Inclined angle dependency on texture formation

We start by showing qualitative features of the texture-formation in a pint glass. Figure 3a shows a snapshot of the texture as the number-density distribution of bubbles in a pint glass 10 seconds after pouring Guinness. Visually, the texture can be easily observed at 30 mm < x < 70 mm, where the wall is inclined due to the tapered shape of the glass with inclined angle of ≈ 15 degrees. Figure 3b shows the temporal change in the phase separation: the black region (liquid of beer), the grey region (bubbly flow), and the white region (head of foam). The bubbly flow region gradually decreases with increasing time and the amount of foam head increases. Finally, the glass of Guinness is completely separated into two contrasting phases (namely, the liquid and the head of foam), corresponding to the phase separation owing to the buoyant rise of each bubble. The texture appearing in a glass propagates downwards with a travelling velocity of ~−35 mm/s (Fig. 3c, Supplementary Movie S1), which is consistent with previous experimental observation5. Because of the differences in the travelling velocity of each wave and the three dimensionality of the flow, the neighbouring waves collide and coalesce, and thus the diagonal stripes in the travelling-wave exhibit the bifurcations and crossovers in the texture.

Figure 3 (a) A snapshot of the bubble-concentration texture of Guinness beer in the pint glass. The corresponding movie is presented in the Supplementary Movie S1. (b), Temporally expanded image of the bubble-concentration distribution at the white dotted vertical line in Fig. 3a (near the centreline of the pint glass). (c) Enlargement of (b) showing travelling waves in the downward direction. The slope of the triangle shown at the upper right corresponds to reference value of −35 mm/s for the maximum downward velocity of the wave5. Full size image

To test the effect of inclination angle on the texture-formation, we repeated the observation of the texture-formation but after adjusting the inclination angle β of the rectangular container. The texture-formation result is shown in Fig. 4a. We found that the bubbles were uniformly distributed at β = 0, 45 and 65 degrees. For 5 degrees < β < 20 degrees, however, the texture appeared and exhibited spatial development: from organised waves to unstructured motions, similar to that of the Tollmien-Schlichting wave or the roll-wave17,18. This finding suggests that the appearance of the texture is triggered by the inclination of the wall and depends on the length of the system. In spatially developing flows, disturbances grow with distance from a flow entrance and can be clearly observed as wave at a sufficient distance to downstream. Although the texture was not clearly found at β > 45 degrees in Fig. 4a, it might possibly appear if the container were taller. We restrict ourselves to the study of the texture-formation in a glass with a height of ~150 mm (the typical size of pint glasses) and focus on the effect of inclination angle. To address the moving direction of bubbles yielding inconsistent results between previous investigations, i.e., whether bubbles move upwards4 or downwards7, we examined the velocity of bubbles near the inclined wall at y ≈ 0.5 mm and x ≈ 80 mm. The corresponding video can be viewed in Supplementary Movie S2. From the superimposed movie of bubble-diameter/velocity, we conclude that this approach provides an excellent measure for both the diameter and the velocity of bubbles. The instantaneous velocities are shown in Fig. 4b, and the time series of the brightness in image B involved with the local-volume-concentration of bubbles and the time series of bubble velocities are shown in Fig. 4c, respectively. The bubbles move downwards with both the vertical and horizontal fluctuations. The cross-correlation between the brightness B and the vertical component of the bubble velocity v x is ≈0.8, exhibiting sufficiently high value. Such a high correlation indicates that bubbles are likely to move downwards more quickly in lower local bubble-volume-concentration regions, and vice versa. These findings suggest that the fluid blobs containing few bubbles descend in the bubble-rich bulk; it is likely that a water film (heavy fluid) falls in air circumstances (light fluid).

Figure 4 Texture-formation and motion of bubbles. (a) Snapshots of the bubble-concentration wave in the rectangular container for various inclination angles β. (b) Microscope image of bubbles in the trapezium container (left), and a superimposed display of bubbles extracted from the image by the template matching method28, showing the bubble-velocity vector obtained via the particle tracking method29 (right). Time-sequential images were captured by a high-speed video-camera mounted on a microscope. The corresponding movie is included as Supplementary Movie S2. (c) Temporal variation in the brightness of image B (top), the vertical v x (middle) and horizontal v z (bottom) bubble velocities. The shaded regions correspond to envelopes given by the local standard deviation of the bubble velocity in the measurement area. Full size image

Here, we consider a particle suspension comprising tap water and hollow glass spherical particles, rather than Guinness containing bubbles, to determine the effect of the volume-concentration of dispersed bodies α. The results for the texture-formation in particle suspension with various values of α are compared with those in Guinness in Fig. 5a. In the particle suspension, as for the novel findings, we can observe the texture, which is comparable with that in Guinness, at all particle concentrations α. The side view of the trapezium container answers the crucial question of where the texture appears in a container. Although bubbles rise in the bulk because of the density difference between the liquid and gas phases, bubbles move downwards in the inclined wall vicinity due to the Boycott effect8 (see accompanying Supplementary Movie S3). As also shown in Fig. 5b, for almost the entire region of the trapezium container including the vertical wall vicinity, the texture cannot be observed in the container, except for the downward flow confined to the inclined wall vicinity. We re-plot this finding in temporally expanded images in Fig. 5c for various distances from the inclined wall: y = 0.1 mm (top), y = 0.5 mm (middle), and y = 1.0 mm (bottom). For both y = 0.5 mm and y = 1.0 mm, we observe a diagonal pattern with a fluctuation in brightness corresponding to the waves travelling downward. For y = 0.1 mm, on the other hand, the image includes a darker region corresponding to the so-called clear-fluid (bubble-free) region7.

Figure 5 Textures in Guinness and a particle suspension. (a) Snapshots of bubble-concentration waves and particle-concentration waves in the trapezium container for various bulk particle concentrations α. (b) Side view of trapezium container exhibiting uniform bubble distribution, except in the inclined wall vicinity. The corresponding movie is included as Supplementary Movie S3. (c) Temporally expanded image of bubble-concentration distribution for various distances from the inclined wall y. (d) Temporal variations of timeline image visualisation of the liquid phase at α = 0.5% (upper) and α = 5% (lower). The lines of molecular tag are formed by an intermittently irradiated laser beam where photo-breaching reactions result in dark (i.e., a non-fluorescent) regions. Full size image

Movement of liquid phase flow

We attempted to determine the exact liquid-phase velocity profile in this thin clear-fluid film. Timeline-image visualisation applying photo-bleaching molecular tagging method was implemented to characterise the liquid-phase velocity at the wall vicinity, as shown in Fig. 5d. The displacements of timelines, which appear as darker regions (molecular tag), exhibit the following behaviours: unsteady descending flow, a maximum descending velocity at a certain location, zero velocity relative to the boundary, and ascending flow in the interior of the container. Furthermore, the number-density of particles significantly decreases below the location at which the magnitude of the liquid velocity is maximised. This formation of the stratified layer causes a gravity current accompanying the global convection (the Boycott effect), as mentioned above8.

The details in the velocity profile of the liquid phase are also obtained via PIV. We apply PIV for Guinness (top) and a particle suspension with α ≈ 5% (bottom). From the instantaneous wall-normal variations of the velocity component along the wall, as shown in Fig. 6b, the descending and ascending motion of the liquid phase exhibit identical feature with that obtained from the photo-bleaching visualisation. To address the velocity fluctuations along the inclined wall, we plot a velocity profile at x = 100 mm in space-time diagram (Fig. 6c). The magnitude of the descending velocity decreases with increasing time because of the reduced amount of the buoyant bubbles, the velocity fluctuations continue to form. Although we found that there are differences in the magnitude and period in the velocity fluctuations between the Guinness and the particle suspension, we judge that these parameters are consistent with the spatio-temporal characteristics of the waves.

Figure 6 Velocity profiles of liquid phase in Guinness and in a particle suspension. (a) A typical snapshot of bubbles or tracer particles. Resin particles with 15 μm in diameter containing laser induced fluorescent (LIF) dye were used as tracer particles. (b) An instantaneous velocity vector field. The colour indicates the velocity along the wall. We treated bubbles in Guinness as behaving like tracer particle because of the small relative velocity between a bubble and the surrounding liquid phase as estimated by Eq. C 1. (c) Space-time diagrams for descending fluctuations. Velocities along the wall are plotted on the same scale as in (b). Full size image

Critical condition for texture formation

To obtain an understanding of the texture-formation in the gravity current, we modelled a stratified flow of two liquids: a clear (heavy) liquid film and a bubble-rich bulk (light) liquid, as shown in Fig. 7a. We consider a thin clear-fluid film that flows down on an infinite flat plate of inclination angle β with respect to the gravity due to the density difference between the clear and bubble-rich fluid. In a stable flow, the gravity or viscous force tends to keep the interface flat (left panel of Fig. 7a). The bubble-texture forms as an interfacial wave (right panel of Fig. 7a) between the two fluids when the fluid motion becomes unstable. The texture-formation is reflected in the velocity fluctuation. We determined the maximum values of the time-averaged velocity along the wall \({|{\bar{u}}_{x}|}_{{\rm{\max }}}\), the maximum root-mean-square values of the velocity-fluctuation component along the wall \({u}_{x,\,{\rm{\max }}}^{^{\prime} }\), and the loci h relative to the wall at which \({|{\bar{u}}_{x}|}_{{\rm{\max }}}\) was achieved via particle image velocimetry (Fig. 7b). Note that it is nontrivial to define the film thickness because of no sharp interface between the clear and bubble-rich fluids due to the discrete bubble distribution. In the present study, the film thickness h is evaluated from the velocity profile instead of the bubble distribution. Following ref.19, in which the particle sedimentation was theoretically studied, the definition of h is the distance from the wall, at which the falling liquid velocity takes the maximum and thus the free-slip condition is likely to be satisfied similar to the conventional roll-wave model18 with a free-slip interface. It should be noticed that from preliminary measurements of time-averaged profiles of the velocity and the particle concentration α, the velocity-based film thickness h has been confirmed to be comparable to the distance from the wall, at which α takes the half of the particle concentration in the bubble-rich fluid, and thus regarded as a natural choice to characterise the flow. Consequently, from these characteristic quantities, we obtain the Reynolds number and the Froude number expressed respectively as

$$Re=\frac{(1-\alpha )\rho \,h\,{|{\bar{u}}_{x}|}_{max}}{(1+\frac{5}{2}\alpha )\mu },$$ (1)

$$Fr=\frac{2{|{\bar{u}}_{x}|}_{{\rm{\max }}}}{3\sqrt{\frac{{\rho }_{0}-{\rho }_{1}}{{\rho }_{0}}\alpha \,h\,g\,\sin \,\beta }},$$ (2)

where g is the gravity acceleration. Re is the ratio of the inertia force to the viscous force and is an essential indicator for the transition from laminar to turbulent motion of flows due to the shear instability20,21. The fluid motion is stable (laminar) at very low Re, whereas it becomes turbulent beyond the critical value Re c , e.g., Re c ≈ 1200 for the boundary layer on a flat plate (Blasius profile)22. Fr is the ratio of the inertia force to the gravity force and is an indicator for the onset of the roll-wave in a liquid-film18. The fluid motion of a falling liquid-film in an inclined open channel is described by the magnitude of Fr. At very low Fr, the fluid motion finds gravity-driven Poiseuille (laminar, non-wavy) flow. At Fr ≥ Fr c , the flow becomes unstable, and thus waves appear at the free surface owing to the instability of the gravity current. Figure 7c shows the scaled velocity fluctuation \({u^{\prime} }_{x,{\rm{\max }}}/{|{\bar{u}}_{x}|}_{{\rm{\max }}}\) as a function of Re and Fr. Note that we performed experiments systematically parametrizing α and β on the basis of α = 5% and β = 10 degrees, but a few semi-systematic parameters were chosen for an extensive investigation. We resolved the velocity profile 10 seconds after pouring to eliminate the flow disturbance induced by pouring a test fluid. An inclined rectangular container was used to minimise the effect of the container shape on the flow. We plot several instances of a value averaged over 10 seconds. Note that both Re and Fr decrease with increasing time in all observations because of the decrease of the buoyant force (convection velocity) due to the accumulation of the dispersed bodies at the free surface (see Fig. 6c). Figure 7c shows that \({u^{\prime} }_{x,{\rm{\max }}}/{|{\bar{u}}_{x}|}_{{\rm{\max }}}\) gradually decreases with increasing Re, and weakly depends on Fr up to Fr ≈ 1, but for Fr > 1, \({u}_{x,\,{\rm{\max }}}^{^{\prime} }/{|{\bar{u}}_{x}|}_{{\rm{\max }}}\) rapidly increases, i.e., the magnitude of the velocity fluctuation increases owing to the texture-formation, implying that the texture-formation is caused by the roll-wave instability rather than the shear instability.