Experimental investigations of liquid falling films flowing under an inclined planar substrate

We report on detailed and systematic experiments of thin liquid films flowing as a result of the action of gravity under an inverted planar substrate. A measurement technique based on planar laser-induced fluorescence (PLIF) was developed and applied to a range of such flows in order to provide detailed space- and time-resolved film-height information. Specifically, the experimental campaign spanned three inclination angles ($\beta =-{15}^{\circ }$, $-{30}^{\circ }$, and $-{45}^{\circ }$, in all cases negative with respect to the vertical), two water-glycerol solutions (with Kapitza numbers of $\text{Ka}=13.1$ and 330), and flow Reynolds numbers covering the range $\text{Re}=0.6–193$. The collection optics were arranged so as to interrogate a spanwise section of the flow extending about $40\mathrm{mm}$ symmetrically on either side the centerline of the film span ($80\mathrm{mm}$ in total), at a distance 330 mm downstream of the flow inlet. A range of flow regimes, typically characterized by strong three dimensionality and pronounced rivulet formation, were observed depending on the imposed inlet flow conditions. In the lower liquid Kapitza number $\text{Ka}(=13.1)$ flows and depending on the flow Re, the free surface of the film was populated by smooth rivulets or regular sequences of solitary pulses that traveled over the rivulets. In the higher liquid $\text{Ka}(=330)$ flows, rivulets were observed typically above $\text{Re}\approx 30,$ depending also on the inclination angle, and grew in amplitude until quasi-two-dimensional fronts developed intermittently that were associated with distinct thin-film regions of varying length and frequency. These regions are of particular interest as they are expected to affect strongly the heat and mass transfer capabilities of these flows. The occurrence of the fronts was more pronounced, with higher wave frequencies, in film flows at smaller negative inclinations for the same flow Re. The rivulet amplitude was found to increase at larger inclinations for the same Re and showed a nonmonotonic trend with increasing Re, reaching a maximum that shifted to higher Re at larger inclinations. Furthermore, in flows that displayed pronounced rivulet formation [i.e., large (negative) $\beta$ and higher Re], the local film-height standard deviation in regions corresponding to the rivulet crests and troughs was reduced compared to the film-height standard deviation calculated over the entire examined film region. The mean rivulet wavelength also increased at larger inclinations, peaking at 26 mm when $\text{Ka}=330$. Based on our experimental results and theoretical arguments, we hypothesize that the formation of rivulets can be attributed, at small $\beta$, to a secondary Rayleigh-Taylor instability mechanism that destabilizes the suspended two-dimensional wavefronts, and at larger $\beta$, to the primary Rayleigh-Taylor instability mechanism of a flat film coating the underside of the inclined plate.

Figure 1

Wave patterns obtained by application of PLIF, with the light sheet arranged along the transverse direction of the flow. The flow Reynolds number, Re, fluid Kapitza number, Ka, and inclination angle, $\beta$, take the following values: (a) $\text{Re}=0.6$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$, (b) $\text{Re}=3.5$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$, (c) $\text{Re}=7.4$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$, (d) $\text{Re}=19$, $\text{Ka}=330,$ and $\beta =-{15}^{\circ }$, (e) $\text{Re}=36$, $\text{Ka}=330,$ and $\beta =-{30}^{\circ }$, and (f) $\text{Re}=118$, $\text{Ka}=330,$ and $\beta =-{45}^{\circ }$.

Figure 2

Schematic of our experimental arrangement showing the orientation of the laser sheet and cameras. [(a), (b)] Two-camera setup relative to the test section from the side and from above. (c) Single-camera setup from above. Angle $\beta$ corresponds to the substrate inclination angle relative to the vertical, and angle $\varphi$ to the angle formed between the excitation plane normal and the view of the cameras.

Figure 3

PLIF images after correction for perspective distortion, taken from flows with (a) $\text{Re}=8.2$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$ (reproduced from Fig. 2 in Ref. [11]) and (b) $\text{Re}=155$, $\text{Ka}=330,$ and $\beta =-{45}^{\circ }$.

Figure 4

PLIF intensity profile across the (a) Rhodamine-stained glass substrate and (b) liquid domain, from which the location of the solid-liquid and gas-liquid interfaces were recovered, respectively.

Figure 5

Mean film-height data, ${\overline{h}}_{\mathrm{PLIF}}$, recovered by application of PLIF and plotted against corresponding Nusselt heights, $h_{\mathrm{N}}$, over the ranges (a) $h_{\mathrm{N}}\approx 0.5–1\mathrm{mm}$ and (b) $h_{\mathrm{N}}\approx 0.5–3\mathrm{mm}$ (reproduced from Fig. 4 in Ref. [11]).

Figure 6

Space- and time-resolved film-height measurements from flows with (a) $\text{Re}=0.6$, (b) $\text{Re}=1.4$, (c) $\text{Re}=2.1$, (d) $\text{Re}=3.5$, (e) $\text{Re}=4.3$, and (f) $\text{Re}=8.2$, in all cases for $\text{Ka}=13.1$ and $\beta =-{30}^{\circ }$. This figure was reproduced from Fig. 5 in Ref. [11].

Figure 7

Space- and time-resolved film-height measurements from flows with (a) $\text{Re}=21$, (b) $\text{Re}=36$, (c) $\text{Re}=52$, (d) $\text{Re}=67$, (e) $\text{Re}=91$, and (f) $\text{Re}=175$, in all cases for $\text{Ka}=330$ and $\beta =-{30}^{\circ }$.

Figure 8

(a) Space-resolved mean film-height, ${\overline{h}}_{\mathrm{x}}$, and (b) film-height standard deviation, ${\sigma }_{\mathrm{x}}$, calculated along the imaged domain for flows with $\text{Re}=21$, 36, 52, 67, 91, and 175, in all cases for $\text{Ka}=330$ and $\beta =-{30}^{\circ }$.

Figure 9

Spatiotemporal film-height map compiled over a 0.35-s recording period from a flow with $\text{Re}=29$, $\text{Ka}=330,$ and $\beta =-{30}^{\circ }$.

Figure 10

Space- and time-resolved film-height measurements from flows with (a) $\text{Re}=38$, (b) $\text{Re}=60$, (c) $\text{Re}=86$, (d) $\text{Re}=110$, (e) $\text{Re}=138$, and (f) $\text{Re}=160$, in all cases for $\text{Ka}=330$ and $\beta =-{45}^{\circ }$.

Figure 11

(a) Space-resolved mean film height, ${\overline{h}}_{\mathrm{x}}$, and (b) film-height standard deviation, ${\sigma }_{\mathrm{x}}$, calculated along the imaged domain for flows with $\text{Re}=38$, 60, 86, 110, 138, and 160, in all cases for $\text{Ka}=330$ and $\beta =-{45}^{\circ }$.

Figure 12

Space- and time-resolved film-height measurements from flows with (a) $\text{Re}=19$, (b) $\text{Re}=40$, (c) $\text{Re}=61$, (d) $\text{Re}=83$, (e) $\text{Re}=108$, and (f) $\text{Re}=133$, in all cases for $\text{Ka}=330$ and $\beta =-{15}^{\circ }$.

Figure 13

(a) Space-resolved mean film-height, ${\overline{h}}_{\mathrm{x}}$, and (b) film-height standard deviation, ${\sigma }_{\mathrm{x}}$, calculated along the imaged domain for flows with $\text{Re}=19$, 40, 61, 83, 108, and 133, in all cases for $\text{Ka}=330$ and $\beta =-{15}^{\circ }$.

Figure 14

Space-resolved mean film height, ${\overline{h}}_{\mathrm{x}}$, and film-height standard deviation, ${\sigma }_{\mathrm{x}}$, calculated along the imaged domain for flows with (a) $\text{Re}=30$, 85, and 115, for $\text{Ka}=330$ and $\beta =-{15}^{\circ }$, (b) $\text{Re}=33$, 95, and 159, for $\text{Ka}=330$ and $\beta =-{30}^{\circ }$, and (c) $\text{Re}=32$, 93, and 154, for $\text{Ka}=330$ and $\beta =-{45}^{\circ }$.

Figure 15

Measured contact angle, $\theta$, as a function of the lateral-rim crest height, ${\overline{h}}_{\mathrm{LR}}$, for all examined flow conditions.

Figure 16

Film-height PDFs for flow conditions spanning the four Re ranges that were investigated, each corresponding to a different liquid Ka and $\beta$ combination. (a) $\mathrm{Ka}=330,\beta =-{15}^{\circ }$, (b) $\mathrm{Ka}=330,\beta =-{30}^{\circ }$, (c) $\mathrm{Ka}=330,\beta =-{45}^{\circ }$, and (d) $\mathrm{Ka}=13.1,\beta =-{30}^{\circ }$.

Figure 17

(a) Mean film height, ${\overline{h}}_{\mathrm{x}}$, and film-height standard deviation, ${\sigma }_{\mathrm{x}}$, plotted as a function of the distance, $x$, along the imaged region of the flow for a film with $\text{Re}=8.2$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$; ${\sigma }_{\mathrm{x}}$ data are shown as error bars. (b) Same plot as in panel (a) but showing the location of the rivulet crests, ${\overline{h}}_{\mathrm{M}}$, and troughs, ${\overline{h}}_{\mathrm{m}}$, and the local standard deviations, ${\sigma }_{\mathrm{h}},\mathrm{M}$ and ${\sigma }_{\mathrm{h}},\mathrm{m}$, respectively; the last two are shown as error bars.

Figure 18

Mean film heights, $\overline{h}$, rivulet-crest heights, ${\overline{h}}_{\mathrm{M}}$, and rivulet-trough heights, ${\overline{h}}_{\mathrm{m}}$, plotted against the flow Re for all examined flow conditions. (a) $\mathrm{Ka}=330,\beta =-{15}^{\circ }$, (b) $\mathrm{Ka}=330,\beta =-{30}^{\circ }$, (c) $\mathrm{Ka}=330,\beta =-{45}^{\circ }$, and (d) $\mathrm{Ka}=13.1,\beta =-{30}^{\circ }$.

Figure 19

(a) Averaged (over all detected rivulets) ${\overline{h}}_{\mathrm{M}}/\overline{h}$ and ${\overline{h}}_{\mathrm{m}}/\overline{h}$ against Re for all examined flow conditions (Re, $\beta$) with $\text{Ka}=330$. (b) Mean relative rivulet amplitude, $({\overline{h}}_{\mathrm{M}}-{\overline{h}}_{\mathrm{m}})/\overline{h}$, against Re for $\mathrm{Ka}=330$.

Figure 20

Film-height standard deviations, ${\sigma }_{\mathrm{h}}$, rivulet-crest height standard deviations, ${\sigma }_{\mathrm{h}},\mathrm{M}$, and rivulet-trough height standard deviations, ${\sigma }_{\mathrm{h}},\mathrm{m}$, plotted against the flow Re for all examined flow conditions. (a) $\mathrm{Ka}=330,\beta =-{15}^{\circ }$, (b) $\mathrm{Ka}=330,\beta =-{30}^{\circ }$, (c) $\mathrm{Ka}=330,\beta =-{45}^{\circ }$, and (d) $\mathrm{Ka}=13.1,\beta =-{30}^{\circ }$.

Figure 21

PSDs for flows with (a) $\text{Re}=19–133$, $\text{Ka}=330,$ and $\beta =-{15}^{\circ }$, (b) $\text{Re}=21–185$, $\text{Ka}=330,$ and $\beta =-{30}^{\circ }$, (c) $\text{Re}=30–160$, $\text{Ka}=330,$ and $\beta =-{45}^{\circ }$, and (d) $\text{Re}=0.6–8.2$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$.

Figure 22

[(a), (c)] Dominant wave frequencies, $f_{\mathrm{t}}$, calculated by employment of FFT. [(b), (d)] PSDs, $\mathrm{PSD}\left(f_{\mathrm{t}}\right)$, corresponding to the peaks in the spectra shown in Fig. 21.

Figure 23

Rivulet wavelengths, $\lambda$, obtained from the mean distance between adjacent rivulet crests and troughs, for flows with (a) $\text{Re}=19–133$, $\text{Ka}=330,$ and $\beta =-{15}^{\circ }$, (b) $\text{Re}=21–185$, $\text{Ka}=330,$ and $\beta =-{30}^{\circ }$, (c) $\text{Re}=30–160$, $\text{Ka}=330,$ and $\beta =-{45}^{\circ }$, and (d) $\text{Re}=0.6–8.2$, $\text{Ka}=13.1,$ and $\beta =-{30}^{\circ }$. (e) Rivulet wavelength obtained by Kharlamov et al. [38] for water films falling down a vertical plane (i.e., with $\beta =0^{\circ }$). The error bars indicate the standard deviation in the calculation of $\lambda$.