Statistical characteristics of falling-film flows: A synergistic approach at the crossroads of direct numerical simulations and experiments

We scrutinize the statistical characteristics of liquid films flowing over an inclined planar surface based on film height and velocity measurements that are recovered simultaneously by application of planar laser-induced fluorescence (PLIF) and particle tracking velocimetry (PTV), respectively. Our experiments are complemented by direct numerical simulations (DNSs) of liquid films simulated for different conditions so as to expand the parameter space of our investigation. Our statistical analysis builds upon a Reynolds-like decomposition of the time-varying flow rate that was presented in our previous research effort on falling films in [Charogiannis et al., Phys. Rev. Fluids 2, 014002 (2017)], and which reveals that the dimensionless ratio of the unsteady term to the mean flow rate increases linearly with the product of the coefficients of variation of the film height and bulk velocity, as well as with the ratio of the Nusselt height to the mean film height, both at the same upstream PLIF/PTV measurement location. Based on relations that are derived to describe these results, a methodology for predicting the mass-transfer capability (through the mean and standard deviation of the bulk flow speed) of these flows is developed in terms of the mean and standard deviation of the film thickness and the mean flow rate, which are considerably easier to obtain experimentally than velocity profiles. The errors associated with these predictions are estimated at $\approx 1.5$% and 8% respectively in the experiments and at $<1$% and $<2$% respectively in the DNSs. Beyond the generation of these relations for the prediction of important film flow characteristics based on simple flow information, the data provided can be used to design improved heat- and mass-transfer equipment reactors or other process operation units which exploit film flows, but also to develop and validate multiphase flow models in other physical and technological settings.

Figure 1

Schematic of the experimental facility and sample wave- and phase-locked average, film height, and flow-field measurements by application of PLIF and PTV (reproduced from Figs. 1 and 4 in Ref [39]). (a) Test-section schematic showing the settling chamber and substrate, and the light-sheet orientation relative to the flow. (b) Wave- and phase-locked average film height near the crest of a solitary wave for a flow with $\text{Re}=21,\text{Ka}=85$, and $f_{\mathrm{w}}=7$ Hz. (c) Wave- and phase-locked average velocity field corresponding to panel (b).

Figure 2

(a) Film-height and (b) bulk-velocity time series plotted over a $\approx 0.8$-s recording-period for a flow with $\text{Ka}=350,\text{Re}=65,$ and $f_{\mathrm{w}}=10$ Hz. Also shown are the film-height and bulk-velocity fluctuations, $h^{\prime }$ and $U^{\prime }$, respectively, the mean film height and bulk velocity, $\overline{h}$ and $\overline{U}$, respectively, and the film-height and bulk-velocity standard deviations, ${\sigma }_{\mathrm{h}}$ and ${\sigma }_{\mathrm{U}}$, respectively.

Figure 3

Bulk-velocity statistics obtained by application of PTV and DNSs of the same flows. (a) Bulk-velocity standard deviation, ${\sigma }_{\mathrm{U}}$, plotted against the flow $\text{Re}$, for the flow conditions provided in Table 1. (b) Coefficient of variation of the bulk-velocity, ${\sigma }_{\mathrm{U}}/\overline{U}$, plotted against the flow $\text{Re}$ for the same flow conditions.

Figure 4

Film-height coefficient of variation, ${\sigma }_{\mathrm{h}}/\overline{h}$, plotted against the bulk-velocity coefficient of variation, ${\sigma }_{\mathrm{U}}/\overline{U}$, over the range of flow conditions presented in Table 1.

Figure 5

Steady terms compiled using both experimental and numerical data for flows with $\text{Ka}=85$, 350, 1800, and 4363, plotted against the flow $\text{Re}$ (based on the results presented in Fig. 13 in Ref [39].). Fits to the four $\text{Ka}$ data sets are provided alongside plots of the mean flow rate against the flow $\text{Re}$, referred to in the legend as “${\nu }_{\mathrm{f}}$ fit.”

Figure 6

Unsteady terms compiled using both experimental and numerical data for flows with $\text{Ka}=85$, 350, 1800, and 4363, plotted against the film-height variance $\left({\sigma }_{\mathrm{h}}^2\right)$ (based on the results presented in Fig. 14 in Ref. [39]). Linear fits to the four $\text{Ka}$ data sets are also provided: (a) Experimental results. (b) Numerical results.

Figure 7

Film height plotted over one spatial period (wavelength) for a flow with $\text{Ka}=350,\text{Re}=45,$ and $f_{\mathrm{w}}=7$ Hz. The DNS results correspond to distances $x=0$, 56, 106, and 256 mm downstream of the flow inlet. The latter coincides with the location where PLIF/PTV data were collected.

Figure 8

Normalized steady, $\overline{U}\overline{h}/\overline{Q}$, and unsteady terms, $\overline{U^{\prime }{h}^{\prime }}/\overline{Q}$, plotted against the downstream distance from the inlet $(x=0.056$, 0.106, 0.156, 0.206, 0.256 m), for all flow conditions that were investigated numerically from Table 1.

Figure 9

Product of the coefficients of variation of the film height and bulk velocity, ${\sigma }_{\mathrm{h}}{\sigma }_{\mathrm{U}}/\left(\overline{h}\overline{U}\right)$, against the downstream distance from the inlet $x=0.056$, 0.106, 0.156, 0.206, and 0.256 m, and of the normalized unsteady term, $\overline{U^{\prime }{h}^{\prime }}/\overline{Q}$, for all flow conditions from Table 1 that were simulated numerically.

Figure 10

Product of the coefficients of variation of the film height and bulk velocity, ${\sigma }_{\mathrm{h}}{\sigma }_{\mathrm{U}}/\left(\overline{h}\overline{U}\right)$, against the normalized unsteady term, $\overline{U^{\prime }{h}^{\prime }}/\overline{Q}$, compiled from both experiments and simulations over all flow conditions in Table 1.

Figure 11

Normalized (by the Nusselt height) mean film height $\overline{h}/h_{\mathrm{N}}$, as a function of (a) the unsteady term, $\overline{U^{\prime }{h}^{\prime }}/\overline{Q}$, and (b) the product of the coefficients of variation of the film height and bulk velocity, ${\sigma }_{\mathrm{h}}{\sigma }_{\mathrm{U}}/\left(\overline{h}\overline{U}\right)$. Linear fits are shown to both experimental and numerical data.

Figure 12

Variation of $\overline{h}/h_{\mathrm{N}}$ as a function of (a) $\overline{U^{\prime }{h}^{\prime }}/\overline{Q}$, and (b) ${\sigma }_{\mathrm{h}}{\sigma }_{\mathrm{U}}/\left(\overline{h}\overline{U}\right)$ obtained by DNS of flows with $\text{Re},\text{Ka},\beta ,$ and $f_{\mathrm{w}}$ as outlined in Table 2.

Figure 13

Plot of the product of the coefficients of variation of the film height and bulk velocity, ${\sigma }_{\mathrm{h}}{\sigma }_{\mathrm{U}}/\left(\overline{h}\overline{U}\right)$, against the relative unsteady terms, $\overline{U^{\prime }{h}^{\prime }}/\overline{Q}$, compiled using numerical data pertaining to all flow conditions of Table 2.

Figure 14

Predicted values of the bulk velocity: (a) mean, ${\overline{U}}^{*}$, and (b) standard deviation, ${{\sigma }_{\mathrm{U}}}^{*}$, normalized by direct experimental and numerical results from the same flow conditions (in Table 3) and plotted as a function of the flow.