction for both the film thickness and the velocity profile.

(a) (b)

Figure Figure8. (a,b 8.) T(ahe,b )PTV The PTVvelocity velocity fitting fitting results results andand boundary boundary identification identification for the for Re the = 1.61, Ref = = 1.61, 0 Hz andf = 0 Re Hz = 2.22, andf Re = 0 Hz= 2.22, f = 0 Hz glycerolglycerol film film flow flow cases cases;; thethe corresponding corresponding adjusted adjustedR2 values R2 values are 0.8317 are and 0.8317 0.8336, and respectively. 0.8336, respectively (c,d) The velocity. (c,d profile) The velocity profile validation and and film film thickness thickness validation validation for the Re for = 1.61,the Re f = 0= Hz1.61, and f Re= 0 = Hz 2.22, and f = 0 Re Hz = glycerol 2.22, f film= 0 flowHz glycerol cases, respectively. film flow cases, respectively.For the For PIV/PLIF the PIV/PLIF results, horizontal results, horizontal error bars illustrate error bars the standardillustrate deviation the standard of the measurementdeviation of data. the measurement data.

The velocity difference in the upper region and the film thickness difference might be caused by the following factors: (i) the edge effect in the film spanwise distribution, which makes the film thicker in the middle and thinner near the edge; and (ii) errors from the liquid property measurements, specifically, the viscosity measurement of the glycerol water solution. Even though the PTV scatter plot for the validation achieved a more accurate result, for the instantaneous velocity distributions, the PIV results were more appropriate, because PIV does not evaluate individual particle images, but ensembles of particle images grouped together in IA [22].

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However, the PIV/PLIF velocity in the near-wall region (0–0.1 mm) is biased for the average effect where a strong velocity gradient near the wall exists [31,32], and the profile from the PTV results avoids the wall effect in PIV. Interestingly, above the near-wall region, the PIV/PLIF results and PTV results coalesced perfectly. For the zone near the liquid–air interface, the experimental velocity was larger than Nusselt’s velocity. Compared to the analytical film thickness, PIV/PLIF and PTV slightly overestimated the film thickness. The velocity difference in the upper region and the film thickness difference might be caused by the following factors: (i) the edge effect in the film spanwise distribution, which Polymers 2021, 13, x FOR PEER REVIEW makes the film thicker in the middle and thinner near the edge; and (ii) errors from the10 of 20 liquid property measurements, specifically, the viscosity measurement of the glycerol water solution. Even though the PTV scatter plot for the validation achieved a more accurate result, for the instantaneous velocity distributions, the PIV results were more appropriate, 3.5.2.because Continuity PIV does Equation not evaluate Validation individual particle images, but ensembles of particle images groupedThe above together validations in IA [22 ].are mainly based on flat films. To directly check the validity of the3.5.2. instantaneous Continuity Equation wavy film Validation velocity field and topology, we resorted to the continuity equation in integral form (Equation (7)) [33], because high-fidelity velocity and topology The above validations are mainly based on flat films. To directly check the validity resultsof the would instantaneous be required wavy to film acquire velocity a satisfying field and topology,reproduction we resorted from the to theexperimental continuity outcome.equation The continuity in integral formvalidation (Equation results (7)) [for33], Re because = 41.05, high-fidelity f = 5 Hz and velocity Re = and 7.43, topology f = 5 Hz are presentedresults wouldin Figures be required 9 and 10, to acquirerespectively, a satisfying as examples, reproduction in which from the the negative experimental value of theoutcome. flow rate The streamwise continuity derivative validation results generally for Re ag =rees 41.05, well f = 5 with Hz and the Re experimental = 7.43, f = 5 Hz thicknessare timepresented derivative, in Figures assuring9 and the10, respectively,fidelity and as accuracy examples, of in whichthe PIV/PLIF the negative measurement value of the techniques.flow rate streamwise derivative generally agrees well with the experimental thickness time derivative, assuring the fidelity and accuracy of the PIV/PLIF measurement techniques. 휕ℎ/휕푡 + 휕푞/휕푥 = 0 (3) ∂h/∂t + ∂q/∂x = 0 (3)

Figure 9. Continuity equation validation with the instantaneous velocity field and film topology for Figure 9. Continuity equation validation with the instantaneous velocity field and film topology the Re = 41.05, f = 5 Hz water film flow case. for the Re = 41.05, f = 5 Hz water film flow case.

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3.5.2. Continuity Equation Validation The above validations are mainly based on flat films. To directly check the validity of the instantaneous wavy film velocity field and topology, we resorted to the continuity equation in integral form (Equation (7)) [33], because high-fidelity velocity and topology results would be required to acquire a satisfying reproduction from the experimental outcome. The continuity validation results for Re = 41.05, f = 5 Hz and Re = 7.43, f = 5 Hz are presented in Figures 9 and 10, respectively, as examples, in which the negative value of the flow rate streamwise derivative generally agrees well with the experimental thickness time derivative, assuring the fidelity and accuracy of the PIV/PLIF measurement techniques. 휕ℎ/휕푡 + 휕푞/휕푥 = 0 (3)

Figure 9. Continuity equation validation with the instantaneous velocity field and film topology Polymers 2021, 13, 1205 11 of 19 for the Re = 41.05, f = 5 Hz water film flow case.

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Figure 10. Continuity equation validation with the instantaneous velocity field and film topology for Figure 10. Continuity equation validation with the instantaneous velocity field and film topology the Re = 7.43, f = 5 Hz glycerol film flow case. for the Re = 7.43, f = 5 Hz glycerol film flow case. 4. Results and Discussions 4. ResultsIn thisand section,Discussions we present the results and discussions based on the experimental velocityIn this field, section, topologies, we present and their the results derivatives. and Consecutivediscussions imagesbased on of filmthe flowexperimental through ve- locitythe field, imaged topologies zone of the, and test their section derivatives. were acquired Consecutive for each images case. It of is film impossible flow through to list the all the moments in this article; therefore, the film flow characteristics are presented only imaged zone of the test section were acquired for each case. It is impossible to list all the with limited snapshots. Images of the same case and at the same time indicate the same momentscondition, in whichthis article could; betherefore used to, correlate the film for flow better characteristics comparison. are presented only with limited snapshots. Images of the same case and at the same time indicate the same condi- tion,4.1. wh Model-Basedich could be Velocity used Field to correlate Reconstruction for better comparison. Our goal of this research was to establish a film model validation method specific to 4.1.the Model results-Based of a Velocity comparison Field between Reconstruction the model-based reconstructed velocity profiles and experimental velocity profiles. The velocity field data have a high temporal resolution Our goal of this research was to establish a film model validation method specific to (0.0005 s) and high spatial resolution (approximately 0.025 mm per vector in the cross- the streamresults direction); of a comparison therefore, between it is reasonable the model to use-based experimental reconstructed flow variables velocity and profiles their and experimentalderivatives forvelocity reconstruction profiles. of T thehe velocityvelocity field field with data model have velocity a high profile temporal assumptions. resolution (0.0005 Thes) and velocity high profiles spatial of resolution the film models (approximately used in the velocity 0.025field mm reconstruction per vector in include the cross- streamNusselt’s direction) velocity; therefore profile, Kapitza’s, it is reasonable velocity profile, to use theexperimental streamwise velocityflow variables profile in and the their derivativestwo equations for reconstruction of the first-order of weighted the velocity residuals field model with (1st model WRM) velocity and the second-orderprofile assumptions.regularized model (RM). The streamwise velocity formulas used are given below: (1)TheThe velocity parabolic profiles velocity of profilethe film in Nusselt’smodels used theory in [ 34the], uvelocity(y): field reconstruction include Nusselt’s velocity profile, Kapitza’s velocity profile, the streamwise velocity profile gh2 cos(β) 2y y 2 in the two equations of the firstu-(ordery) = weighted residuals− model (1st WRM) and the(4) sec- 2µ h h ond-order regularized model (RM). The streamwise velocity formulas used are given below: where g is the acceleration due to gravity, h is the local film thickness, ρ represents the (1) Thefluid parabolic density, velocityµ stands profile for the in dynamic Nusselt’s viscosity, theory and [34],β is푢 the(푦): inclination angle [34]. With the formula given, a parabolic profile could be readily constructed with local film heights. 푔ℎ2 cos(훽) 2푦 푦 2 푢(푦) = [ − ( ) ] (4) (2) Self-similar velocity profile assumption2휇 based onℎ localℎ film thickness and flow rates, which is given by Kapitza et al. In this paper, we refer to that equation as Kapitza’s where gvelocity is the acceleration profile. This due velocity to gravity, profile corresponds ℎ is the local to a film first-order thickness, correction  represents of Nus- the fluid density, 휇 stands for the dynamic viscosity, and 훽 is the inclination angle [34]. With the formula given, a parabolic profile could be readily constructed with local film heights. (2) Self-similar velocity profile assumption based on local film thickness and flow rates, which is given by Kapitza et al. In this paper, we refer to that equation as Kapitza’s velocity profile. This velocity profile corresponds to a first-order correction of Nusselt’s theory, which could adapt the velocity to the deformations of the free surface and changes in the flow rate [35–37].

푞 푦 1 푦 2 푢(푦) = 3 [ − ( ) ] (5) ℎ ℎ 2 ℎ From Equation (5), a self-similar velocity can be acquired with an elementary experimental velocity field and film topology.

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selt’s theory, which could adapt the velocity to the deformations of the free surface and changes in the ﬂow rate [35–37].

q y 1 y 2 u(y) = 3 − (5) h h 2 h

From Equation (5), a self-similar velocity can be acquired with an elementary experimental velocity field and film topology. (3) The streamwise velocity profile in the first-order weighted residuals model is a series expansion, and the profile finally adopted in the 1st WRM is given below:

q 2 1 8 4 1 u(y) = 3 f + a − f + f − f + a f − f + f − f (6) h 0 1 5 0 1 3 2 3 35 0 3 2 3 5 4

1 1 q2 a = 3Re − h∂ q − h∂ , (7) 1 2 t 2 x h 3 q a = 3Re − h3q∂ , (8) 3 20 x h3 j + 1 f = yj+1 − yj+2, j = 0, 1, 2, 3, 4, (9) j j + 2

where the test functions fj, amplitudes aj, j = 0, 1, 2, 3, 4 and reduced variable y = y/h formed the velocity profiles. (4) The second-order regularized model reduced the four-equation first-order weighted residual model (2nd WRM) by eliminating the independence of s1 and s2 [2]. The streamwise velocity profile in the regularized model is given by the same formula as the 2nd WRM assumption.

3 s s u(y) = (q − s − s )F + 45 1 F + 210 2 F (10) h 1 2 0 h 1 h 2 1 F = y − y2, (11) 0 2 17 7 7 F = y − y2 + y3 − y4, (12) 1 6 3 12 13 57 111 99 33 F = y − y2 + y3 − y4 + y5 − y6, (13) 2 2 4 8 16 32 1 19 74 s = 3Re h2∂ q − q2∂ h + hq∂ q , (14) 1 210 t 1925 x 5775 x 2 2 s = 3Re q2∂ h − hq∂ q , (15) 2 5775 x 17325 x

where Fj, j = 0, 1, 2 is an orthogonal basis that is constructed through the Gram– Schmidt orthogonalization procedure, and s1, s2 are inertia corrections for the velocity distribution. Reconstructed streamwise velocity profiles and velocity fields with different numerical models were compared to experimental results. However, the four equations of the 2nd WRM need four independent variables (h, q, s1, and s2) to support the reconstruction, among which the independent s1 and s2 are beyond this experimental data range. Hence, the 2nd WRM was not adopted in velocity profile construction and comparison.

4.2. Comparison of Instantaneous Velocity Profiles In the presented comparisons, the experimental instantaneous velocity profiles were plotted against different model-reconstructed velocity profiles, and in addition to the velocity profile, different time slices are listed to include typical zones in solitary waves. Polymers 2021, 13, 1205 13 of 19

For each snapshot, the velocity profiles are plotted against the model-reconstructed profiles in three subfigures. In each subfigure of the profile comparison, the y-axis represents y, which is normalized by the local film thickness to unity, and different colors indicate different profile positions. Moreover, the free surface of the film determined from PIV/PLIF images is displayed in pictograms. They do not cover the entire region Polymers 2021, 13, x FOR PEER REVIEW 13 of 20 because the boundary might introduce some drawbacks when calculating derivatives. In pictograms, different velocity profile positions are highlighted with correlating colored lines to show their streamwise positions in the film topologies. The streamwise velocity streamlines are also given for the RM results and experimental PIV/PLIF results to present field streamlines are also given for the RM results and experimental PIV/PLIF results to a more direct comparison. However, in the following section, only some of the canonical present a more direct comparison. However, in the following section, only some of the moment snapshots are presented here. Figures 11 and 12 present the typical responses of canonical moment snapshots are presented here. Figures 11 and 12 present the typical re- the Re = 41.05, f = 5 Hz and Re = 7.43, f = 5 Hz cases to disturbances. Conclusions can be sponsesdrawn of after the Rethe = careful 41.05, examination f = 5 Hz and of Re Figure = 7.43,s 11 f = and 5 Hz 12 cases: to disturbances. Conclusions can be drawn after the careful examination of Figures 11 and 12:

Figure 11. Cont.

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Figure 11. Cont.

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FigureFigure 11. 11Comparison. Comparison between between the the experimental experimental instantaneous instantaneous velocity velocity profiles profiles and and corresponding corresponding model-reconstructed model-reconstructed Figure 11. Comparison between the experimental instantaneous velocity profiles and corresponding model-reconstructed velocity profiles for the Re = 41.05, f = 5 Hz water film flow case. velocityvelocity profiles profiles for the for Rethe = Re 41.05, = 41.05, f = f 5 = Hz 5 Hz water water film film flow flow case. case.

Figure 12. Cont.

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Figure 12. Cont.

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Figure 12. Comparison between the experimental instantaneous velocity profiles and the corresponding model- Figure 12. Comparison between the experimental instantaneous velocity profiles and the corresponding model-recon- reconstructed velocity profiles for the Re = 7.43, f = 5 Hz glycerol film flow case. structed velocity profiles for the Re = 7.43, f = 5 Hz glycerol film flow case. Nusselt’s profile underestimated the velocity magnitude for Re = 7.43, f = 5 Hz and Nusselt’s profile underestimated the velocity magnitude for Re = 7.43, f = 5 Hz and overestimated the velocity magnitudes in the case of Re = 41.05, f = 5 Hz. Except in the overestimated the velocity magnitudes in the case of Re = 41.05, f = 5 Hz. Except in the fast-changing zones, the fronts of the main hump and capillary wave crests were similar. fast-Kapitza’schanging velocityzones, the profile, fronts the of 1stthe WRMmain hump velocity and profile, capillary and wave the RM crests velocity were profilesimilar. work veryKapitza’s well forvelocity the Re profile, = 7.43, the f = 51st Hz WRM glycerol velocity cases profile because, and Kapitza’s the RM velocity velocity profile profile (Equationwork very (10)) well correlatesfor the Re with= 7.43, the f = first5 Hz term glycerol of the cases 1st because WRM (Equation Kapitza’s (11))velocity and profile RM (Equation(Equation (15)) (10) velocity) correlates profiles, with and the furtherfirst term correction of the 1st did WRM not yield (Equation a vital difference(11)) and RM in the(Equation aspects ( of15 the)) velocity velocity profiles, profile basedand further on flow correction rates and did film not thickness yield a vital for thedifference glycerol in waterthe aspects solution of cases.the velocity profile based on flow rates and film thickness for the glycerol waterThe solution 1st WRM cases. and RM are both two-equation models; the RM is reduced from the four- equationThe WRM, 1st WRM however and theseRM are two both models two currently-equation give models; very similarthe RM velocity is reduced fields. from From the thefour reconstructed-equation WRM, velocity however profiles, these no cases two or models data in currently this study give could very induce similar significant velocity differences.fields. From the reconstructed velocity profiles, no cases or data in this study could induce significantCompared differences. to the experimental velocity field, the 1st WRM and RM fields were not able toCompared precisely describe to the experimental the actual velocity velocity field field, in the the film 1st flowWRM for and Re =RM 41.05, fields f = were 5 Hz not in frontable ofto theprecisely main humpdescribe and the the actual first capillary velocity wavefield in troughs. the film flow for Re = 41.05, f = 5 Hz in front of the main hump and the first capillary wave troughs. 5. Summary 5. SummaryIn this film flow experiment, the instantaneous velocity field and film topology of disturbedIn this falling film liquid flow filmsexperiment, were measured the instantaneous simultaneously velocity using field the and PIV/PLIF film topology technique of todisturbed determine falling the hydrodynamic liquid films we characteristicsre measured simultaneously of excited thin film using flows. the PIV/PLIF The applied tech- experimentalnique to determine measurement the hydrodynamic method could ch providearacteristics a high of resolutionexcited thin in film both flows. the temporal The ap- dimensionplied experimental and the spatial measurement dimension. method An experimental could provide validation a high resolution method for in theboth film the flow tem- modelporal velocitydimension field and was the established spatial dimension. by comparing An experimental the experimental validation velocity method fields for and the constructedfilm flow model velocity velocity fields basedfield wa ons the est modelablished and by elementary comparing experimental the experimental data. velocity fieldsAn and experimental constructed methodology velocity fields validation based on wasthe model performed, and elementary mainly by experimental streamwise velocitydata. profile and film thickness comparisons; experimental PIV/PLIF results were comparedAn experimental to Nusselt’s solution methodology and experimental validation wa PTVs performed results. A cross, mainly comparison by streamwise suggests ve- thelocity overall profile accuracy and film of the thickness experimental compa PIV/PLIFrisons; experimental results despite PIV/PLIF the inherent results limitations were com- ofpared PIV andto Nusselt’s uncertainties solution in the and determined experimental flow PTV conditions. results. OwingA cross tocomparison the high temporal suggests

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and spatial resolution, creditable derivatives are derived. The integral continuity equation is reproduced, which validates the instantaneous velocity field. Based on the film response to the inlet disturbance, solitary wave cases were chosen to exhibit the flow field hydrodynamic characteristics. To validate the film models, model- predicted velocity fields were calculated based on the experimental flow rate, film thickness and their derivatives; the validated models included Nusselt’s theory, Kapitza’s theory, and the 1st WRM and RM models. After a comparison between the experimental results and reconstructed velocity profiles, the following conclusions could be drawn: (i) under the majority of conditions, Nusselt’s theory underestimates the velocity in the glycerol water solution but overestimates the velocity in the water film cases, except for rapidly changing parts such as capillary wave troughs. (ii) Kapitza’s model could be referred to as the first term of the velocity profile, which already performs well to depict the flow field for the glycerol film cases. For the water film cases, the Kapitza model failed to produce such complex transient velocity fields. (iii) The 1st WRM was very close to the 2nd RM with identical flow rates and surface boundaries for all the cases considered in this research. The 1st WRM could reconstruct the velocity field well for most zones, except the front of the main hump and the first capillary wave troughs, where the velocity profiles were rapidly changing. Our research, experimental methods and experimental data should be of interest for film-related experiments and simulation studies.

Author Contributions: Conceptualization, R.D. and R.W.; methodology, R.W.; software, R.D.; validation, R.D. and R.W.; formal analysis, R.D. and R.W.; investigation, R.W.; writing—original draft preparation, R.W.; writing—review and editing, R.D.; visualization, R.W.; supervision, R.D. and H.J.; project administration, R.D. and H.J.; funding acquisition, R.D. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China (Grant No. 51779126). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors would like to acknowledge the support from the National Natural Science Foundation of China (Grant No. 51779126). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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