An experimental study of spatiotemporally resolved heat transfer in thin liquid-film flows falling over an inclined heated foil

Abstract This paper describes the development of an experimental technique that combines simultaneous planar laser-induced fluorescence (PLIF) and infrared (IR) thermography imaging, and its application to the measurement of unsteady and conjugate heat-transfer in harmonically forced, thin liquid-film flows falling under the action of gravity over an inclined electrically heated-foil substrate. Quantitative, spatiotemporally resolved and simultaneously conducted measurements are reported of the film thickness, film free-surface temperature, solid–liquid substrate interface temperature, and local/instantaneous heat flux exchanged with the heated substrate. Based on this information, local and instantaneous heat-transfer coefficients (HTCs) are recovered. Results concerning the local and instantaneous HTC and how this is correlated with the local and instantaneous film thickness suggest considerable heat-transfer enhancement relative to steady-flow predictions in the thinner film regions. This behaviour is attributed to a number of unsteady/mixing transport processes within the wavy films that are not captured by laminar, steady-flow analysis. The Nusselt number Nu increases with the Reynolds number Re; at low Re values the mean Nu number corresponds to 2.5, in agreement with the steady-flow theory, while at higher Re, both the Nu number and the HTC exhibit significantly enhanced values. Evidence that the HTC becomes decoupled from the film thickness for the upper range of observed film thicknesses is also presented. Finally, smaller film thickness fluctuation intensities were associated with higher HTC fluctuation intensities, while the amplitude of the wall temperature fluctuations was almost proportional to the amplitude of the HTC fluctuations.


Introduction
Liquid films falling under the action of gravity are classical free-surface flows that are either directly encountered in or representative of a wide range of industrial applications, including wetted-wall absorbers, heat exchangers, condensers, evaporators and reactors. The practical interest in these flows arises primarily from their high surface-to-volume ratios and excellent heat and mass transfer characteristics. It therefore comes as no surprise that extensive theoretical [1][2][3][4], numerical [5][6][7][8][9] and experimental [10][11][12][13] efforts have been devoted to their study over the past decades. The extraction of reliable detailed experimental data has, however, proven particularly challenging owing to a number of challenges inherent to these flow systems, such as the restricted fluid domains under observation (often sub-mm) and the intermittent nature of the moving and wavy interface. The present investigation is motivated by the strong demand for detailed, spatiotemporally resolved hydrodynamic and heat-transfer data in planar falling-film flows. This information is necessary for the purpose of improving our fundamental understanding of these important flows, and also for furthering the development and validation of advanced analytical tools and numerical modelling codes, by providing sophisticated closures for accurate and reliable predictions. Beyond this fundamental interest, the availability of the information can also prove valuable in our efforts to appreciate the transport capabilities of these flows and our in-depth understanding of the associated underlying heat-transfer mechanisms, which can then be harnessed in practical situations to improve performance and reduce the size and cost of equipment.
This paper describes a simultaneous application of opticaldiagnostic techniques for the provision of the aforementioned spatiotemporally resolved heat-transfer information in thin liquid-films flowing down an inclined heated foil. Optical techniques are recognized as advanced measurement tools, capable of providing detailed, non-intrusive multi-dimensional information on concentration, phase, temperature, velocity and other scalar and/or vector fields with high spatial and temporal resolution. As a result, they are becoming increasingly prevalent in scientific and engineering research and development fields. In particular, the heat-transfer community is moving gradually towards experiments that can combine heat-transfer measurements such as of temperature, heat flux, heat-transfer coefficient (HTC) and in a few cases the velocity field within the fluid domains, thus moving towards a comprehensive understanding of their associated interactions leading to a particular heat-exchange performance.
Specifically, the experimental technique reported here, namely PLIF-IR, involves a combination of laser-induced florescence (PLIF) and infrared (IR) thermography, and was developed in order to enable simultaneous measurements of the film free-surface height relative to the solid substrate (film thickness), the film free-surface and solid-liquid substrate interface temperatures, and the instantaneous heat flux exchanged between the liquid film and the heated substrate. By extension, the acquired data allow the estimation of local and instantaneous HTCs, and an investigation of their variation as a function of important flow parameters. Another important aspect of this effort is our specific interest in developing an experimental methodology that allows the explicit study of unsteady and conjugate heat-transfer [14,15]. In these problems the temperature and heat flux at the solid-liquid interface (wetted substrate side) both time-vary, and it is necessary to measure directly these variations. To address this challenge, the employment of a thin metal foil as a substrate enables IR measurements at its underside (un-wetted substrate side) to provide temperature and heat flux information on the wetted solid-liquid surface.
A considerable body of previous research has been dedicated to the development of a variety of experimental measurement techniques and their adaptation to the study of isothermal film-flows. Relevant experimental methodologies can broadly be classified as conventional or advanced, depending on the practice employed towards retrieving the desired data, primarily focusing on recovering the interface shape (film thickness and wave dynamics) and mass-transfer characteristics (entrainment, reposition, etc.). Examples of the former, (i.e. conventional, non-optical methods) are hotwire anemometry (HWA) [16,17], electrical conductivity [18,19] and capacitance probes [20,21]. Such measurement approaches have played an important role in improving our understanding these flows and have contributed to many advancements made in this field, yet they are associated with certain challenges; in some cases they are intrusive, can suffer from spatial or temporal resolution limitations, are by and large point-measurements, and often involve complex and cumbersome calibration procedures, thus progressively favouring the implementation of non-intrusive optical methods.
Amongst a plethora of relevant optical-diagnostic techniques, fluorescence-based methods have proven to be very effective and are therefore popular amongst researchers engaging in film-flow investigations. These methods employ a tracer (either occurring naturally in the flow or an added chemical substance) that is excited using a laser source. Fluorescence-based methods can be further classified as laser-induced fluorescence (LIF) imaging or planar laser-induced fluorescence (PLIF) imaging. The primary difference between the two is that LIF utilizes the fluorescence intensity, measured at or near the incident light direction, to quantify the film thickness along a line, over a one-dimensional (1-D) or two-dimensional (2-D) domain. PLIF on the other hand relies on planar illumination across the flow field and imaging from the side, and can be used to directly identify the extent of the liquid domain, and to resolve with a higher level of detail, resolution and accuracy 1-D spatiotemporal film thickness variations along the illuminated plane. Consequently, one or both film boundaries need to be identified in raw PLIF images in order to measure the spatial extent of the film. Another advantage of the latter approach (PLIF) is that the fluid-flow domain is visualized directly, allowing for spatiotemporally resolved velocity or temperature data to be obtained simultaneously with the film thickness. Examples of experimental investigations employing LIF imaging are reported in Refs. [22,23], and more recently by Alekseenko et al. [24]. PLIF-based studies are presented in Refs. [14,[25][26][27] in various multiphase/ interfacial flows.
Beyond isothermal film-flow investigations, fluorescence-based methods using temperature-sensitive markers have been employed when evaluating the thermal performance of microchannel devices and other microfluidic systems [28,29], as well as plate heatexchangers in the presence of oscillating flows [30,31,9]. Temperature measurements with an absolute accuracy of 0.5 K, or better, have been reported [28,32,33]. Compared to isothermal film-flows, the study of diabatic (heated/cooled) film-flows which entail liquid-temperature measurements is more demanding, owing to the fact that the liquid is often in physical and thermal contact with a hot solid-substrate which is typically metallic and opaque. Illumination of the liquid domain from the solid side is unfeasible in such a configuration, with the (intermittent and disturbed) liquid freesurface providing the only optical access to the liquid domain. In this case, refraction at the wavy interface results in strong beam-steering that must be accounted for by post-processing correction algorithms. Such measurements are also subjected to the restrictions noted earlier, and any available studies are consequently limited to single-point measurements.
Amongst others, the excellent efforts at RWTH Aachen University [34,35] to conduct simultaneous film-thickness and temperature measurements by employing diacetyl fluorescence and phosphorescence imaging are worthy of mention. In more detail, time-varying local HTCs were reported at the measurement position (point), suggesting a strong correlation between the film thickness and heat transfer (þ80% enhancement ahead of the wave and À40% deterioration below the wave crest). Unfortunately, the particular experimental effort was restricted to a narrow flow parameter range (Re ¼ 126), and the role of important experimental parameters on the wave regime and HTC performance was not within its scope. Furthermore, the results were recovered on the assumption that the velocity within the film is well-described by the theoretical parabolic Nusselt profile (for 1-D, laminar and fully developed steady-flow dominated by the balance of gravity and viscosity only). Deviations from this profile on account of twodimensional (2-D) and three-dimensional (3-D) unsteadiness have, however, been confirmed [10,27], which suggests that additional experiments to confirm these findings are necessary.
Further improvements to our insight into the complex and unsteady flow phenomena associated with thin films and their link to heat-transfer performance can be attributed to the extensive body of experimental research carried out at the Kutateladze Institute of Thermophysics of the Siberian Branch of the Russian Academy of Sciences [36,12,[37][38][39] and at the Institute of Thermophysics of the Technical University of Darmstadt [40][41][42][43]. The experimetnal efforts at Kutateladze aimed at examining the process of rivulet formation due to thermocapillary convection, employing a wide range of optical techniques including IR thermography, particle velocimetry, shadowgraphy and LIF. In Ref. [12], a range of locally (non-uniformly) heated falling film flows between Re ¼ 300 and Re ¼ 500. Even when low heat fluxes were imparted onto the vertically falling water films, significant thermocapillary effects were observed. Non-uniform heating was also employed in Ref. [36] to induce the formation of a horizontal bump-shaped free-surface deformation across the film, which was found to transition to a rivulet pattern above a critical heat input. By depositing Aluminium particles onto the liquid surface, a highly non-linear 3-D flow was revealed, while a stagnation point was identified at the top of ''horseshoe-like" structures. The separation between adjacent rivulets in experiments carried out with uniform heating was shown to depend weakly on the film inclination angle above 15° [37], while positive temperature gradients in the flow direction resulted in an increase in the wave amplitude. The evolution of 3-D waves into rivulet structures along with the dependence of the rivulet wavelength on the Re and applied heat flux were studied in Refs. [38,39]. Shorter wavelengths ensued when increasing the heat flux, while longer wavelengths were observed when increasing the Re. The work at Darmstadt has been dedicated to the characterization of the thermal performance of a variety of film flows. For example, experimental studies of the heat transfer (including by evaporation) of laminar water falling-films over smooth, as well as patterned plates, are reported in Refs. [40,41], using IR thermography and confocal chromatic imaging (CCI). Significant heat transfer enhancements were noted compared to a smooth-plate arrangement when grooved plates were employed. Furthermore, different mechanisms of film rupture were also investigated. Other heat transfer systems that have been extensively investigated (in a similar fashion) by the same research group include micro-channel and micro-tube flows (see, for example, Refs. [42,43]). It is noted that this work is mostly dedicated to reporting time-averaged HTCs and characterizing the overall heat transfer performance of the examined systems, while the aforementioned work at Kutateladze focuses mainly on the characterization of hydrodynamic flow-regimes.
The experimental work presented here is specifically associated with the application of the PLIF-IR technique, which employs fluorescence in conjunction with IR thermography. As noted earlier, the free-surface temperature of liquid films has been measured extensively by various researchers with IR imagers, sometimes with filter-equipped lenses that block the wavelengths over which the working fluid is transparent. The key aspect of this measurement tactic is the absorptivity of the fluid, which is required to be sufficiently high at the wavelengths used for imaging. This ensures the observation of a thin layer of fluid near the interface (i.e. of the free-surface), without contamination from the liquid bulk deeper within the film, or from the solid substrate. For example, Refs. [44,45] used an IR camera to measure the interfacial temperature of silicone-oil films; in this case the absorption coefficient allowed 99% of the radiation to be absorbed within a thin (10)(11)(12)(13)(14)(15) lm) layer at the surface. IR thermography has also been extensively employed in water film-flows (e.g. Refs. [46,47]), as well as many others.
The present paper is an extension of the experimental investigation reported in Mathie and Markides [15]. Contrary to that work, which focused exclusively on the recovery of the so-called 'augmentation ratio' in unsteady flows including film flows in the same apparatus with the same PLIF-IR technique, and only in select experimental conditions, the focus of the present work is on the full, detailed characterization of unsteady heat-transfer over a range of thin liquid-film flow-conditions in this apparatus, thus exploring the effects of important variables and parameters on the obtained heat transfer. The paper is structured as follows: in the next sections (Sections 2 and 3) the experimental apparatus is presented briefly, along with the employed optical measurement techniques, namely PLIF imaging and IR thermography. Also included is a description of the relevant data processing methodologies. In Section 4 the experimental results are presented. The results are organized in two sections, one dedicated to noteworthy qualitative observations regarding any interfacial thermal features and the phenomenology and dynamics of the examined film flows, and the second pertinent to the quantitative heat-transfer characterization. Finally, the paper closes with a summary of the main conclusions (Section 5) derived from the present study.

Flow apparatus
A series of experiments were conducted in order to measure the instantaneous temperature and heat flux across wavy films developing over a main test-section that comprises an inclined, electrically heated Titanium foil of thickness a ¼ 50 lm [15]. Heating of the test-section substrate was achieved by holding the foil taught between two 10 mm-thick copper electrodes and applying a current using a DC power supply rated to 10 V and 1000 A (i.e. 10 kW), such that a maximum heat-flux of _ q ¼ 4:2 W cm À2 over the 1200 cm 2 surface area of the foil was possible. Beyond the part of the apparatus that was employed for generating the heated substrate over which the flows were considered, the experimental arrangement also comprised a preparation section where the film flows were initiated, developed and delivered to the substrate and a closed flow-loop via which the liquid circulated to the test section ( Fig. 1). This was preferred to an open-loop arrangement, as it allowed for the fluorescent dye concentration to be preserved throughout the duration of the experiments. When in operation, the liquid initially passed through a counterflow heat exchanger supplied by Alfa Laval, in order to modulate its temperature before being controlled, measured and entering the preparation and test sections. The cooling stream used for the heat exchanger was supplied by the water mains, with the rate of heat extraction regulated by using Valve V1 and temperature indications on both sides of the heat exchanger from 1 mm K-type thermocouples. A second valve, Valve V2, was used in conjunction with a flow-recirculation loop (as illustrated in Fig. 1) for bypassing part of the flow back to the tank, thus allowing a lower flow-rates to be achieved in the test section with the same pump; this maintained higher flow-rates through the pump and the heat exchanger, which were operated more efficiently and closer to their design specifications.
In the present work, pure water and water-ethanol solutions were used as the liquid phase, the former during the development and testing of the experimental setup, and the later during the main heat transfer experiments reported in this paper. Water has a high surface tension, resulting to partial dewetting even when modest heating is applied to the foil. This phenomenon not only hampers the reliability and reproducibility of the experimental results, but also presents a threat to the foil as unwetted regions will heat up disproportionately and deform. In order to prevent this outcome, an ethanol-water solution (20% ethanol and 80% water by volume) was preferred, which compared to pure water displays substantially higher wettability, allows for higher heat inputs to be applied, and reduces the risk of damaging the setup.
The density q f , kinematic viscosity m f and surface tension r f of the employed solution at 20°C correspond to 978 kg m À3 , 1.53 Â 10 À6 m 2 s À1 and 4.2 Â 10 À2 N m À1 respectively.
In order to selectively trigger the rapid growth of waves and obtain developed wave-regimes within a relatively short testsection, the flow was split downstream of the bypass valve (V2) by diverting a portion of it through a rotating throttle valve (oscillator valve, V3 in Fig. 1). Torque to drive the oscillator valve was provided by a 5 V, 1 A, 1.8°step-angle stepper-motor controlled by an in-house developed controller, thus achieving a maximum rotational (i.e. wave) frequency of approximately 11 Hz. After the oscillator valve, the pulsating supply was recombined with the steady supply that was also controlled by an independent valve (V4). The total flow-rate was then measured by using an ultrasonic flowmeter (UF25B by Cynergy3 Components). In more detail, the recorded pulse-sequence (one TTL pulse generated for each mL of flow) was used to recover instantaneous and cumulative flowrate time-traces, from which the mean flow-rates and intensity of flow-rate pulsations were calculated. The rotational speed of the pump, and therefore the film flow-rate, was modulated by using a 1200 V A variable autotransformer (Carroll & Meynell). Finally, the flow was distributed uniformly across the heated area of the foil by a flow-preparation section of width w ¼ 0:3 m, that featured a series of grids, meshes and a honeycomb, as well as a knife-edge slot along with a set of micrometer stages that were used to adjust and set the initial height of the liquid films as these were introduced over the heated-foil test-section.
A complete view of the apparatus is given in Fig. 2. Measurements were performed at two streamwise positions; at an 'upstream' location a distance x ¼ 185 mm from the distribution box knife-edge at the flow inlet, and at a 'downstream' location x ¼ 285 mm from the inlet. In the arrangement shown in Fig. 2 (a) the IR camera is positioned underneath the test section with the laser sheet-optics illuminating a vertical slice along the direction of the flow, whereas in a second arrangement ( Fig. 2(b)), the two PLIF cameras are positioned to observe a horizontal slice across the film, excited by the laser sheet-optics that were positioned on top of the test section (at the top of the image). Experiments with the IR camera in this position were also performed. A fluorescent target used to correct for pulse-to-pulse laser-intensity variations can also be seen, located just above the foil. The Titanium foil and distribution box were installed on an aluminium support-frame with the inclination angle fixed to b ¼ 40°to the horizontal. The distribution box, which comprised grids, meshes and a honeycomb, functioned as a settling chamber, breaking down and dissipating secondary flows, large-scale eddies and turbulence, and creating a uniform flow at the test-section inlet. Finally, a type-K thermocouple was installed inside the chamber allowing for the liquid temperature to be monitored. Using the mean flow-rate measurement from the ultrasonic flowmeter, a Reynolds number can be defined at the flow inlet, as: In this expression, U stands for the local bulk-fluid velocity, D for the channel depth at the inlet (20 mm), C for the flow rate per unit channel width w, and m f for the kinematic viscosity of the liquid.

Measurement techniques: PLIF imaging
A 2-D PLIF imaging technique was used to characterize the local and instantaneous film thickness and film free-surface temperature simultaneously. To that end, a temperature-sensitive fluorescent dye (Rhodamine B) was dissolved in the liquid phase (here, pure water or a water-ethanol solution), forming a low concentration ($1 g/L), perfectly mixed aqueous-dye solution. Amongst others, the well-characterized strong dependence of Rhodamine B fluorescence to temperature variations was a primary reason for employing this particular tracer in these present experiments.
The aqueous liquid-phase was excited by a 0.1 mm-thick lasersheet, generated by a dedicated sheet optics and lens combination manufactured by LaVision that was positioned over the wavy interface and pointing downwards. Furthermore, a linear transition-stage was employed, so that the location of the excitation plane could be adjusted with a high degree of accuracy. The sheet optics were connected via a guiding arm to a dual-cavity, frequency-doubled Nd:YAG laser (Nano-L-50-100PV by Litron Lasers), capable of approximately 50 mJ/pulse at the selected excitation wavelength (532 nm) and a frequency of 100 Hz per head. Each head could be fired separately by the software, allowing a maximum continuous sampling of 200 Hz, and therefore, a temporal resolution down to 5 ms.
The resulting fluorescent light was collected by a pair of synchronized LaVision VC-Imager Pro HS 500 cameras equipped with 1280 Â 1024 pixel CMOS sensors and Sigma 105 mm f/2.8 Macro lenses. Two cameras were used, positioned next to each other, in order to expand the field-of-view (FoV) of the measurement (by employing image stitching), while maintaining a high spatial-resolution (approximately 20-30 lm/pixel). In addition to the employment of extension rings in achieving the desired magnification, a Scheimpflug adaptor was installed on one camera (the one positioned at an angle to the excitation plane) to focus the entire film domain accurately and sharply onto the image. Finally, optical long-pass 540 nm dichroic/interference filters were installed on both camera lenses, in order to block any scattered laser-light while allowing the red-shifted fluorescence emission to be collected by the detectors, thus improving the signal-tonoise ratio (SNR). Both cameras and lasers were synchronized by a LaVision High-Speed Controller (HSC), operated by using the LaVision DaVis software.

Measurement techniques: IR thermography
Along with the liquid-film thickness and free-surface temperature measurements extracted by PLIF imaging, the foil (un-wetted) underside temperature was measured simultaneously by an IR camera. The corrected measurement was then used to recover the (wetted) solid-liquid substrate temperature and the heat flux exchanged between the liquid film and the heated substrate, based on the knowledge of the imposed volumetric (Joule) heating-rate. Two different IR cameras were employed, a Cedip Titanium and a FLIR SC3000. With each laser pulse, the laser (PLIF) system provided the triggering signal for the IR camera, as well as the data acquisition (DAQ) system used to record the flowmeter and thermocouple output signals. The recorded thermocouples were those positioned inside the flow preparation/distribution box, at the film inlet and in thermal contact with the underside of the foil, the latter in a series of separate experiments conducted for IR calibration and validation purposes.
In reference to the setup schematic presented in Fig. 2, the total incident radiation-flux seen by the IR camera, I cam , is a linear superposition of the radiation emitted by the foil underside, which is at a temperature T u , and any direct or reflected background radiation I bg : where r ¼ 5:67 Â 10 À8 W m À1 K À4 is the Stefan-Boltzmann constant, and is the emissivity of the Titanium foil. Bare, polished Titanium has a very low emissivity, of approximately 0.1, whereas the thin oxide-layer that quickly forms on its surface causes its emissivity to increase to about 0.3; either way, the IR radiation collected by a detector positioned at right angles to the foil is influenced by reflections. Consequently, in the present work, the IR camera was in fact positioned at an offset angle to the foil, rather than directly opposite the foil surface, thus preventing it from capturing a reflected image of itself. Furthermore, in order to suppress any background IR radiation and prevent imaging of the reflections of other thermal sources, the region that would have otherwise been in the line-of-sight of the camera following specular reflections from the foil was obstructed by a screen constructed from thick black cardboard (with near-unity emissivity). The screen was further backed by aluminium foil which contributed to keeping its temperature even by uniformly dispersing heat.
Having suppressed any background signals and reflections, the IR radiation originating from the screen is solely the result of its temperature T bg . Thus, the total IR radiation measured by the camera I cam , is the sum of the background radiation associated with the screen temperature T bg , and the direct emission from the foil I u : such that the apparent temperature measured by the IR camera T app is given by: Eq. (4) can only be used to evaluate the temperature at the foil underside provided the employed IR thermograph integrates the radiation emitted over the entire IR spectrum. Typically, however, IR cameras detect radiation over a limited spectral band, in this case 3.6-5.1 lm and 8.0-9.0 lm for the Cedip and FLIR cameras respectively, and thus, the absolute temperature of the foil cannot be directly determined from the Stefan-Boltzmann law (Eq. (3)). I cam can instead be linked to temperature measurements via calibration curves provided by the camera manufacturer that consider, aside from the wavelength dependence of the detector (relative spectral response), data conversion settings such as the integration time and gain, and the detector optics (lens and filters). Using the relevant calibration curve, and provided that the background temperature and emissivity of the foil are known (the first using thermocouples, the latter by comparison to a blackbody of the same temperature), I bg and I u can be calculated sequentially. In these experiments, Eq. (3) was used to evaluate the temperature on the underside of the foil from raw IR images following a series of calibration and validation efforts carried out using thermocouples, as well as thermal targets consisting of squares of electrical tape with known, high emissivity. In more detail, simultaneous measurements of the temperature of the tape with the IR camera and the thermocouples were used to find the correct foil temperature, and thus the correct local emissivity of the foil. As a final validation exercise, independent measurements of the film and foil temperatures were conducted by both the IR camera and a series of fine (75 lm diameter) unsheathed, butt-welded thermocouples. The root mean square (RMS) deviation between IR camera and thermocouple measurements amounted to 0:1°C, which we consider as a representative estimate the error associated with the IR thermography measurement.

Heating conditions
The deviation in the power supplied to the foil from changes to the foil resistance and stability of the DC power supply over a run was measured and found to be <0.1% of the nominal supplied power. In order to provide uniformly distributed heating over the foil surface-area, the use of copper electrodes that were much thicker than the foil itself, and therefore with a significantly lower electrical resistance, was necessary. The power losses in the cables, based on direct resistance measurements in which the nominal resistance of the connecting cables and the total resistance of the foil were found to be 540 lW and 18.3 mW respectively, were estimated to amount to 3% of the power delivered to the foil. The voltage distribution over the foil was examined by conducting point measurements with a multimeter. The observed maximum deviation in this test amounted to no more than 0.1% over the 300 Â 400 mm foil area, which is expected to be the same as the non-uniformity in the imposed volumetric heating rate in the foil. Moreover, a uniform temperature distribution was observed over the foil during dry tests in which the foil was heated without establishing a film flow, and its temperature was measured with the IR camera. The standard deviation of the temperature during these dry tests did not exceed 0:1°C.
Results from tests concerning the average temperature over the foil obtained at different heat fluxes are shown in Fig. 3. A noteworthy observation stemming from these particular tests was that heat convection to the air was small, peaking at $1% of the of heat flux delivered to the liquid during the falling film experiments. Specifically, without a film flow, no more than _ q ¼ 0:042 W cm À2 or 50 W of thermal power were necessary to sustain a steady foil temperature of 60°C, or about 40°C above ambient; by comparison, nearly 5 kW were required to obtain the same foil temperature once a typical film flow was established. Given that the heat flux into the film from the foil's (wetted) top surface is three orders of magnitude higher than any heat losses to the environment from the (un-wetted) underside, the underside of the foil can therefore be considered an adiabatic boundary during the flow experiments.
During the actual experimental runs, the liquid temperature was kept below 40°C in order to minimize evaporation and focus the investigation on forced convective heat-transfer in liquid film flows. In a test near the upper range of achievable heat fluxes (corresponding to the highest heat losses), _ q ¼ 3:2 W cm À2 of heating power was applied to an 8 L/min water film flow, across the w ¼ 0:3 m wide test-section. Assuming complete heat transfer from the foil to the water (no heat losses to the surroundings), a streamwise temperature rise of 17.2 K m À1 can be estimated; however, a gradient of 16.7 K m À1 was recovered with the IR camera from the thermal profile of the fully developed portion of the flow. This corresponds to heat losses of $0.1 W cm À2 , which amount to less than 3% of the total heat input. If we conservatively assume that all of this heat was used to evaporate liquid from the film surface, which is unlikely as other heat transfer processes take place including convective heat-transfer to the air at the film's free surface [37], the maximum evaporative flow rate would be less than 3 mL min À1 . This is 3-4 orders of magnitude smaller than the bulk film flow rate. Finally, the HTC between the foil and the liquid film for this flow was measured at about 2400 W K À1 m À2 , whereas the HTC between the liquid film and the surrounding air was an order of magnitude lower at $200 W K À1 m À2 (from direct measurements). These results are in general agreement with the Nusselt theory for film flows, as well as typical HTC values obtained by other investigations in the literature (see, for example, Ref. [38]).

PLIF data processing
In order to extract film thickness and free-surface temperature data from raw PLIF images, a series of post-processing steps were implemented. Initially, raw images were corrected for perspective distortion by employment of a calibration graticule and a pinhole projection model available in DaVis. In more detail, a precisionmachined block of acrylic with markings arranged in a regular rectangular pattern (calibration target) was carefully located at the excitation plane using micrometer stages. The accuracy of the correction procedure was approximately 50 lm, which equates to less than twice the nominal resolution of the imaging setup. A perspective distortion corrected PLIF image is presented in Fig. 4.
Having corrected the raw PLIF images geometrically, a number of methods were devised to identify systematically the film freesurface. Of these, a gradient method was found to produce the most accurate film thickness estimates as well as the smoothest thickness profiles. Specifically, each image column (i.e. series of intensity values along the vertical/y-direction) was considered independently, so that the PLIF signal intensity was treated as a function of the y-position alone. Both the position and magnitude of the maximum gradient in each intensity profile (each x-pixel or image column) were then extracted, and the location of the gas-liquid interface was taken as the intercept of a tangential first order polynomial fitted through that location and a completely dark background with zero intensity I ¼ 0 (see Fig. 4(b)): y max;g ¼ arg y max @Iðx; yÞ @y ; and; ð5Þ where y max;g is the location of the maximum gradient and y int is the resulting interface location estimate. A similar gradient-based interface identification technique was also used in Ref. [27] with excellent results. The fluorescence intensity near the free-surface of the film was used to measure the interface temperature in a spatially resolved manner, following a series of calibration experiments. The technique was calibrated using a number of independent PLIF acquisitions, whereby the fluid temperature was adjusted between each  run and PLIF data were collected with no power input to the foil, and thus a uniform temperature throughout the liquid film. The measured PLIF signal intensities over each run were then averaged (over a number of images), normalized and plotted against the mean temperature measurement from thin (0.2 mm) thermocouples. This generated the calibration curve shown Fig. 5, which was later used to convert direct measurements of the fluorescence intensity near the free-surface to evaluations of the film's surface temperature. The relationship between the fluorescence intensity and temperature appears to be linear with a gradient of À0.015 normalized fluorescence units per°C, or À1.5%°C À1 .
Along with the mean PLIF signal intensities used to derive the calibration curve in Fig. 5, a plot of the corresponding standard deviations of the same PLIF intensity signals (one point for each calibration run associated with a single liquid-film temperature, and hence thermocouple measurement) are presented in Fig. 6(a) as a function of temperature, while the resulting probability density function (PDF) of the difference between the PLIF and thermocouple temperature measurements is shown in Fig. 6(b). The calculated standard deviation between the thermocouple measurements and the PLIF measurements was AE0.5°C, a value used as an estimate of the error associated with the PLIF temperature measurement. This level of accuracy is rendered satisfactory, especially considering that the manufacturer's quoted accuracy for the thermocouples employed in the calibration was AE0.1°C.
Additional errors associated with the PLIF-based temperature measurement technique ensue due to light distortion when the interface is strongly curved. In such cases, regions with significant film curvature typically appear brighter due to local refraction and lensing. Despite the fact that the emission intensity at shallow depths in the immediate vicinity of the interface is less sensitive to this effect, local fluorescence intensities are also relatively low in these regions, reducing the measurement signal-to-noise ratio (SNR). Therefore, the depth from the free surface to the position inside the liquid domain over which a spatially averaged fluorescence intensity was extracted when evaluating an ''interfacial" fluorescence intensity value was identified manually (this varied over the experimental runs) as a compromise between measurement noise and measurement accuracy. The suitability of this selection can be gauged by comparing final, processed PLIF temperature data with direct simultaneous measurements of the free-surface temperature made by the IR camera. This was done in a series of independent quality-check runs in which the IR camera was placed over the films looking down, i.e. from the same side as the PLIF measurement.
An example simultaneous measurement is shown in Figs. 8 and 9. The mean absolute deviation between the two measurements is found to be 1.3 K, and therefore it is possible to deduce that the interfacial curvature introduces an absolute error of approximately 0.8 K to the PLIF temperature measurement. This is significantly lower than the temperature differences across the films that we are attempting to measure, which on average are in the range 10-20 K but can be locally and instantaneously 30 K, or higher, depending on the flow condition and imposed heat flux (e.g. see Figs. 9-11).

IR thermography data processing
As with the PLIF measurements, perspective distortion corrections were implemented to raw IR images by employment of an affine transformation method (see, for example, Ref. [48]). Such a method is fully constrained when a minimum of four coordinates on the image plane are known. Towards that end, a graticule consisting of a thick rigid sheet covered by a reflective foil layer and marked by electrical tape was constructed and imaged, producing a strongly contrasting pattern on resulting IR images (see Fig. 7). The mapping error was reduced further by using nine rather than four control points and solving the over-constrained set of equations by means of a least squares approach. The resulting spatial alignment accuracy between simultaneous PLIF and IR images was better than 2 mm.
Following the employment of Eq. (3) in order to recover the instantaneous temperature distribution over the foil underside and the implementation of the aforementioned geometric corrections, the next processing step was to recover the temperature and heat flux distribution on the wetted foil surface (solid-liquid interface). The foil thickness was selected purposefully to be over an order of magnitude thinner than any thermal feature observed on its surface with the IR camera, and consequently, it can be assumed that heat transfer is dominated by 1-D thermal conduction through its thickness (for a detailed account of the spatial resolution of IR thermography, refer to Ref. [49]). By further assuming a constant thermal diffusivity, a model for the 1-D thermal conduction with constant heat generation by resistive heating can be used to describe the thermal response of the foil to temperature fluctuations imposed on one of its sides, in this case the top side in contact with the liquid. A detailed description of the model can be found in Refs. [14,15]. This response will depend upon the imposed boundary condition (B/C) on the foil underside, which as discussed in Section 3.1, can effectively be assumed to adiabatic. The thermal response that relates the solid-liquid interface temperature T w (inner wall ''w") to the measured temperature fluctuations on the underside of the foil T u for a given frequency is given by: while the corresponding heat flux response is given by: (c) Joint-PFD of instantaneous film thickness data (from Fig. 9(b)) plotted against the corresponding instantaneous film free-surface temperature data (from Fig. 9(a)) compiled over the same spatial region (line indicated in Fig. 8), x = 285 mm downstream of the flow inlet. A power law fit to the data was also generated using a least-squares approach and has been included to the plot.

Phenomenological observations, interfacial features and dynamics
Selected instantaneous results pertinent to the formation and interaction of thermal rivulet structures on the film free-surface are presented in Figs. 8-10. Recall that for the generation of these, and all subsequent results, the liquid employed in the experimental campaign was a mixture of 20% ethanol and 80% water by volume; compared to pure water, this displays higher wettability and allows higher heating fluxes to be applied prior to film rupture. In addition, the foil inclination angle was fixed to b ¼ 40°.
Close observation of Fig. 8, generated by positioning the IR imager on the same side as the PLIF cameras over the films pointing downwards, reveals regularly spaced, finger-like structures forming close to the flow inlet and developing in the flow direction (x-direction, þve from top to bottom). Near the inlet, these features are narrower and their temperature is comparable to the surrounding liquid, while further downstream they become progressively hotter and wider, in some cases abruptly (e.g. the large finger just to the left of the image midline). Another noteworthy observation     concerns the relatively discontinuous temperature profile of these rivulets compared to the adjacent fluid which is much cooler. The instantaneous data in Fig. 9(c) correspond to the 'downstream' PLIF measurement position identified by the horizontal line near the bottom of Fig. 8. In fact, the IR surface temperature profile in Fig. 9(a) has been extracted at this location directly from Fig. 8. This measurement position is located at a streamwise distance x ¼ 285 mm downstream of the distribution box knife-edge at the flow inlet. Measurements at a second 'upstream' PLIF position, x ¼ 185 mm from the inlet, were also performed (e.g. see Figs. 17 and 18). The length (spanwise width) of the measurement line was 40-60 mm depending on the experimental run. The PLIF data displayed here cover a 40-mm spanwise length across the flow. The flow and imposed heating conditions correspond to Re ¼ 179 and _ q ¼ 3:5 W cm À2 , respectively. Fig. 9(a) presents quantitative free-surface temperature data recovered simultaneously by IR thermography and temperature-dependent PLIF, while Fig. 9(b) gives an indication of the film height obtained from the same PLIF data. In Fig. 9(c), the PLIF results (thermography and film height) are plotted against each other, demonstrating the observed relationship between the two quantities.
The following remarks can be made from Fig. 9: 1. The agreement between the two optical techniques with regard to the surface temperature measurement (1.3 K, as stated earlier) is satisfactory, with the main thermal features identified and adequately resolved by both. The PLIF measurement does have a lower SNR while also offering a comparatively limited field of view. 2. The liquid film is thicker where the surface temperature is lower and vice versa (see Fig. 9(c)), with the two troughs in the film thickness measurement corresponding to the two larger peaks in the surface temperature measurement. We also note, however, that the shorter temperature peak near the centre of the measurement domain does not correspond to a film thickness trough. This observation is also apparent in Fig. 9(c), by the spread of free-surface temperature values associated with the highest film-thickness values included in the plot (leftmost region). The identification of this clearly discernible thermal feature, which was captured by both imaging techniques, was encountered consistently in these experiments. 3. From direct observations of many such image sequences, it appears that the advent of flat hot rivulet structures, such as the one discussed above, manifests itself in the development of Marangoni flows at the film surface, driven by temperature-induced surface tension gradients. In other words, local surface tension variations are believed to trigger film thickness instabilities, and hence the ensuing (Marangonidriven) mass transfer, characterized by accumulation of fluid in the cooler regions and thinning at the hotter ones. The rivulets, whose origin can be found further upstream (i.e. the larger structures seen in Fig. 8), might have formed and evolved in this manner. 4. The sharp peak on the film free-surface temperature profile in Fig. 8 coincides with a less prominent peak on the foil temperature profile. The temperature difference between the foil and film free-surface at this location is drastically reduced, suggesting a significantly higher heat transfer coefficient compared to the surrounding liquid. This observation will be revisited extensively in the upcoming sections, and linked to the local and instantaneous film thickness. Figure 10 proceeds to show the temperature difference across the liquid film at the same 'downstream' measurement position as above, obtained by repositioning the IR camera to observe the temperature of the underside of the foil, instead of the surface of the liquid film; the latter now measured by the LIF technique. A sharp peak (rising $15 K from the background) can be observed in the fluid surface-temperature at À10 < z < 0, which coincides locally with a slight increase ($2-3 K) in the temperature of the foil. The temperature difference between the foil and fluid surface at this point is significantly reduced, by a factor of two, which implies that there must be a considerably higher local HTC at that point compared to the surrounding fluid. We will return to consider this in detail in Section 4.2.
The spatial evolution of these free-surface thermal features is examined in Fig. 11, where spanwise temperature profiles obtained with the IR camera are displayed at different axial locations along the streamwise direction of the flow. In more detail, instantaneous temperature profiles across the surface of the liquid film (z-direction) were extracted from the same image (and thus at the same time instant), at different streamwise distances x from the inlet. The specific x locations are indicated in the figure. The observed temperature variations across the film free-surface increase with increasing distance from the inlet, reaching a maximum near the bottom-most measurement location (x ¼ 285 mm) of nearly 20 K cm À1 . Furthermore, for all examined axial locations, the temperature rise between the cool fluid and the rivulet peaks along the spanwise direction of the flow appears nearly linear, with temperature differences across the liquid films reaching values up to $30 K.  In light of the presented results and aforementioned observations, we propose the following mechanism for the formation and evolution of thermal rivulets and the associated film-height modulation across the spanwise direction of the flow. We noted earlier that the rivulet effects are most probably attributed to the emergence of Marangoni flows induced due to temperature gradients, and consequently, surface tension gradients. For such flows to develop, an initial temperature disturbance at the interface or perturbation in the flow must trigger a surface-tension inhomogeneity or gradient. This surface tension gradient then drives a spanwise flow component at the surface, which reinforces the temperature profile by the advection of hotter liquid from within the film bulk (near the foil) and towards the gas-liquid interface. A heat transfer enhancement should also ensue as a consequence of the fluid motion. We have also shown that this effect is initially (closer to the inlet) limited, manifesting itself as only a modest temperature variation at the interface. At some distance further downstream, however, this spatially limited hot-fluid region expands abruptly and hotter fluid appears to be ''spilling" over to cooler, neighbouring regions. The ensuing accumulation of liquid in those regions causes the formation of rivulets (spanwise film height modulation) downstream of the initially observed thermal structures. Gravity acts as a restoring force, countering the growth of rivulets or even completely suppressing it. In such cases, significant spanwise return flow towards the lower layers of the film should be anticipated. Furthermore, gravity effects are expected to introduce a significant inclination angle dependance to the topological characteristics of these rivulets, the flow field underneath them, and by extension the heat transfer performance of these films. It should finally be noted that in order to verify the validity of the proposed mechanism, knowledge of the flow field underneath the rivulet strictured is necessary. Experiments that combine the (a) Comparison between the film depth d measured at 185 mm downstream of the flow inlet by PLIF, free-surface temperature T LIF measured at the same location by PLIF, and foil underside temperature T IR measured by IR thermography at 285 mm and 185 mm downstream of the flow inlet, for a flow Re = 179 with no inlet forcing.
(b) Comparison between the film depth d measured at 185 mm downstream of the flow inlet by PLIF, free-surface temperature T LIF measured at the same location by PLIF, and foil underside temperature T IR measured by IR thermography at 285 mm and 185 mm downstream of the flow inlet, for a flow Re = 251 with no inlet forcing. optical techniques employed in the present study, alongside velocimetry techniques such as particle image velocimetry (PIV) and particle tracking velocimetry (PTV) could contribute greatly towards uncovering the underlying, complex flow phenomena and rivulet formation, evolution and interaction mechanisms. The temporal evolution of the thermal features is considered in Fig. 12. The rivulets display an oscillatory behaviour with respect to the streamwise direction of the flow, an effect that would be expected to contribute to heat-transfer enhancement through stronger mixing. The high temperature fluid is observed to move towards and away from the inlet with an approximate period of 1 s, with the originally thin rivulet growing wider (0.0-0.5 s) and its distance from the inlet reaching a minimum at around 0.5 s. Finally, it gets washed back down with its edge becoming blunter until the cycle repeats (0.9-1.0 s).
Greater insight into the effects of the applied heat flux and liquid flow rate on the thermal features developing over both the film free surface and solid-liquid interface, the latter as a result of conjugation, is provided by inspection of the ensuing temperature distributions in Figs. 13 and 14. Here, the flow Re was varied between 107 and 251, while the imposed heat-flux _ q was adjusted in the range 0.1-3.6 W cm À2 . The reader will note some missing panels at higher heat fluxes and lower flow rates (Fig. 13, top right). For the conditions it was not possible to operate the experiment; following film rupture and de-wetting, there was considerable risk of damaging the foil due to the rapidly rising local temperatures. It should also be noted that the presented temperature distributions of the foil underside and liquid film free-surface were not collected simultaneously. Instead, the (single) IR camera was relocated once the desired range of experimental conditions was investigated, and the same conditions were reproduced.
Regarding the operating flow conditions (flow rate and heat flux combinations) that were successfully investigated, the presence of the previously discussed streak-like thermal features (rivulets) on the film free-surface was observed, once again, as a regular equispaced pattern along the streamwise direction of the flow. In the lower Re and heat flux experiments, no rivulet oscillations were observed. At the higher Re and heat flux flows, however, the hotter ''fingers" raced up and down the x À z domain with a spatial amplitude of 5-10 cm and a frequency between 0.5 and 2 Hz. The mean distance of these features from the inlet was also affected by the flow rate and imposed heat flux, with higher flow rates pushing the oscillating structures further downstream, and higher heat fluxes moving them further upstream. The corresponding foil temperature distributions (Fig. 14) are neither uniform, nor constant, with the footprints of the free-surface rivulets clearly discernible. In this case, it is apparent that increasing either the flow rate or the imposed heat flux results to both the formation of a higher number of rivulets along the spanwise direction of the flow, as well as an increase in their amplitude relative to the background IR signal (i.e. the rivulets become hotter compared to the surrounding fluid).
Inlet flow pulsation was also instigated in some experiments by diverting a portion of the total flow through a rotating throttle valve, thus generating hydrodynamic waves which travel down the film. The purpose behind this tactic was to obtain more coherent and higher amplitude film thickness fluctuations, and therefore, better modulated HTC fluctuations. Noise-driven interfacial instabilities emerge shortly downstream of the inlet, evolve into a two-dimensional wave sequence comprising highly asymmetric solitary waves (frequency content centred around the fastest growing mode), and eventually break down into three-dimensional structures. In contrast, the imposition of monochromatic perturbations at the inlet results in wave sequences which adopt the forcing frequency, while wave growth saturates within a few wavelengths [50]. The generation of coherent wave structures (in terms of shape, amplitude and wavelength), thus, allows us to isolate and study specific aspects of the heat transfer characteristics of these waves. In addition, given the thermal boundary condition selected for these experiments (uniform substrate heating), the substrate length necessary to reach fully developed wave regimes would be prohibitive in terms of the required heating power. Finally, we would like to note that pulsed-flow arrangements have been employed by many researchers in the past, for similar reasons as in this study (Refs. [22,51,52]). Figures 15 and 16 explore the effect of heat flux (the same four values examined earlier) and pulsation amplitude (4% and 10% of the mean flow rate) on the film free-surface and foil temperature distributions respectively. In these measurements, the mean flow Re was fixed at 179, while the forcing (wave) frequency was set to 1.8 Hz. Careful inspection of Fig. 16 reveals that the flow is now much more unsteady following the introduction of hydrodynamic waves. Only for the lowest pulsation amplitude and heat flux combination does the temperature distribution appear relatively regular, with little variation in the streamwise direction where typically higher temperature rivulets develop. At higher heat fluxes, more irregular higher-dimensional temperature patterns emerge, rushing down the foil due to the imposed flow rate perturbations (waves). On the film surface, the effect of pulsation on the temperature distribution is more pronounced. At the lowest heat flux, the temperature distributions appear rather uniform, with the wave spacing and smaller capillary waves just distinguishable. As the imposed heat flux is increased, regions of high surface temperature form much like before. The passing of each wave front, however, enforces strong mixing between the hotter regions and surrounding cooler liquid. The spatial extent of the hot zones along the spanwise direction of the flow seems to increase with increasing distance from the inlet, while with increasing heat flux and/or pulsation amplitude, a clear mixing enhancement is discernible by the higher number of strongly skewed, intertwined thermal rivulets.

Thermal response coupling to the film thickness
The spatiotemporal evolution of the film thickness d (far left) and the film free-surface temperature T LIF (second from left) measured simultaneously by PLIF at the 'upstream' measurement position x = 185 mm downstream of the flow inlet over a 1.8 s time interval are plotted in Fig. 17 alongside the foil temperature T IR (far right) measured by the IR camera at the same location for two flow conditions: (a) Re ¼ 179; and (b) Re ¼ 251. In addition, the foil temperature T IR measured by the IR camera further downstream at x ¼ 285 mm from the flow inlet (second from right) is also presented in this figure. For the particular experimental runs, the two PLIF cameras were arranged so as to image a region along the film span, generating data equivalent to the film thickness and film free-surface sample measurement presented earlier (Fig. 9). The imposed heat-flux was set to _ q ¼ 3:6 W cm À2 . To the best knowledge of the authors, the generation of such spatiotemporally resolved (local and simultaneous) conjugate (fluid-solid) heat-transfer data is the first of its kind.
As was reported earlier in relation to Fig. 9, high film free-surface temperatures (light colour regions in Fig. 17, second from left columns) are observed in thin film-trough regions which are resolved in both space and time (dark colour regions in Fig. 17, leftmost columns) and vice versa. Moreover, the local film thinning shows a high degree of correlation with the darker (lower temperature) regions on the foil underside ( Fig. 17(a) and (b), last column from the left) which are, however, observed with a slight delay compared to the corresponding trough and high temperature regions. This delay, which is more clearly observable in Fig. 17 (b), appears to be around 0.1 s, and is most probably attributed to the thermal inertia of the foil. An additional observation that can be made here, is that locally higher HTCs ensue in thin film regions (troughs), as more heat is extracted from the substrate. Finally, the film thickness map of Fig. 17(a) shows troughs developing in the region z ¼ 5-20 mm between t ¼ 0:1 s and t ¼ 0:6 s and reappearing at around t ¼ 1:3 s. This oscillatory behaviour, also observed in Fig. 17(b), is believed to correspond to the one noted earlier in Fig. 12. While the troughs oscillate up and down along the streamwise direction of the flow, faint ''bow-like" features appear ahead of and along their peripheries. Figure 18 shows the evolution of the local and instantaneous heat flux _ q, heat-transfer coefficient h and Nusselt number Nu in the same experiment described in Fig. 17. As expected, Nu and HTC are highest where the film is thinnest, while the aforementioned ''bow-like features are once again evident. The heat flux variation amounts to approximately 4% of the imposed mean heat-flux ( _ q ¼ 3:6 W cm À2 ) and the mean Nu amounts to about 2.8. The HTC variations are in the range 1-3 kW K À1 m À2 .

Quantitative heat-transfer characterization: Mean performance
The above observations suggest that the local and instantaneous HTC h is found here to be augmented in thinner film regions, with small amplitude fluctuations encountered even in capillary wave regions. A coupling between the film thickness and the HTC was also reported in a separate, earlier effort by the present authors (Ref. [15]) that considered measurements only at a single spanwise location, along the streamwise direction of the flow and over a 40 mm domain centered at x ¼ 185 mm downstream of the inlet. In Ref. [15], joint probability density functions (PDFs) of the instantaneous and local HTC against the instantaneous and local film thickness were compiled from data generated over 12 measurement runs, which had the same flow conditions (Re ¼ 300, forcing frequency of 5 Hz) but with the heat input varied from 200 W to 5 kW. A series of noteworthy observations were presented and are briefly reproduced here in order to facilitate the upcoming analysis whose aim is to consider the role of unsteadiness and the impact of the flow Re on the instantaneous and local HTC.
Firstly, one important finding from the previous effort was that the HTC, even in a time-varying instantaneous sense, was largely independent of the imposed heat flux, at least over the range of heat fluxes examined in that (and the present) experimental campaign [15]. This permits data across experimental runs with different heat fluxes (but the same otherwise flow conditions, i.e. Re, etc.) to be grouped together. Secondly, the experimental results were compared to theoretical HTC variations expected from known film thickness variations based on Nusselt flow (corresponding to Nu ¼ 2:5); such a comparison has been included also in the present results in Fig. 19 (solid line). The Nusselt solution [53] describes the steady, 1-D, fully developed, laminar film flow over an inclined foil, and therefore constitutes a useful benchmark assessment tool for the hereby investigated wavy films, allowing the effect of unsteadiness to be examined relative to the steady laminar case. The film thickness and HTC for a Nusselt flow can be obtained by considering the momentum (Navier-Stokes) and energy equations respectively, and implementing the relevant simplifications (terms are neglected based on the aforementioned assumptions). The relevant derivation can be found in Ref. [15]. It should be noted that non-negligible deviations between the analytical Nusselt predications and the experimental findings are in fact expected, Fig. 19. PDFs of the HTC, h, plotted against film thickness, d, for data acquired under three different flow conditions: Re ¼ 68; Re ¼ 720 and Re ¼ 1300.
since the films described in the current contribution are not steady, one-dimensional, and neither fully developed nor even laminar in certain cases. In Ref. [15] an enhancement in the heat-transfer performance of wavy films relative to the steady case was observed, as anticipated, due to unsteadiness, increased mixing and turbulence. Despite the fact that the experimental results followed the general trend described by the Nusselt relationship reasonably well (i.e. the HTC experiences an enhancement when the films become thinner), the magnitude of the HTC enhancement (mean HTC relative to the HTC at the mean film thickness) was observed to be significantly higher than the Nusselt value, by up to a factor of 2.
New insight into the effects of unsteadiness on the observed HTC enhancement can be obtained by examining joint-PDFs generated from separate sets of data corresponding to different Re (Fig. 19). For the lowest Re flows (Re ¼ 68), the majority of the experimental data are in close proximity to the theoretical Nusselt description, but the film thickness does not show considerable variations (d % 0:5 mm) while the HTC varies by AE40% around a value of h % 3200 W m À2 K À1 . As the Re increases to 720, the trend described by the Nusselt solution is reproduced, with the experimentally determined HTC values being, however, globally higher than the Nusselt trend-line. Moreover, the HTC dependence on the film thickness becomes weaker than the analytical Nusselt solution at higher film thicknesses. Further increasing the Re to 1300 boosts the observed HTC values even further, and accentuates the previously noted observations, with the HTC appearing almost decoupled from the film thickness, attaining a value around h % 3000 W m À2 K À1 above approximately d % 1:2 mm. This decoupling at higher film thicknesses (which presumably correspond to regions adjacent to the wave crests) and high Re may be potentially linked to flow recirculation in the reference frame moving with the wave speed. As the Re increases, the flow velocity surpasses the wave velocity, resulting in flow recirculation underneath the wave crest (where the flow velocity is typically maximized). This phenomenon has been reported in the past for both laminar and turbulent falling films [54,55], and has been linked to local heat transfer enhancements [5]. In downwards annular gas-liquid flows, multiple recirculation zones have been reported underneath large wave structures, referred to as disturbance waves [56].
In Refs. [34,35], the heat transfer enhancement experienced due to waviness was studied for a laminar water-film flowing down a heated plate inclined at 2°to the horizontal. HTCs were measured along the wave topology and normalized by the mean value at the examined axial distance away from the flow inlet (230 mm). Unfortunately, this excellent experimental effort was limited to a narrow flow-parameter range (Re ¼ 126), and therefore any comparison will have to be restricted to this aforementioned condition. The results indicated that an þ80% enhancement compared to the mean HTC was experienced at the wave trough, while a À40% deterioration occurred below the wave crest. Behind the wave crest, and while the local film thickness exceeded the mean film thickness, HTC values lower the mean were consistently encountered. Ahead of the wave trough, and despite the fact that local film thicknesses were lower than the mean value, local HTCs once again did not exceed the mean value. Another noteworthy observation from this data is that the HTC variation followed closely (and inversely) the film thickness variation along the wave topology, a behaviour that was also observed in our experiments [15].
In reference to our present results shown in Fig. 19, we measured HTC values in the range 2100-5100 W m À2 K À1 with a mode of 3150 W m À2 K À1 for the flow with Re ¼ 68. These values correspond to a relative (normalized to the flow mean) HTC deterioration of (À30%) topologically emanating from below the wave crests, and a (+60%) enhancement stemming from the thinner film regions ahead of the wave crest (i.e. the wave troughs). Hence, close quantitative agreement is observed with the results presented in Refs. [34,35], despite the different experimental conditions and methods.
The formerly noted deviations from the Nusselt theory are re-examined in Fig. 20, where the mean flow Nu is plotted against the mean flow Re over the full range of tested pulsation amplitudes and frequencies. Along with the experimentally determined data points, a correlation produced by Wilke [57], linking the HTC, Prandtl number Pr and Re in heated falling films is also presented. Inspection of Fig. 20 reveals that whereas the lower Re data points (below approximately 200) fit closely the value predicted by the Nusselt theory (Nu ¼ 2:5), intermediate and high Re flows display significantly higher Nu numbers, matching Wilke's correlation closely when the latter is expressed in terms of the mean flow Nu number. A possible explanation for this substantial heattransfer enhancement, as observed in both experimental investigations [58] and also verified by modelling efforts [59], is stronger mixing and enhanced convective effects in the main wave regions, associated with the emergence of recirculation zones.

Quantitative heat-transfer characterization: Fluctuating performance
Nevertheless, the strength of the developed experimental technique is that it allows one to consider, beyond the mean heat-transfer performance of these flows, also the magnitude of the spacial and temporal variations around this mean behaviour. normalized by the mean film thickness d, for two pulsation frequencies: 1.8 Hz and 3.6 Hz. These quantities (fluctuation over mean) are referred to as fluctuation intensities. Referring to either pulsation frequency data, it can be ascertained that higher HTC fluctuation intensities are associated with lower film thickness fluctuation intensities. This trend can be described for both pulsation frequency data sets by fitting power functions, suggesting that stronger waviness dampens the oscillations in the HTC relative to the mean value. By comparing the two data sets amongst each other, it can be observed that overall lower HTC fluctuation intensities ensue in the higher pulsation frequency runs. Moreover, with Fig. 20. Mean Nu against the mean Re, compiled using data from different flow conditions spanning a range of pulsation amplitudes and frequencies for Pr ¼ 5:4. Along with the experimentally derived data, the correlation produced by Wilke [57] is also plotted for the examined Re range.
increasing film thickness fluctuation intensities, the disparity in the HTC behaviour between the higher wave frequency data and the lower wave frequency data diminishes.
Further, the effect of the HTC fluctuation intensity r h =h on the normalized fluctuation amplitude of the fluid interface-wall temperature difference r DT =DT is presented in Fig. 22. The temperature difference across the liquid film is observed to increase monotonically and near-linearly with the amplitude of the HTC fluctuation. The plotted data also reveals that the normalized temperature fluctuation is consistently about 25% smaller than the driving HTC fluctuation intensity. This relationship suggests that, in these particular flows, the heat flux fluctuations are small relative to those of DT and h.
Finally, it is worth considering briefly the effect of pulsation frequency on the interfacial wave velocity, the latter calculated from cross-correlations across successive film thickness profiles. This was investigated for three different flow Re in the range 140-350 and two forcing frequencies, 1.8 and 3.6 Hz. The results are presented in Fig. 23. As the mean flow Re increases from 140 to 350, the mean interfacial wave velocity increases in the range 0.5-0.9 ms À1 . Moreover, a near-linear trend can be identified despite the limited number of data points; a result that is expected given that the fluid viscosity and foil inclination angle remain unaltered, and consequently, the flow Re scales with the bulk flow velocity. Another interesting observation is that the relationship does not go through the origin. The underlying reason for this phenomenon is believed to be the superposition of the wave velocity with the bulk flow velocity. By comparing the two data sets amongst each other, it can finally be noted that the lower forcing frequency can be associated with higher interfacial velocities. This observation can be explained on the basis that higher excitation forcing frequencies produce shallower, smoother and slower waves compared to the larger and higher-dimensional waves associated with lower frequencies.

Conclusions
A combined PLIF and IR thermography technique was developed and employed for the detailed inspection and experimental characterization of unsteady and conjugate heat-transfer in planar, gravity-driven liquid films flowing down an inclined heated metal foil. The main challenge associated with these measurements, and in particular in the presence of finite heat-transfer conjugation effects, is that space and time-resolved information is required of both interfacial temperature variations (i.e at the solid-liquid and gas-liquid interfaces), which must be measured directly and simultaneously. Thus, simultaneous measurements were pursued and are reported herein for the local and instantaneous film thickness, free-surface temperature, foil/substrate temperature, heat flux exchanged between the heated foil and liquid film, and finally the HTC (and Nusselt number) over a range of flow conditions (i.e. Reynolds numbers, pulsation amplitudes and frequencies, heat fluxes).
The formation, interaction and spatial and temporal evolution of thermal rivulet structures on the film free-surface is examined and associated with film thickness variations. In some cases, flat thermal rivulets (i.e. not associated with film thickness fluctuations) were also identified and linked to the development of Marangoni flows on the film free surface. In more detail, local surface tension variations are believed to trigger film thickness instabilities and the ensuing (Marangoni-driven) mass transfer, characterized by accumulation of fluid in cooler regions and thinning in hotter ones. The introduction of flow pulsation had a pronounced effect on these features, instigating a clearly discernible mixing enhancement and resulting to the formation of a higher number of strongly skewed, intertwined thermal rivulets.
Following a qualitative assessment of the thermal features observed on the film free-surface and foil underside, a quantitative analysis was carried out with the ultimate goal of associating the observed interfacial wave dynamics with the heat-transfer characteristics of these flows (such as the instantaneous and local HTC variation with the mean film thickness and film thickness   fluctuation intensity). These are of paramount significance as they can be used to design equipment with far greater heat and mass transfer capabilities through understanding and harnessing enhancement mechanisms. Despite describing the HTC enhancement at lower film thickness values reasonably well and also capturing the overall HTC variations due to fluctuating film thicknesses, the results presented herein suggest that the magnitude of the true HTC can be significantly higher than HTC predictions based on Nusselt theory, by up to a factor 3 in some cases. A number of unsteady flow phenomena linked to the interface waviness of are believed to contribute towards this enhancement. The average Nu number increased with the mean flow Re, while at low values (lowest flow rates), a Nu of around 2.5 was observed, in agreement with the steady, laminar flow theory. Evidence that the HTC becomes decoupled from the film thickness for the upper range of encountered film thicknesses, has also been presented. Finally, lower wave amplitude intensities were associated with higher HTC fluctuation intensities (relative to their respective means), while the amplitude of the wall temperature fluctuation was found to be almost linearly dependent on the amplitude of the fluctuation of the HTC.