4.1. Spatio-temporal characterisation of Kapitza, ISW and merged mode

In this subsection, we will characterise the spatio-temporal linear responses associated with the Kapitza, ISW and merged instability modes, via spatio-temporal linear stability calculations, following the continuation procedure outlined in § 3.4.

Before that, we discuss figure 10, where we have plotted results from temporal linear stability calculations for the conditions used by Ishimura et al. (Reference Ishimura, Mergui, Ruyer-Quil and Dietze2023), i.e. water/air ( ${\textit{Ka}}$ = 3174), $\beta ={5}^\circ$ , $H^\star ={13}\,\textrm {mm}$ and ${\textit{Re}}_{{L}}$ = 43.1. The figure represents dispersion curves of the temporal growth rate, $\omega _i$ (figure 10 a), and wave speed, $c_r$ (figure 10 b), versus the wave number, $k_r$ . These curves obtained by continuation agree well with the Chebyshev calculation of Ishimura et al. (Reference Ishimura, Mergui, Ruyer-Quil and Dietze2023). Different symbols correspond to different values of the gas Reynolds number, ${\textit{Re}}_{{G}}$ . At low values of $|{{\textit{Re}}_{{G}}}|$ , the Kapitza mode (dashed black curves with open symbols) and the ISW mode (solid blue curves with filled symbols) are separated and distinct. Upon increasing $|{{\textit{Re}}_{{G}}}|$ , the two modes coalesce to form an unstable merged mode (thin solid red curves with open diamonds and pentagons). This merged mode is concatenated from the short-wave portion of the ISW mode and the long-wave portion of the Kapitza mode (a second, stable, merged mode, not represented in figure 10, results from the coalescence of the remaining opposing portions). At ${\textit{Re}}_{{G}}$ = –5750 (diamonds), the merged mode displays two maxima that are comparable in magnitude. Then, at ${\textit{Re}}_{{G}}$ = –6620 (pentagons), the short-wave maximum becomes dominant. As shown in figure 10(b), both the ISW mode and the merged mode are associated with negative values of the linear wave speed, $c_r$ .

Figure 11.

SP curves obtained from spatio-temporal linear stability calculations. Parameters according to figure 10: water/air ( ${\textit{Ka}}$ = 3174), $\beta ={5}^\circ$ , $H^\star ={13}\,\textrm {mm}$ , ${\textit{Re}}_{{L}}$ = 43.1. Thick, physical SPs satisfying the Briggs collision criterion (Briggs Reference Briggs1964); thin dotted, unphysical SPs. Circles, ${\textit{Re}}_{{G}}$ = −4700; squares, ${\textit{Re}}_{{G}}$ = −5200; diamonds, ${\textit{Re}}_{{G}}$ = −5750; pentagons, ${\textit{Re}}_{{G}}$ = −6620. (a) ISW mode; (b) Kapitza mode. For ${\textit{Re}}_{{G}}$ = −5750 (diamonds) and ${\textit{Re}}_{{G}}$ = −6620 (pentagons), the Kapitza mode splits into three segments, two of which are represented here via dashed and dot-dashed curves. See figure 25 of Appendix C for all three segments.

At this point, it is convenient to recap how the curves for the ISW mode in figure 10(a) were obtained via our numerical continuation procedure (see § 3.4). We focus on the blue curve with filled squares, which corresponds to ${\textit{Re}}_{{G}}$ = –5200. To obtain this curve, we first construct a curve for a greater value of $|{{\textit{Re}}_{{G}}}|$ , where the merged mode exists, e.g. ${\textit{Re}}_{{G}}$ = –5750 (diamonds in figure 10 a). The merged mode originates at $k_r$ = $k_i$ = $\omega _i$ = 0, where an analytical solution exists in the laminar limit ( $\tilde {l}$ = 0). We start at that solution and then activate the turbulent stresses via a Boolean control parameter. Next, we increase $k_r$ at fixed ${\textit{Re}}_{{G}}$ , thus following the red thin solid curve with diamonds in figure 10(a), until reaching the second neutral point, where $\omega _i$ = 0. After that, we fix $\omega _i$ = 0 and reduce $|{{\textit{Re}}_{{G}}}|$ until its target value, ${\textit{Re}}_{{G}}$ = −5200. Finally, we vary $k_r$ at fixed ${\textit{Re}}_{{G}}$ to obtain the blue curve with filled squares in figure 10(a). As explained in § 3.4, the dispersion curves in figure 10(a), obtained from temporal stability calculations, yield the starting points for our spatio-temporal calculations, which we discuss next.

Figure 11 represents SP curves obtained from spatio-temporal linear stability calculations for the same parameters as in figure 10. These curves represent the temporal growth rate, $\omega _i$ , associated with perturbations that remain at long time in the reference frame of a moving observer ( $\omega _k$ = 0) versus the speed of that observer, $V_{\textit{ray}}$ . Figure 11(a) corresponds to the ISW mode and figure 11(b) to the Kapitza mode. In figures 11(a) and 11(b), the thick solid, dashed and dash-dotted curve segments correspond to physical SPs, where the Briggs collision criterion (Briggs Reference Briggs1964) is satisfied, whereas the thin dotted segments correspond to unphysical SPs. The boundary between physical and unphysical SPs was determined by manually checking the Briggs collision criterion, i.e. by checking how spatial branches ( $\omega _i$ = const., $V_{\textit{ray}}$ = const.) collide in the $k_i$ – $k_r$ plane at a given SP (see discussion of figures 13 and 14). For the inclination angle considered here, $\beta ={5}^\circ$ , no merging of the ISW and Kapitza modes occurs, in contrast to the temporal analysis reported in figure 10. We will observe a merged mode for larger values of $\beta$ , as will be discussed later in this subsection (see figure 15).

From figure 11(a), we conclude that the ISW mode is always upward convective, i.e. physical SPs with $\omega _i\gt 0$ only exist for $V_{\textit{ray}}\lt 0$ . However, positive growth rates are observed up to slightly negative values of $V_{\textit{ray}}$ . Also, we observe that the physical segments of the SP-curves (solid blue curves) retain their bell-shaped form upon increasing $|{{\textit{Re}}_{{G}}}|$ , whereby their maximum growth rate increases.

According to the Briggs criterion, there can be only one physical instability mode for any given $V_{\textit{ray}}$ . Thus, the Kapitza mode must cover the range from slightly negative to large positive values of $V_{\textit{ray}}$ , i.e. it is absolute ( $\omega _i\gt 0$ , $V_{\textit{ray}}=0$ ), as evidenced by the dashed curves in figure 11(b). This mode undergoes several topological changes. At low $|{{\textit{Re}}_{{G}}}|$ values, the SP-curves are bell-shaped and their maximum increases with increasing $|{{\textit{Re}}_{{G}}}|$ . Then, at ${\textit{Re}}_{{G}}$ = −5750 (diamonds), the Kapitza mode splits into several SP curves that each, via their physical portions, cover a certain range of $V_{\textit{ray}}$ . For ease of representation, only two of these segments are represented, i.e. via the dashed and dash-dotted thick curves. Figure 25 in Appendix C shows all three physical SP-curve segments. We point out that, due to the multiplicity of SP curves in figure 11(b), the growth rate, $\omega _i$ , can jump discontinuously upon varying $V_{\textit{ray}}$ .

After a further increase of $|{{\textit{Re}}_{{G}}}|$ , i.e. at ${\textit{Re}}_{{G}}$ = −6620 (pentagons in figure 11 b), the dashed SP-curve segment has lost its maximum and has taken a shape that is better adjusted to the corresponding segment of the ISW mode in figure 11(a) (solid blue curve with filled pentagon there). Even though this adjustment of the ISW and Kapitza modes does not constitute a true merging, it does lead to a rather smooth transition from one mode to the other upon varying $V_{\textit{ray}}$ . This can be clearly seen in figure 15(a), where we have plotted SP curves for the ISW and Kapitza modes in the same graph. Finally, comparing figures 11(a) and 11(b), we observe that the ISW mode becomes dominant at large $|{{\textit{Re}}_{{G}}}|$ .

Figure 12.

Wave measures for the physical portions of the SP curves in figure 11 (thick lines there). Symbols mark the locus of SPs with maximum $\omega _i$ . (a) Wave number; (b) wave speed in the laboratory reference frame; (c) imaginary part of the wave number, $k_i$ , allowing to determine the spatial growth rate; (d) wave speed in the laboratory reference frame versus the wave number. Line styles and symbols as in figure 11.

In figure 12, we have plotted different wave measures corresponding to the physical portions of the SP curves in figure 11 for both the ISW mode (blue solid curves with filled symbols) and the Kapitza mode (dashed and dash-dotted black curves with open symbols). These measures correspond to waves with group velocity $V_{\textit{ray}}$ in the laboratory reference frame. The symbols identify the SPs with maximum growth rate, $\omega _i$ , in figure 11, i.e. the waves of a given mode that are most likely to emerge in an experiment.

Figure 12(a) plots the wave number, $k_r$ , versus the ray velocity, $V_{\textit{ray}}$ , confirming that the ISW mode is indeed associated with short waves (large $k_r$ ), whereas the Kapitza mode is associated with long waves (small $k_r$ ), at least for $V_{\textit{ray}}\lt 50$ , where $\omega _i{\gt 0}$ in figure 11(b). Figure 12(b) shows the wave speed in the laboratory frame, $\omega _r$ / $k_r+V_{\textit{ray}}$ , versus $V_{\textit{ray}}$ . Here, we see that the ISW mode is mostly associated with negative wave speeds, whereas the Kapitza mode is always associated with positive wave speeds. Figure 12(c) plots the imaginary part of the wave number, $k_i$ , versus $V_{\textit{ray}}$ , allowing to determine the spatial growth rate. For both the ISW (filled symbols) and Kapitza (open symbols) modes, the SPs marked by symbols, which correspond to the growth rate maxima in figures 11(a) and 11(b), coincide with $k_i{=0}$ . Thus, in combination with the curves for $\omega _r$ / $k_r+V_{\textit{ray}}$ in figure 12(b), we conclude that unstable waves associated with the ISW mode grow spatially as they move upward and unstable waves associated with the Kapitza mode grow spatially as they move downward. Finally, figure 12(d) plots the wave speed in the laboratory reference frame versus the wave number, confirming that the most-amplified upward-travelling waves associated with the ISW mode (filled symbols) are short and that the most-amplified downward-travelling waves associated with the Kapitza mode (open symbols) are long.

Before discussing the effect of different control parameters on the ISW and Kapitza modes, we demonstrate via figures 13 and 14 how we have checked the Briggs collision criterion (Briggs Reference Briggs1964) for the SP-curves in figure 11. Figure 13(a) represents SP curves corresponding to the ISW and Kapitza modes for ${\textit{Re}}_{{G}}$ = −5200 (curves with squares in figure 11). Figure 13(b) represents the corresponding dispersion curves obtained from temporal stability analysis. These allow to identify the maximum of $\omega _i$ , up to which the evolution of spatial branches must be tracked. For the current parameters, we observe $\omega _i^{\textit{max}}=15.1$ .

Figure 13.

Selected stability calculations for the ISW (solid blue) and Kapitza (dashed black) instability modes. Parameters according to figure 11(a) (squares there): ${\textit{Re}}_{{G}}$ = –5200. (a) SP curves, $\omega _k$ = 0. Thick, physical SPs; dotted, unphysical SPs. Symbols mark SPs used for checking Briggs collision criterion (Briggs Reference Briggs1964) in figure 14. (b) Temporal stability analysis for conditions in panel (a): $k_i$ = $V_{\textit{ray}}$ = 0. The maximum for $\omega _i$ is $\omega _i^{\textit{max}}$ = 15.1.

Figure 14.

Verification of the Briggs collision criterion (Briggs Reference Briggs1964) for SPs marked by symbols in figure 13(a). Evolution of spatial branches (fixed $\omega _i$ and $V_{\textit{ray}}$ ) as $\omega _i$ is reduced from $\omega _i\gt \omega _i^{\textit{max}}$ (dash-dotted curves) to $\omega _i\lt \omega _i^{\textit{SP}}$ (dashed curves), where $\omega _i^{\textit{max}}=15.1$ is the maximum temporal growth rate observed in figure 13(b). The value of $\omega _i$ decreases from the dash-dotted line, via the dotted line, until the dashed line. (a) Physical SP on the ISW branch (filled square in figure 13 a): $V_{\textit{ray}}$ = −57, $\omega _i^{\mathrm{\textit{SP}}}=6.96$ , $\omega _i$ = $[15.3, 7.73, 6.01]$ ; (b) unphysical SP on the Kapitza branch (open square in figure 13 a): $V_{\textit{ray}}$ = −44, $\omega _i^{\textit{SP}}=$ −41.0, $\omega _i$ = [15.3, −39.5, −42.9]; (c) unphysical SP on the ISW branch (square with vertical division in figure 13 a): $V_{\textit{ray}}$ = 63, $\omega _i^{\mathrm{\textit{SP}}}=0.882$ , $\omega _i$ = $[15.3, 5.92, 0.687]$ ; (d) physical SP on the Kapitza branch (square with horizontal division in figure 13 a): $V_{\textit{ray}}$ = 47, $\omega _i^{\mathrm{\textit{SP}}}=$ −8.05, $\omega _i$ = [15.3, 3.78, −8.24].

The symbols in figure 13(a) mark SPs for which we have checked the Briggs collision criterion in figure 14 and, as in figure 11, thick solid and dashed lines correspond to physical SPs, where the collision criterion is satisfied, and thin dotted lines correspond to unphysical SPs. Figure 14 represents spatial branches, i.e. curves with $\omega _i$ = const. and $V_{\textit{ray}}$ = const., associated with these different SPs. Figure 14(a) corresponds to the filled square in figure 13(a), i.e. a physical SP on the ISW branch at $V_{\textit{ray}}$ = −57 and $\omega _i^{\textit{SP}}$ = 6.96. As shown in figure 14(a), the spatial branches represented by dash-dotted ( $\omega _i$ = 15.3), dotted ( $\omega _i$ = 7.73) and dashed ( $\omega _i$ = 6.01) magenta curves emanate from opposite half-planes (in terms of $k_i$ ) at $\omega _i\gt \omega _i^{\textit{max}}$ = 15.1 (dash-dotted curves), approach the SP (filled square) without intersecting one another (dotted curves, $\omega _i\gt \omega _i^{\textit{SP}}$ ), collide (not shown) and then move away from one another (dashed curves, $\omega _i\lt \omega _i^{\textit{SP}}$ ). For all values of $\omega _i$ up to the collision, an initial integration path with $k_i=0$ at $\omega _i\gt \omega _i^{\textit{max}}$ can be deformed and translated such as to avoid intersecting any spatial branches. Thus, the Briggs collision criterion is satisfied.

Next, figure 14(b) corresponds to the SP marked by an open square in figure 13(a), i.e. an unphysical SP on the Kapitza branch at $V_{\textit{ray}}$ = −44 and $\omega _i^{\textit{SP}}$ = −41.0. The spatial branches colliding at that SP in figure 14(b) both emanate from the upper half-plane ( $k_i\gt 0$ ) and, thus, the Briggs collision criterion is not satisfied.

The opposite is observed for the SPs marked by half-filled squares in figure 13(a), for which figures 14(c) and 14(d) represent the evolution of spatial branches up to and beyond collision. In this case, the SP corresponding to the ISW mode (blue square with vertical division in figure 13(a), $V_{\textit{ray}}$ = 63, $\omega _i^{\textit{SP}}$ = 0.882) is unphysical, as the spatial branches colliding at that point in figure 14(c) both emanate from the lower half-plane ( $k_i\lt 0$ ). By contrast, the SP corresponding to the Kapitza mode (black square with horizontal division in figure 13 a, $V_{\textit{ray}}$ = 47, $\omega _i^{\textit{SP}}$ = −8.05) is physical, as the spatial branches colliding at that point in figure 14(d) emanate from opposite half-planes and do not intersect before colliding.

Figure 15.

Effect of the inclination angle, $\beta$ , on the ISW (thick blue solid), Kapitza (thick black dashed and dash-dotted), and merged (thin red solid) instability modes. Parameters according to figure 11. Spatio-temporal stability formulation: $\omega _k$ = 0. Dotted (blue, black and red) lines correspond to unphysical SPs, where the Briggs collision criterion is not satisfied. (a) $\beta$ = ${5}^\circ$ . Circles, ${\textit{Re}}_{{G}}$ = −4700; diamonds, ${\textit{Re}}_{{G}}$ = −5750; pentagons, ${\textit{Re}}_{{G}}$ = −6620. (b) $\beta$ = ${10}^\circ$ . Circles, ${\textit{Re}}_{{G}}$ = −6000; diamonds, ${\textit{Re}}_{{G}}$ = −6600; pentagons, ${\textit{Re}}_{{G}}$ = −8000. (c) $\beta$ = ${15}^\circ$ . Circles, ${\textit{Re}}_{{G}}$ = −7135; diamonds, ${\textit{Re}}_{{G}}$ = –7600; pentagons, ${\textit{Re}}_{{G}}$ = −8500. For ${\textit{Re}}_{{G}}$ = −8500, there are no physical instability modes for $V_{\textit{ray}}\gt 12.5$ . (d) $\beta$ = ${90}^\circ$ . Circles, ${\textit{Re}}_{{G}}$ = −5000; diamonds, ${\textit{Re}}_{{G}}$ = −9000; pentagons, ${\textit{Re}}_{{G}}$ = −12 000.

We now discuss how the spatio-temporal linear response of a gas-sheared falling liquid film is affected by the inclination angle, $\beta$ . In particular, we wish to know what happens when the falling liquid film is vertical. Figure 15 represents SP curves obtained for the water/air system and four different values of $\beta$ . Figure 15(a) corresponds to the inclination angle studied up until now, i.e. $\beta ={5}^\circ$ . Here, we have plotted selected SP curves from figures 11(a) and 11(b) in the same graph. Once again, thick solid blue curves correspond to the ISW mode and thick dashed and dash-dotted black curves to the Kapitza mode, whereas thin dotted curves highlight unphysical SPs. For the highest value of $|{{\textit{Re}}_{{G}}}|$ (pentagons), the physical portions of the Kapitza and ISW modes seem to connect rather smoothly, but they do not truly merge.

This changes at larger inclination angles, e.g. at $\beta ={10}^\circ$ in figure 15(b). Here, the SP curves for the two modes collide at ${\textit{Re}}_{{G}}$ = −6600 (diamonds) and, for larger values of $|{{\textit{Re}}_{{G}}}|$ , they merge to form two merged modes. The thin red curves marked by pentagons, which correspond to ${\textit{Re}}_{{G}}$ = −8000, represent these two merged modes. The first mode (filled pentagon) results from a merging of the physical portions of the ISW and Kapitza SP curves. This mode remains physical over a wide range of $V_{\textit{ray}}$ and becomes unphysical only for $V_{\textit{ray}}\gt 75$ , where $\omega _i\lt 0$ . Thus, we will designate it as physical merged mode. It is unstable over a range of $V_{\textit{ray}}$ spanning from large negative values, via $V_{\textit{ray}}$ = 0, up to large positive values, i.e. it is absolute. The second merged mode (open pentagon) results from a merging of the unphysical portions of the ISW and Kapitza modes. It is physical only for $V_{\textit{ray}}\gt 75$ , where $\omega _i\lt 0$ . Thus, it is never associated with an instability.

Figure 15(c) represents SP curves for an even larger inclination angle, i.e. $\beta ={15}^\circ$ . In this case, merging of the ISW and Kapitza modes occurs before the ISW mode can become unstable on its own, i.e. at ${\textit{Re}}_{{G}}$ = −7135, corresponding to the curves with circles. Nonetheless, the physical merged mode resulting from this merging inherits one of the main features of the ISW mode, i.e. its capacity to generate upward-travelling ripples. This is evidenced by the thin solid red curves with diamonds and pentagons, which exhibit a wide region of $V_{\textit{ray}}\lt 0$ , where $\omega _i\gt 0$ .

Finally, figure 15(d) represents SP curves for a vertically falling liquid film. Here, merging of the ISW and Kapitza modes occurs when the ISW mode is very far from instability ( $\omega _i{\ll }0$ ). This may explain why Vellingiri et al. (Reference Vellingiri, Tseluiko and Kalliadasis2015) and Schmidt et al. (Reference Schmidt, Náraigh, Lucquiaud and Valluri2016) did not observe the ISW mode in their temporal linear stability calculations for vertically falling liquid films. The interfacial modes observed by these authors, which can display growth rate maxima at very large wave number values (see e.g. figure 8), seem to be examples of our merged mode. Another consequence is that the signature of the ISW mode becomes apparent in the unstable portion of the merged mode only at large $|{{\textit{Re}}_{{G}}}|$ (solid red curve with filled pentagon in figure 15 d, where ${\textit{Re}}_{{G}}$ = −12 000). Further, the maximum $\omega _i$ is now observed in the range $V_{\textit{ray}}\gt 0$ , making it more likely to observe downward travelling waves as opposed to upward-travelling ripples in an experiment. As a general conclusion, we may deduce from figures 15(a)−15(d) that the relevance of the ISW mode (or of its signature) is confined to increasingly large $|{{\textit{Re}}_{{G}}}|$ as the inclination angle, $\beta$ , is increased.

Figure 16.

Role of turbulence in shaping the ISW, Kapitza and merged instability modes. Parameters according to figures 15(a) and 15(d). Comparison of SP curves for turbulent (curves with regular symbols) and laminar (curves with large symbols, $\tilde {l}_{\mathrm{t}}$ = 0 in (3.11g )) primary-flow gas velocity profiles at equivalent $T^\star _{{{G}} 0}$ . (a) $\beta$ = ${5}^\circ$ , $T^\star _{{{G}} 0}={3170}\,\textrm {N} \,\textrm {m}^{-2}$ (figure 15 a). Regular diamonds, ${\textit{Re}}_{{G}}$ = −5750; large diamond, ${\textit{Re}}_{{G}}$ = −26 233. (b) $\beta$ = ${90}^\circ$ , $T^\star _{{{G}} 0}=10\,100 \,\textrm {N} \, \textrm {m}^{-2}$ (figure 15 d). Regular pentagons, ${\textit{Re}}_{{G}}$ = −12 000; large pentagons, ${\textit{Re}}_{{G}}$ = −90 712.

In figure 16, we discuss the role of turbulence in shaping the ISW, Kapitza and merged instability modes. In figures 16(a) and 16(b), we compare selected SP curves from figures 15(a) ( $\beta ={5}^\circ$ , ${\textit{Re}}_{{G}}$ = −5750, diamonds) and 15(d) ( $\beta ={90}^\circ$ , ${\textit{Re}}_{{G}}$ = −12 000, pentagon) with corresponding SP curves (large diamonds in figure 16 a and large pentagon in figure 16 b) obtained under the assumption of a laminar primary-flow gas velocity profile ( $\tilde {l}_{\mathrm{t}}$ = 0 in (3.11g )) at equivalent $T^\star _{{G}}$ . Due to the absence of Reynolds stresses in the laminar limit, much greater values of $\left |\textit{Re}_{G}\right |$ are required to reach the same value of $T^\star _{{G}}$ as in the turbulent case, i.e. ${\textit{Re}}_{{G}}$ = −26 233 in figure 16(a) and ${\textit{Re}}_{{G}}$ = −90 712 in figure 16(b). This may explain why Schmidt et al. (Reference Schmidt, Náraigh, Lucquiaud and Valluri2016), who considered laminar flow, observed their interfacial mode, with a growth rate maximum at large wave number values (see e.g. figure 8), only at very large values of $|{{\textit{Re}}_{{G}}}|$ , and why negative wave speeds were observed only for a moderate density ratio.

Comparing the turbulent (curves with regular symbols) and laminar (curves with large symbols) calculations in figures 16(a) and 16(b), we observe quite significant differences. For the lower inclination angle, $\beta$ = ${5}^\circ$ (figure 16 a), there is a qualitative difference between the two data sets, as merging of the ISW and Kapitza SP curves occurs in the laminar case, whereas it does not in the turbulent case. Also, the maximum temporal growth rate, $\omega _i^{\textit{max}}$ , is five times greater in the laminar limit. For the case of a vertically falling liquid film, $\beta$ = ${90}^\circ$ (figure 16 b), differences between the turbulent and laminar predictions are only quantitative, yet they remain very significant. Thus, we may conclude that the role of turbulence in shaping the different instability modes is not only due to the larger gas stresses (for fixed ${\textit{Re}}_{{G}}$ ) it generates at the liquid–gas interface, but also due to the shape of the turbulent velocity profile in the gas, which modifies the linear stability problem, as compared with the laminar limit, via ${\varPsi _0^{\prime \prime \prime}}$ in (3.21c ).

Figure 17.

Role of the linear gas stress perturbations, $\check {T}_{{G}}$ and $\check {P}_{{G}}$ , defined in (3.23), in shaping the ISW and Kapitza modes via (3.7d ) and (3.7e ). Saddle-point curves for one set of parameters from figure 11: ${\textit{Re}}_{{G}}$ = −5200. Solid lines with squares, full model; dashed with crosses, $\hat {T}_{{G}}$ = 0 in (3.7d ); dash-dotted with asterisks: $\hat {P}_{{G}}$ = 0 in (3.7e ). (a) ISW mode; (b) Kapitza mode.

To conclude this section, we discuss figure 17, which quantifies the role of the linear gas stress perturbations, $\check {T}_{{G}}$ and $\check {P}_{{G}}$ , defined in (3.23), in shaping the ISW (figure 17 a) and Kapitza (figure 17 b) instability modes. Conditions correspond to one set of parameters from figure 11, i.e. ${\textit{Re}}_{{G}}$ = −5200. The solid curves marked with filled and open squares in figure 17 were obtained by solving the full linear stability problem, and correspond exactly to the two curves marked with squares in figures 11(a) and 11(b). In figure 17, the dashed curves marked with crosses ( $\hat {T}_{{G}}$ = 0) and the dash-dotted curves marked with asterisks ( $\hat {P}_{{G}}$ = 0) represent different limits, wherein either the perturbation amplitude of the tangential gas shear stress, $\hat {T}_{{G}}$ , which enters via (3.7d ), or that of the gas pressure, $\hat {P}_{{G}}$ (asterisks), which enters via (3.7e ), is inactive. According to figure 17(a), both $\check {T}_{{G}}$ and $\check {P}_{{G}}$ are responsible for rendering the ISW mode unstable, the effect of $\check {P}_{{G}}$ being somewhat stronger. In the case of the Kapitza mode (figure 17 b), the effect of $\check {T}_{{G}}$ is stabilising, whereas that of $\check {P}_{{G}}$ is destabilising. For convenience, we have not checked for the unphysical portions of the dashed and dash-dotted SP curves in figures 17(a) and 17(b) via the Briggs criterion.

4.2. Neutral stability bound of the ISW mode

In the current subsection, we establish the range of parameters for which the ISW mode is unstable, i.e. under what conditions upward-travelling ripples are possible in an experiment. A convenient way to identify this limit via numerical continuation is to track the neutral points of the ISW, Kapitza and merged modes in terms of the main control parameters. By neutral points, we mean points where $\omega _i$ = 0.

In figures 18(a) and 18(b), we have replotted the physical portions of the SP curves from figures 15(a) and 15(b) in an $\omega _i$ versus $k_r$ diagram. In addition, in figure 18(a), we have plotted an additional SP curve (marked by an asterisk) corresponding to the critical ${\textit{Re}}_{{G}}$ at the onset of the ISW mode, where $\omega _i^{\mathrm{\textit{max}}}$ = 0. Magenta crosses in figures 18(a) and 18(b) mark neutral points obtained from corresponding temporal stability calculations ( $k_i$ = $V_{\textit{ray}}$ = $\omega _i$ = 0). We observe that the neutral points are invariant between the temporal ( $k_i$ = $V_{\textit{ray}}$ = $\omega _i$ = 0) and spatio-temporal ( $\omega _i$ = $\omega _k$ = 0) calculations, i.e. the corresponding physical wave number, $k_r$ , obtained from both types of calculations is the same. Tracking these neutral points ( $\omega _i$ = 0) in terms of different control parameters will allow us to identify which of the three instability modes are unstable for a given set of conditions. For convenience, we will perform these calculations using the temporal formulation.

Figure 18.

Physical portions of the SP curves from figures 15(a) and 15(b) replotted in the $\omega _i$ versus $k_r$ plane. Crosses mark neutral points ( $\omega _i$ = 0) obtained from corresponding temporal stability calculations ( $k_i$ = $V_{\textit{ray}}$ = 0). (a) $\beta ={5}^\circ$ (figure 15 a). Circles, ${\textit{Re}}_{{G}}$ = −4700; asterisks, ${\textit{Re}}_{{G}}$ = −4915; squares, ${\textit{Re}}_{{G}}$ = −5200; diamonds, ${\textit{Re}}_{{G}}$ = −5750; pentagons, ${\textit{Re}}_{{G}}$ = −6620. (b) $\beta ={10}^\circ$ (figure 15 b). Circles, ${\textit{Re}}_{{G}}$ = −6000; diamonds, ${\textit{Re}}_{{G}}$ = −6600; pentagon, ${\textit{Re}}_{{G}}$ = −8000.

Before discussing the results of these calculations, two caveats regarding the equivalence of neutral points obtained from temporal and spatio-temporal calculations must be mentioned. First, some SP curves associated with the Kapitza mode (dashed black curves) in figures 18(a) and 18(b) possess an additional neutral point that does not have an analogue in the temporal formulation (points of intersection with the $\omega _i$ = 0 axis that do not coincide with a magenta cross). However, such points are not physically relevant, because they lie on the lower branches of SP curves possessing a limit point. This means that another SP, with $\omega _i\gt 0$ , exists at the same value of $k_r$ . Second, we have previously shown (compare figures 10 a and 15 a) that merging of the ISW and Kapitza modes is not necessarily observed in both the temporal and spatio-temporal formulations. As merging results in a loss of neutral points, it is conceivable that this discrepancy may yield a different behaviour of such points in the temporal and spatio-temporal formulations. However, as shown in figure 15(a), neutral points disappear in the spatio-temporal formulation even in the absence of merging, due to a loss of physicality (violation of the Briggs criterion) of SP curve portions (dotted portions in figure 15 a). As we will show in figure 19(a), the loss of neutral points due to merging (temporal formulation) coincides with the loss of neutral points due to the truncation of unphysical SP curve portions (spatio-temporal formulation), at least up to the accuracy of our calculations.

Figure 19(a) represents the typical plot type that we will use to characterise the onset of the ISW instability mode. Here, we have plotted the evolution of the neutral points ( $\omega _i$ = 0, marked by magenta crosses in figure 18 a) under a variation of ${\textit{Re}}_{{G}}$ at fixed ${\textit{Re}}_{{L}}$ for the water/air system. We focus on the black curve with circles, which corresponds to the conditions in figure 18(a). The onset of the ISW mode is given by the limit point (LP) marked by a filled black circle. Here, the two neutral points ( $\omega _i$ = 0) associated with the ISW mode collapse at the same value of $k_r$ , i.e. at the SP marked by the asterisk in figure 18(a). For lower values of $\left |{Re_G}\right |$ , the neutral curve in figure 19(a) displays only one neutral point (in addition to $\omega _i$ = 0 at $k_r$ = 0, which is not shown) and, thus, only the Kapitza mode is unstable, as evidenced by the curves with circles in figure 18(a). At the lower end of the plotted $\left |{Re_G}\right |$ range, the curves in figure 19(a) are bounded by the turbulent/laminar transition, beyond which our immobile-interface assumption eventually breaks down. For large values of $\left |{Re_G}\right |$ , i.e. to the left of the filled circle in figure 19(a), there are two possible situations. Either there are three neutral points, in which case both the ISW and Kapitza modes are individually unstable (see e.g. curves with squares in figure 18 a, where ${\textit{Re}}_{{G}}$ = −5200), or there is only one neutral point, corresponding to the largest neutral $k_r$ of the ISW mode (see e.g. curves with diamonds in figure 18 a, where ${\textit{Re}}_{{G}}$ = −5750).

Figure 19.

Effect of the channel height, $H^\star$ , on the ISW mode. Parameters based on figure 11: $\beta$ = 5 $^\circ$ , ${\textit{Re}}_{{L}}$ = 43.1. (a) Neutral point curves ( $\omega _i$ = 0) from temporal formulation associated with the ISW (thick solid), Kapitza (dashed) and merged (thin solid) instability modes: ${\textit{Ka}}$ = 3174 (water/air). Filled symbols, instability threshold of the ISW mode; open symbols, merging point (MP); crosses, flooding point (FP). From circles to triangles, $H^\star$ = $[13,19,25,35,50]$ mm. Asterisks, spatio-temporal calculation. Thin cyan curve with plus sign tracks ISW neutral bound. (b) ISW neutral bound in terms of $H/h_0$ versus ${\textit{Re}}_{{G}}$ for different liquids from table 1 in contact with air. From crosses to pentagons, ${\textit{Ka}}$ = $[17.78,121.4,509.5,963.2,301.3,3174,1988]$ , $\varPi _\mu$ = $[0.0009,0.004,0.0058,0.0078,0.0076,0.018,0.031]$ .

The transition point from three neutral points to one single neutral point is identified by open symbols in figure 19(a) and throughout the manuscript. In the temporal formulation, such a transition is associated with a merging of the ISW and Kapitza modes (e.g. figure 10 a). In the spatio-temporal formulation, it is either associated with the same type of merging (curves with pentagons in figures 15 b and 18 b) or with a concatenation of physical SP curve segments associated with the ISW and Kapitza modes (curves with diamonds in figures 15 a and 18 a). In the latter case, the transition point is not necessarily identical for the temporal and spatio-temporal formulations. However, we have found no significant discrepancies in our calculations. This is illustrated by the black asterisks in figure 19(a), which correspond to the neutral point curve at $\beta$ = ${5}^\circ$ obtained with the spatio-temporal formulation. This curve agrees closely with the neutral point curve obtained from the temporal formulation (black solid curve with circles). Consequently, in the remainder of the manuscript, we will refer to the transition point marked by open symbols in figure 19(a) as the merging point (MP), a term that will encompass both true mode merging (figure 15 b) and the concatenation of physical SP curve portions stemming from different modes (figure 15 a). Further increase of $|{{\textit{Re}}_{{G}}}|$ past this point eventually leads to the loss of SP solutions, owing to a breakdown of the primary flow solution. The corresponding point is designated as flooding point (FP), following Kushnir et al. (Reference Kushnir, Barmak, Ullmann and Brauner2021), and is marked by crosses in figure 19(a) and similar figures.

Figure 20.

Effect of the liquid Reynolds number, ${\textit{Re}}_{{L}}$ , on the onset of the ISW mode. Neutral point curves for two fluid combinations from table 1: $H^\star ={13}\,\textrm {mm}$ , $\beta ={5}^\circ$ . From pentagons to triangles, ${\textit{Re}}_{{L}}$ = $[5,10,20,43.1,80]$ . (a) ${\textit{Ka}}$ = 3174 (water/air); (b) ${\textit{Ka}}$ = 17.78 (silicone oil II/air).

Next, we discuss via figures 19, 20 and 21 how the instability threshold of the ISW mode is affected by the main control parameters, i.e. the channel height, $H^\star$ , the liquid and gas Reynolds numbers, ${\textit{Re}}_{{L}}$ and ${\textit{Re}}_{{G}}$ , the inclination angle, $\beta$ , and the liquid properties, represented in particular by the viscosity ratio, $\varPi _\mu$ , and the Kapitza number, ${\textit{Ka}}$ . In figure 19(a), the different neutral curves plotted correspond to different values of $H^\star$ at fixed ${\textit{Re}}_{{L}}$ = 43.1 and $\beta ={5}^\circ$ for the water/air system. We observe that the critical value of $|{{\textit{Re}}_{{G}}}|$ corresponding to the ISW instability threshold (LPs marked by filled symbols) increases strongly as the channel confinement is decreased (moving from circles to triangles). In all cases, mode merging occurs as $|{{\textit{Re}}_{{G}}}|$ is increased.

The thin cyan curve in figure 19(a), which corresponds to the water/air system, tracks the LPs marking the ISW instability threshold (filled symbols). Similar curves, which we designate as neutral bounds of the ISW instability mode, are represented in figure 19(b). Here, we have plotted neutral bounds in terms of the critical relative confinement, $H$ / $h_0$ , versus the gas Reynolds number, ${\textit{Re}}_{{G}}$ , for the seven real liquids listed in table 1 in contact with air. All other parameters are set to the same values as in figure 19(a), in particular, ${\textit{Re}}_{{L}}$ = 43.1. Between the different liquids, both the Kapitza number, ${\textit{Ka}}$ , and the viscosity ratio, $\varPi _\mu$ , change, but it is in terms of $\varPi _\mu$ that the neutral bounds are organised. We observe that, the lower the value of $\varPi _\mu$ (moving from pentagons to crosses in figure 19 b), the stronger the relative confinement needed for the onset of the ISW mode. Conversely, for fixed $H$ / $h_0$ , the lower the value of $\varPi _\mu$ , the larger the critical value of $|{{\textit{Re}}_{{G}}}|$ . This is due to the stabilising effect of viscous dissipation in the liquid, which increases with decreasing $\varPi _\mu$ at fixed $\mu _{{G}}$ . In contrast, the effect of ${\textit{Ka}}$ is non-monotonous, e.g. we observe a delay of the ISW onset when moving from methanol (pentagons, ${\textit{Ka}}$ = 1988) to water (plus signs, ${\textit{Ka}}$ = 3174), whereas the opposite is observed when moving from isopropanol (squares, ${\textit{Ka}}$ = 301.3) to water (plus signs, ${\textit{Ka}}$ = 3174).

We now turn to figure 20, which reports the effect of the liquid Reynolds number, ${\textit{Re}}_{{L}}$ , on the neutral point curves for the water/air (figure 20 a) and silicone oil II/air (figure 20 b) systems. We observe that the critical value of $|{\textit{Re}}_{{G}}|$ for the instability threshold of the ISW mode (LPs marked by filled symbols) is almost insensitive to ${\textit{Re}}_{{L}}$ in the case of water (figure 20 a), whereas it increases significantly with decreasing ${\textit{Re}}_{{L}}$ in the case of silicone oil (figure 20 b). Quantitatively speaking, $\left |{Re_G}\right |$ increases by a factor of 2 when reducing ${\textit{Re}}_{{L}}$ by a factor of 8 in figure 20(b). Thus, the occurrence of upward travelling ripples is increasingly likely the lower the value of ${\textit{Re}}_{{L}}$ , which seems to be in line with the trend of the flooding threshold reported in the experiments of Zapke & Kröger (Reference Zapke and Kröger2000) and Drosos, Paras & Karabelas (Reference Drosos, Paras and Karabelas2006). The difference in sensitivity of the ISW instability threshold with respect to ${\textit{Re}}_{{L}}$ between large- ${\textit{Ka}}$ (figure 20 a) and low- ${\textit{Ka}}$ (figure 20 b) liquids may be understood in terms of the competition between the viscous and capillary stabilisation mechanisms acting within the liquid film. As ${\textit{Ka}}$ is decreased, viscous stabilisation becomes comparable to surface-tension-driven stabilisation and, thus, the role of the film thickness, $h_0$ , which decreases with decreasing ${\textit{Re}}_{{L}}$ and controls viscous dissipation within the liquid film, becomes more important.

Finally, we discuss figure 21, which demonstrates the effect of the inclination angle, $\beta$ , on the threshold of the ISW mode. This figure is complementary to figure 15, where we have discussed the effect of $\beta$ on the nature of the spatio-temporal response of a gas-sheared falling-liquid film. Figure 21(a) represents neutral point curves for different values of $\beta$ , all other parameters being fixed according to figure 11. We observe that the ISW segment of these curves (thick solid lines) disappears as $\beta$ is increased, i.e. merging of the ISW and Kapitza modes (MPs marked by open symbols) occurs before the ISW mode can become unstable (see discussion of figure 15 c). This is the case e.g. for the curve with a diamond in figure 21(a) ( $\beta ={15}^\circ$ ), which corresponds to the conditions in figure 15(c). Nonetheless, even at large $\beta$ , neutral waves ( $\omega _i$ = 0) associated with the merged mode (large $\left |{Re_G}\right |$ on curves with pentagons in figure 21 a, 21 c and 21 d) are characterised by large wave numbers (figure 21 a), as well as negative wave speeds (figure 21 c) and large angular frequencies (figure 21 d). This confirms our earlier observation from figure 15(d) that upward-travelling ripples are possible also for strongly inclined falling liquid films.

Figure 21.

Effect of the inclination angle, $\beta$ , on the onset of the ISW mode. Parameters based on figure 11: water/air, $H^\star ={13}\,\textrm {mm}$ . (a) Neutral point curves for different values of $\beta$ : ${\textit{Ka}}$ = 3174, ${\textit{Re}}_{{L}}$ = 43.1. From circle to pentagon, $\beta$ = $[5,10,15,90]^\circ$ . (b) ISW neutral bound in terms of $\beta$ versus ${\textit{Re}}_{{G}}$ for different values of ${\textit{Re}}_{{L}}$ : ${\textit{Ka}}$ = 3510. Solid curves with filled symbols, linear stability calculations. Filled symbols mark (physical or numerical) end of ISW LP tracking. Downward triangle, ${\textit{Re}}_{{L}}$ = 10; pentagon, ${\textit{Re}}_{{L}}$ = 20.5; upward triangle, ${\textit{Re}}_{{L}}$ = 43.1; diamond, ${\textit{Re}}_{{L}}$ = 46; square, ${\textit{Re}}_{{L}}$ = 103; circle, ${\textit{Re}}_{{L}}$ = 167.5. Open symbols, ISW threshold obtained from our experiments (§ 2). Pentagon, ${\textit{Re}}_{{L}}$ = 20.5; diamonds, ${\textit{Re}}_{{L}}\in [44,48]$ ; squares, ${\textit{Re}}_{{L}}$ = 103; circles, ${\textit{Re}}_{{L}}\in [167,168]$ . (c) Neutral wave speed corresponding to panel (a). (d) Neutral angular frequency corresponding to panel (a).

Figure 21(b) represents the neutral bound of the ISW mode for the water/air system in terms of the inclination angle, $\beta$ , versus the gas Reynolds number, ${\textit{Re}}_{{G}}$ , for different values of the liquid Reynolds number, ${\textit{Re}}_{{L}}$ . Other parameters are based on those in figure 11, only that we have evaluated fluid properties at the temperature of the corresponding experiments. Solid curves represent our linear stability calculations. Their end points (filled symbols) mark the conditions at which the LP of the ISW mode can no longer be tracked, either because the ISW mode disappears (blue square, black circle, magenta upward triangle), or for numerical reasons. Open symbols mark the ISW instability threshold observed in our experiments (see § 2). Agreement between our experiments and our linear stability calculations is very good. All data sets show that an increase in the inclination angle, $\beta$ , greatly increases the counter-current gas flow rate required to produce upward-travelling ripples (increasing $\left |{Re_G}\right |$ ). Further, we see that the ISW threshold is almost insensitive to the value of ${\textit{Re}}_{{L}}$ , which confirms our earlier observation from figure 20(a).

Figure 22.

Comparison of the ISW and AI thresholds for parameters according to figures 19(b) and 21(b). Dashed curves with open symbols, AI threshold ( $\omega _i$ = $V_{\textit{ray}}$ = 0); solid lines with filled symbols, ISW threshold. (a) Effect of the channel height, $H^\star$ , for parameters from figure 19(b). From diamonds to pentagons, ${\textit{Ka}}$ = $[121.4,3174,1988]$ , $\varPi _\mu$ = $[0.004,0.018,0.031]$ . (b) Effect of the inclination angle, $\beta$ , for parameters from figure 21(b). Magenta triangles, ${\textit{Re}}_{{L}}$ = 43.1; blue squares, ${\textit{Re}}_{{L}}$ = 103.

To conclude this section, we compare the threshold of the ISW mode with the AI threshold. We do this based on figure 22, where we have reproduced several ISW neutral bounds (solid lines with filled symbols) from figures 19(b) and 21(b), which demonstrate the effect of the channel height, $H^\star$ , and the inclination angle, $\beta$ , respectively. Dashed curves with open symbols in figure 22 represent the AI threshold, which we have obtained by numerical continuation of the SP ( $\omega _k$ = 0) with $\omega _i$ = $V_{\textit{ray}}$ = 0. In figure 22(a), the AI limit always lies beyond the ISW threshold, meaning that short-wave ripples occur before temporal growth can be observed in the system. In contrast, the solid and dashed curves in figure 22(b) intersect. At large inclination angles, the AI limit lies below the ISW threshold, whereas the opposite holds for small inclination angles. We point out that the dashed curves in figure 22(b) are truncated where the SP at $V_{\textit{ray}}$ = 0 becomes unphysical.