2. Faster Taylor bubbles

We artificially increase the distance between the tube wall and the lateral surface of the bubble by constructing a ‘cage’ with an array of thin vertical rods regularly arranged on a circle coaxial with the tube as shown in figure 1. With a suitable spacing of the rods, surface tension is able to contain the gas within the core of the tube bounded by the cage. The wider annular space thus created facilitates the downward flow of liquid, and the bubble rise velocity is correspondingly increased, as can be seen in figure 2, in which the horizontal dashed line is the dimensionless bubble rise velocity in an empty tube (i.e. without the rod cage) according to the correlation of Viana et al. (Reference Viana, Pardo, Yánez, Trallero and Joseph2003). For all the cases simulated, the rise velocity of the bubble is larger than that in the empty tube, except for cases $K$ and $L$, which are for the somewhat different arrangement shown in figure 9$(b)$ and are addressed later in § 4.

Figure 1. $(a)$ The Taylor bubble rises inside a ‘cage’ of thin vertical rods (blue) arranged on a circle concentric with the tube axis. The images on the right are views of the rising bubble from above, $(b)$, and from below, $(c)$. The gas protrudes between the rods at the top, but as the pressure in the annulus increases downward, the bottom of the bubble becomes circular.

Figure 2. Dimensionless bubble rise velocity, or Froude number (2.1), for some of the cases presented in this paper. The pertinent parameters for each case are given in table 1. The horizontal dashed line is the value predicted by the correlation of Viana et al. (Reference Viana, Pardo, Yánez, Trallero and Joseph2003) for the parameters of case $A$.

Table 1. Parameter values for the cases shown in this paper; for all simulations, $Mo = 2\times 10^{-6}$. For cases $H$ (figure 7$a$) and $I$ (figure 8), the gas eventually gets out of the cage and the rise velocity fails to stabilize. Case $J$ is for a bubble in the annular space outside the cage, as shown in figure 9$(a)$. Cases $K$ and $L$ are for grooved tubes as shown in figure 9$(b)$.

The quantity shown in figure 2 is the Froude number, defined by

(2.1)\begin{equation} Fr = \frac{U_B}{\sqrt{gD}} . \end{equation}

The parameters characterizing each simulation are given in table 1; they are the Galilei and Morton numbers,

(2.2a,b)\begin{equation} Ga = \frac{\sqrt{gD^{3}}}{\nu} , \quad Mo=\frac {g \nu^{4}\rho^{3}} {\sigma^{3}}, \end{equation}

the number $N_r$ of rods, the radius $R_c/(D/2)$ of the cylinder on which the rods are arranged and the ratio $d/D$ of the rod to the tube diameter. In (2.2a,b) $\nu$ and $\rho$ are the liquid kinematic viscosity and density, and $\sigma$ is the surface tension coefficient. The fraction of area blocked by the rods, which may be called the solidity of the cage, is $\alpha _s = N_r d/(2{\rm \pi} R_c)$. Our choice of parameters is strongly influenced by computational considerations and may not reflect cases of greater practical relevance in which $Ga$ would be larger and $d/D$ and, especially, $Mo$, would be significantly smaller. While we have included in the supplementary material available at https://doi.org/10.1017/jfm.2021.432 the results of some preliminary simulations for $Mo=2.5\times 10^{-11}$, a value close to that of water, it appears that such situations would be most efficiently studied in a laboratory rather than numerically.

We have found that the fluid mechanics of our system differs in a non-trivial way from that of the standard Taylor bubble case. In order to illustrate the differences and gain some understanding of the present results it is useful to look at the velocity and pressure fields. In describing some typical results, we focus on case $A$, for which the non-dimensional bubble volume is $V_{gas}/(({{\rm \pi} }/{6})D^{3})=1.66$ and the bubble length is $H/D=2.75$ or $H/2R_c=3.93$. With these values the Froude number of the correlation of Viana et al. (Reference Viana, Pardo, Yánez, Trallero and Joseph2003) is 0.335. Our simulation for an empty tube gave $U_B/\sqrt {gD}=0.328$ with a difference of 2.1 %. The bubble rise velocity with the cage in place is $U_B/\sqrt {gD}=0.56$, nearly 70 % greater than in the empty tube. The shape of the bubble for this case is shown in figure 1$(a)$; horizontal views of the tube cross-section from above and from below the bubble can be seen in panels $(b)$ and $(c)$. The gas penetrates somewhat into the gaps between the rods in the upper part of the bubble (panel $b$), but the extent of these protrusions decreases with depth until the bubble becomes nearly circular toward the bottom (panel $c$).

Figure 3 shows profiles of the flow fields along three horizontal rays passing through the middle of the gap between adjacent rods and extending from the axis at $r=0$ to the inner surface of the tube at $2r/D=1$; the shaded area indicates the radial extent occupied by the rods. The vertical positions at which these results are taken are indicated by the three small black arrows along the tube in figure 4$(c)$. From left to right, the three panels show the normalized pressure, horizontal (radial) liquid velocity $u_r$ and vertical (axial) liquid velocity $u_z$, negative downward. The solid lines are for the uppermost ray at the level of the bubble tip; the dashed lines are for a slightly lower ray, at the level at which the bubble cap has expanded to occupy the available core space inside the cage; and the dash-dotted line is at a level slightly below the bubble bottom. The velocity data for the intermediate level start at the bubble surface, since the rest of the cross-section is occupied by the gas. For the ray at this level the horizontal portion of the pressure line in the core is actually the constant gas pressure inside the bubble.

Figure 3. Normalized pressure $(a)$, horizontal velocity $(b)$ and vertical velocity $(c)$ along horizontal rays from the tube axis, $r=0$, to its inner surface, $r=D/2$. The solid lines are at a level slightly above the bubble tip, the dashed lines at the base of the cap at the bubble top and the dash-dotted lines slightly below the bubble bottom. These levels are indicated by the small black arrows on the left of figure 4$(c)$.

Figure 4. Normalized pressure (solid line in $a$), horizontal velocity (red line in $b$) and vertical velocity (blue line in $b$) along vertical lines corresponding to the midpoint of the gap between adjacent rods and the midpoint of the annulus between the tube wall and the circle of rods. The dashed line in panel $(a)$ is the pressure along the tube axis, which is constant inside the bubble. The small black arrows at the left of panel $(c)$ show the levels at which the radial profiles shown in the previous figures are taken.

The pressure at the two upper levels is essentially constant in the core and greater than in the surrounding annulus, with the pressure gradient pushing the liquid out of the core to allow the bubble to rise. The situation is reversed under the bubble, where liquid must flow from the annulus into the core to claim the space vacated by the bubble. The radial velocity, figure 3$(b)$, is largest (in modulus) at the narrowest point of the gap between rods and is particularly large at the intermediate level where the bubble has first expanded to its largest diameter. The vertical velocity, figure 3$(c)$, is positive above the bubble, although it is significantly lower than the bubble rise velocity. In this region the liquid is pushed laterally into the annulus through the rod cage rather than upward. The vertical velocity is also positive under the bubble, and larger than the rise velocity of the latter. This circumstance indicates the presence of a recirculating wake in the bubble rest frame. At the intermediate level this velocity component is very small in the gap, owing to the no-slip condition at the rod surfaces. In the annulus, it resembles a downward channel-flow behaviour with a small degree of slip at the edge of the rod region.

Corresponding results for pressure and velocity along vertical lines are shown in figures 4$(a)$ and 4$(b)$, respectively. The horizontal thin dashed lines mark significant levels along the bubble, namely the location of the bubble tip, the location where the bubble cap has grown to occupy the tube core and the location of the bubble bottom. The small black arrows along the tube on the left identify the positions at which the radial profiles shown in the previous figure are taken.

The dashed line in figure 4$(a)$ is the pressure along the axis of the tube, which is constant inside the bubble. The other results in panels $(a)$ and $(b)$ are all taken along a vertical line at the middle of the annular region between the rods and the tube wall. Near the top of the bubble the pressure in the annulus is significantly lower than in the core region, as already shown in figure 3. This pressure difference actually markedly and rapidly increases as the cross-section occupied by the bubble reaches its maximum. The core–annulus pressure difference appears localized in a relatively short region above the bubble, while it is smaller and extends downward much farther under the bubble.

The slope of the pressure distribution in the annulus is essentially hydrostatic far ahead of, and far behind, the bubble. However, over much of the bubble length, the pressure gradient is nearly constant and less steep than hydrostatic, indicating the presence of an adverse pressure gradient in the annulus. A clear indication of this effect (which is in marked contrast with the standard Taylor bubble case, for which the pressure in the annulus is essentially constant) is that the gas tends to invade the gaps between the rods more at the top than at the bottom of the bubble, as can be seen in figure 1$(b{,}c)$. This indicates a pressure increase with depth. The origin of this feature must be sought in the opposite liquid flows out of the tube core near the tip of the bubble and back into the core under it. Since the core pressure just above and just below the bubble is nearly the same (see the black dashed line in figure 4$a$), these opposite flows require the pressure in the annular region to increase with depth. That this is the correct explanation is supported by our simulations with a thinner annulus (cases $F$ and $G$). In these cases the bubble rise velocity is just a little bit larger than for an empty tube, so that the horizontal flow out of and into the cage is reduced. As a consequence, the pressure difference necessary to drive this flow is also smaller and the corresponding adverse pressure gradient less pronounced. This observation suggests that, by reducing the pressure drop across it, a cage made of thin rods or wires could significantly reduce the pressure gradient in the annulus, thus improving the ability of surface tension to confine the gas inside the cage. It may well be worth pursuing this possibility experimentally, because a smaller adverse pressure gradient would enable much longer bubbles to remain contained within the cage.

The blue (left) and red lines in figure 4$(b)$ are the vertical and horizontal velocities. The latter vanishes far above the bubble, then rapidly grows, reaching a sharp maximum in the short length in which the bubble cross-section increases. Behind the bubble the behaviour is more gradual, with the radial velocity small and negative, as will be seen more clearly later. The vertical velocity increases significantly in the annulus as the bubble is approached from above, remains fairly large and negative along most of the tube section occupied by the bubble, and then slowly decreases back to zero. The comparison with the nearly vanishing horizontal velocity over much of the tube highlights the quasi-fully-developed channel-like behaviour of the flow through a large portion of the annulus, to which we return below.

A different perspective on the velocity fields is offered by figure 5, which shows, in panels $(a)$ and $(b)$, the radial and axial velocity distributions in vertical cuts of the tube and in two horizontal sectors, one near the top and one just below the bottom of the bubble. The horizontal velocity is quite large in the region above the bubble, becoming nearly jet-like in the gap between the rods and quickly dissipating in the annular region beyond. Under the bubble, the radial velocity is mostly negative as liquid is drawn into the core. The vertical velocity, in panel $(b)$, is negative in the annulus both at the top and under the bubble as liquid is drawn down and into the core, and is large and positive in the core as already seen in figure 3$(c)$. The negative vertical velocity persists for a long distance under the bubble and a shorter one above, giving yet another indication of the difference between the two regions (see also figure 4).

Figure 5. Horizontal $(a)$ and vertical $(b)$ velocity distributions along the tube and on horizontal sectors near the top and bottom of the bubble.

3. Theoretical considerations

The constancy of the pressure gradient in the annular space along the bubble shown in figure 4 suggests that the flow in this region can be approximated as fully-developed, an impression strengthened by the near vanishing of the horizontal velocity and the shape of the vertical velocity distribution.

The tube wall imposes a no-slip condition, while, on average, the condition on the circle bounding the cage may be approximated as a partial slip in the form

(3.1)\begin{equation} u_z(r=R_c) = s \left. \frac{\partial u_z}{\partial r}\right|_{r=R_c} , \end{equation}

with $s$ a suitable slip length and the $z$ axis, coincident with the tube axis, positive upward. The pertinent form of the Navier–Stokes equations in the axial direction is readily solved (see e.g. Goldsmith & Mason Reference Goldsmith and Mason1962), and the following expression for the liquid flow rate is obtained after imposing the appropriate boundary conditions:

(3.2)\begin{align} Q ={-} {\rm \pi}\frac{\rho g +\partial_z p}{ 8\mu} \left[(R_t^{2}-R_c^{2})^{2} -\frac{R_t^{2}-R_c^{2}+2sR_c}{\log (R_t/R_c)+s/R_c} \left[R_t^{2}-R_c^{2}-2R_c^{2}\log (R_t/R_c) \right]\right] , \end{align}

with $R_t=D/2$ the tube radius. By conservation of volume, $Q$ must balance the rate at which the bubble, rising with velocity $U_B$, makes room available under its base, approximately given by ${\rm \pi} R_c^{2} U_B$, so that ${\rm \pi} R_c^{2}U_B = -Q$.

Equation (3.2) is appropriate along the body of the bubble, where there is little if any exchange of liquid between the inside and outside of the cage, so that $Q$ is essentially constant. However, the flow rate through the annulus in the regions above and below the bubble cannot be constant, as liquid must be pushed out of the cage by the rising top and pulled back into the cage by the rising bottom. As already noted, these processes require suitable pressure gradients in the horizontal direction which we have seen earlier in figure 3$(a)$. If we approximate the flow in the gap between the rods as a Poiseuille flow in a two-dimensional channel with an effective width $h_{eff}$, we can estimate the local flow rate per unit height through each gap as

(3.3)\begin{equation} q = \frac{h_{eff}^{3}}{12\mu} \frac{p_{in}(z)-p(z)}{d} , \end{equation}

where $p_{in}(z)$ and $p(z)$ are the pressures inside and outside the cage; the fraction in this equation approximates the pressure drop across the rod cage. This flow affects the flow rate in the annulus, $\textrm {d}Q/\textrm {d}z = N_r q$. By making a quasi-equilibrium approximation, we use (3.2) for $Q$ in this equation, finding

(3.4)\begin{equation} \frac{\partial^{2}p}{\partial z^{2}} - \ell^{{-}2} p ={-}\ell^{{-}2}p_{in}(z) . \end{equation}

This relation identifies the characteristic length scale for the effect of the flow into and out of the cage as

(3.5)\begin{equation} \ell = \sqrt{\frac{3{\rm \pi} d}{2N_rh_{eff}^{3}}\left[R_t^{4}-R_c^{4} -\frac{(R_t^{2}-R_c^{2})^{2}}{\log (R_t/R_c)}\right]} . \end{equation}

The slip length can be estimated to be very small in our case and has been neglected here. By approximating the pressure inside the cage above the bubble as hydrostatic on the basis of figure 4 and other results (not shown), the solution of (3.4) decreasing far upstream is readily found to be

(3.6)\begin{equation} p(z) = p_{in}(z)+[p(z_t)-p_{in}(z_t)] \exp({-(z-z_t)/\ell}) , \end{equation}

with $p_{in}(z)=p_{in}(z_t) -\rho g (z-z_t)$, $z_t$ corresponding to the highest point of the bubble top. The difference $\varDelta p=p-p_{in}$ is indeed exponential for a significant distance above the bubble, as shown in figure 6$(a)$. The characteristic length of the exponential in this case is found to be $\ell /D=0.4$, and it becomes equal to the value in (3.5) upon taking the effective gap width $h_{eff}/h\simeq 2.16$, with $h$ the minimum width of the inter-rod gap. Application of (3.4) to the region under the bubble is more difficult, because the behaviour of the pressure in the core region cannot be approximated as simply as above, as is evident from figures 4 and 6$(b)$.

Figure 6. Logarithm of the normalized core–annulus pressure difference, above $(a)$ and below $(b)$ the bubble, as a function of vertical position. The vertical dashed lines mark the position of the bubble top in $(a)$ and of the bubble bottom in $(b)$. The characteristic length of the exponential in (3.6) has the value $\ell /D= 0.4$. In $(a)$, the portion of the line to the left of the dashed line is the region of the bubble top.

Consideration of the results presented thus far also permits us to draw some general conclusions on the effects of the bubble length and of the annulus thickness. Concerning the former, suppose that the rise velocity and the adverse pressure gradient in the annulus were to remain the same as the bubble length is increased. This would create a bigger inner–outer pressure difference at the top and/or bottom of the bubble, increasing the flow out of and/or into the annulus. This effect would make it impossible for the rise velocity to remain unchanged. Suppose then that the rise velocity were to remain constant with the adverse pressure gradient in the annulus becoming smaller. This would result in a bigger downward flow rate as the effect of gravity becomes less hindered, which again would be incompatible with the assumption of a constant rise velocity. If the rise velocity of a longer bubble were smaller than that of a shorter one, this would require a stronger adverse pressure gradient, causing a bigger pressure difference at the top and/or bottom of the bubble, which, again, would require a faster rise velocity. Thus, unlike the standard Taylor bubble case, as the bubble length increases, so does its rise velocity. This increased velocity must be compensated by a greater downward flow in the annulus, which, in turn, requires a weaker adverse pressure gradient. Thus it may be concluded that longer bubbles rise faster with a weaker adverse pressure gradient in the annulus.

Consider now the effect of the thickness of the annulus. If the adverse pressure gradient were to remain the same as the thickness is increased, the downward liquid flow rate would increase and, with it, so would the bubble rise velocity. But the flow into and out of the cage, which determines the rise velocity, is little affected by the annulus thickness, which would be incompatible with an increase in velocity. Thus, it must be concluded that increasing the annulus thickness increases the adverse pressure gradient and the bubble rise velocity. We will see a confirmation of these expectations in the next section.

4. Some additional results

The acceleration of the rise velocity of a Taylor bubble described in this work depends on too many parameters to permit an exhaustive investigation. Nevertheless it is interesting to look at some other results that give some indication of what might be expected with other parameter values.

Figure 7 shows different computed bubble shapes for several cases. The middle image in panel $(a)$ is the same as that shown in figure 1. The images flanking it differ only in the gas volume and, therefore, the bubble length. In agreement with the considerations at the end of the previous section, the shorter bubble $C$ rises more slowly than the longer bubble $A$. A steady rise velocity for the longest bubble $H$ is actually not reached, because, while it is still growing beyond the velocity of bubble $A$, at some point the gas breaks out of the cage. This confirms the earlier statement (§ 2) about the possibility that the pressure gradient in the annulus could limit the length of bubbles that can be contained for a given surface tension. It can indeed be seen in panel $(a)$ how, at the top of the bubbles, the gas increasingly penetrates the gap between the rods as the bubble length increases.

Figure 7. Some additional examples of computed bubble shapes; the letters labelling each image correspond to those labelling the Froude numbers in figure 2 and to the cases in table 1. The middle bubble in $(a)$ is the same as that shown in figure 1; the two bubbles flanking it differ only in the amount of gas they contain. The left bubble in panel $(b)$ rises in a cage with a larger number of thinner rods and a larger porosity. For the right bubble in $(b)$ the vertical rods of the cage are connected by evenly spaced collars with the same diameter as the vertical rods. The second and third bubbles in panel $(c)$ rise in tubes with diameters 3.3 % and 8.3 % smaller than that of the first bubble.

An example of what might happen when containment fails is shown in figure 8 (case $I$), in which the gas has formed ‘belts’ outside the cage enveloping the rods, and the rise velocity does not stabilize. The same eventually happens for the longest bubble of figure 7$(a)$. The left bubble in figure 7$(b)$ is for case $B$, in which the cage has a larger porosity. In this case the characteristic length $\ell$ defined in (3.5) is smaller than in case $A$, which implies a stronger axial pressure gradient, which pushes on the bubble top, increasing its radius of curvature. The right bubble in figure 7$(b)$ rises in a cage in which evenly spaced horizontal circular collars, or hoops, connect the rods. In this case, the presence of two, rather than one, finite radii of curvature helps surface tension contain the bubble, and, indeed, the figure shows that the gas pushes out of the cage less than in the cases of panel $(a)$. The influence of the annulus thickness is shown in figure 7$(c)$, in which the first bubble is the same as in figure 1 while the other two are for annuli thinner by 11.1 % and 27.8 % (corresponding to tube diameters smaller by 3.3 % and 8.3 %) with the same cage radius. As discussed in § 2, in these cases the pressure difference in the annulus between the top and bottom of the bubble is smaller because of the reduced rise velocity, and the gas protrudes significantly less into the gap between the rods.

Figure 8. Two views of the bubble of case $I$, for which surface tension is unable to contain the gas within the rod cage; in this case the rise velocity does not stabilize.

Figure 9$(a)$ shows another possible implementation of the present idea (case $J$). Here the gas is confined within the annular space between the cage of rods and the tube wall. As can be seen in figure 2, the resulting increase in the bubble rise velocity is quite significant in spite of the relatively small gas volume. Yet another possible implementation is shown in figure 9$(b)$, in which the downward flow of liquid along the bubble occurs in grooves etched along the wall. With the parameters used for these simulations (cases $K$ and $L$ in table 1), the Reynolds number for the groove flow is relatively small, and the liquid flow is too slow to result in an increase of the rise velocity. In these cases the rise velocity is actually somewhat smaller than the value predicted by the correlation of Viana et al. (Reference Viana, Pardo, Yánez, Trallero and Joseph2003) for a round tube having the same (outer) diameter.

Figure 9. Two alternative implementations of the principle described in this paper. $(a)$ An arrangement in which the bubble is outside the cage, occupying the annular space between the cage and the tube wall (case $J$). This configuration is also successful in increasing the rise velocity, as shown in figure 2 and table 1. $(b)$ The case $K$ bubble rising in a tube with grooves etched in the tube wall to help promote the downward flow of liquid.

5. Numerical aspects

The three-dimensional numerical simulations were carried out by using the volume-of-fluid Navier–Stokes solver interIsoFoam embedded in the open-source software OpenFOAM v2012. This solver is designed for two incompressible, isothermal immiscible fluids with the surface tension force between phases modelled as a continuum surface force (CSF; Brackbill, Kothe & Zemach Reference Brackbill, Kothe and Zemach1992). The isoAdvector phase-fraction-based interface-capturing approach (Roenby, Bredmose & Jasak Reference Roenby, Bredmose and Jasak2016) is used to enhance the sharpness of the gas–liquid interface. The flow fields were updated in time with the second-order backward scheme, while the upwind-biased second-order scheme was used for the advection term. The phase fraction field was updated with the Crank–Nicolson method, and the van Leer limiter was used for the advection term.

In most cases, a sector of the tube containing only one rod was simulated with symmetry boundary conditions along the radial surfaces and a no-slip condition on the solid surfaces. The rod was assumed to be fully wetted with a zero solid–liquid contact angle. The simulated tube had a length of $L/D \simeq 12$ or larger with its top and bottom ends closed (no-slip condition). The bubble was released near the bottom and evolved to a steady shape and constant rise velocity which began to be affected by the domain top only when the bubble was in close proximity to it. All the data reported in the paper were taken well before this time. We set both the density and the kinematic viscosity of the gas to be 1000 times smaller than those of the liquid. On the basis of previous experience with related problems we are confident that the gas flow had a negligible effect on the phenomenon simulated. The pressure in the bubble was taken as the reference pressure.

The typical mesh consisted of about 4.8 million cells and was refined in the vicinity of the rod in order to resolve the flow in the gap between the rods. For our base case $A$ we also used a coarser mesh with 1.0 million cells, finding a difference of only 1.1 % in the calculated Froude number. Nevertheless, we used the finer mesh for the data shown in the figures. As mentioned in § 2, we also simulated the rise of the bubble in an empty tube (i.e. without the cage) with comparable cell size and distribution; here we found good agreement with the correlation from the literature, which was built on an extensive data bank.