CARACTERISTICA PARA OBRA MECANICA PARA LA CONSTRUCCION DEL SISTEMA DE TUBERIAS DE PE

Increase in fluid viscosity by approximately by a factor of 2 increases the Morton number by 10 (fourth power). Increase in fluid viscosity will reduce the bubble deformation through less fluid motion. In this section, we present results for M o = 0.01. Because the qualitative effects of confinement are similar to those at lower M o, we present these results and discuss them only briefly. The plots shown are similar to figures 5.4to5.7. Figure 5.10 shows the three-dimensional perspective of the bubble deformation for Bo = 1, 10, 50 and 100 and M o = 0.01 at various times. The bubble dynamics at this Morton number is similar to observations made for M o = 0.001. However, the deformation is smaller at higher Morton number. This is more clear from results at Bo = 100. For low Morton number (Fig. 5.4) there were several tiny satellite bubbles forming due to insufficient local resolution and low viscosity. No such pattern can be seen in Fig. 5.10. The rest of the features are similar at both Morton numbers and all Bond numbers.

Time Bo 0 2.0 2.5 3.5 5.0 1 10 50 100

Figure 5.10Transient bubble shapes for confinement ratio of 4 and M o = 0.01.

Figure 5.11 shows the two-dimensional transient shapes and positions of the bubbles at different times on the center plane for all three confinement ratios, all four Bond numbers and M o = 0.01. It can be seen that the bubbles again rise in a straight line for all Bond numbers and confinement ratios. The bubble goes through a range of deformations such as spherical, ellipsoidal, spherical cap and disk-like structures. The shapes at M o = 0.01 are similar but, as expected quantitative differences are seen.

For Bo = 1, it can be observed from Fig. 5.11a that bubble remains spherical. In 10 time units the bubble rose by 1.31, 1.96 and 2.31 diameter units respectively for confinement ratios of 2, 3 and 4. By comparing results between figures 5.5a and 5.11a, we can say that the as the Morton

For Bo = 10 (Fig. 5.11b), the terminal shapes of the bubble varied from asymmetric for C_r = 2 to symmetric ellipsoidal bubble for Cr = 4. For the case of Cr = 2 it can be seen that in the

steady state the bubble is asymmetric with respect to it’s major axis. The bottom surface is flatter than the top surface. It can also be seen the distance between top surface and major axis is more than the distance between bottom surface and the major axis. Discrepancies in the shapes of top and bottom surfaces of the ellipsoid decrease with confinement ratio and for the Cr = 4 case the

symmetry along the major axis is almost restored. Similar to observations made in Fig. 5.5b, the distance traveled by bubble increases with increasing confinement ratio. The bubble rose by 4.26, 5.55 and 6.00 diameter in 10 time units respectively in ducts with confinement ratios of 2, 3 and 4.

(a) (b)

(c) (d)

Figure 5.11Bubble shapes for confinement ratios of 2, 3 and 4 and M o = 0.01: a) Bo = 1, b) Bo = 10, c)

Bo = 50 and d) Bo = 100.

At larger Bond numbers the bubble deforms to a spherical cap at intermediate times and relaxes to a disk-like structure in the steady state. The trends are similar for both Bo = 50 and Bo = 100 at all confinement ratios. In the steady state a significant difference in the thickness of the middle portion of the bubble can be observed between Cr = 2 and Cr = 4. Terminal shape of the bubble

for Cr = 4 is flatter and wider than that for Cr = 2. However no such distinctions can be made

between Cr = 3 and Cr = 4 at this stage. For Bo = 50 (Fig. 5.11c), in 10 time units the bubble

Bo = 100 (Fig. 5.11d), in the steady state the bubble shape is hemispherical for confinement ratio of 2. But for confinement ratio of 3 and 4 the steady shapes are flattened moons. In this case in 10 time units the bubble rises by 5.04, 5.90 and 6.26 diameter units respectively for confinement ratios of 2, 3 and 4.

Figure5.12 shows the transient rise velocities for the three confinement ratios, four Bond numbers and M o = 0.01. In the case of Bo = 1 there are small oscillations in the rise velocities at Cr = 3

and Cr = 4. These oscillation can be seen in Fig. 5.12a. For Cr = 3, these oscillations are less

than 2% of the terminal velocity hence can be neglected. In the case of Cr = 4 the oscillations are

systematic and periodic in nature. The period and amplitude of oscillation are ∆t = 1.78 and 1.75% respectively. For Bo = 1 the terminal velocities of the bubbles are 0.13, 0.20 and 0.23 respectively for confinement ratios of 2, 3 and 4. The terminal velocities of the bubbles for Bo = 10 are 0.44, 0.58 and 0.63 respectively and these are shown in Fig. 5.12b. The oscillation in rise velocity is observed for Bo = 10 as well, however, the oscillations are significant only for Cr = 2.

(a) (b)

(c) (d)

Figure 5.12 Rise velocity for confinement ratios of 2, 3 and 4, and M o = 0.01: a) Bo = 1, b) Bo = 10, c)

Bo = 50 and d) Bo = 100.

Rise velocities for Bo = 50 and Bo = 100 are shown in Fig. 5.12c and Fig. 5.12d. It can be seen that oscillations in the rise velocities at higher Bond numbers are far more significant than that at lower Bond numbers. These oscillations are related to the oscillations of the bottom surface of the bubble during it’s ascent. As reported in the section 5.3.1the deformation and oscillation of the bubble is considerably large at higher Bond numbers. From rise velocities of Bo = 50 and Bo = 100

we can deduce that bottom surface goes through two oscillations before it reaches steady state. In the steady state the rise velocities are 0.506, 0.603 and 0.644 respectively for Bo = 50. In the case of Bo = 100 the terminal rise velocities are 0.512, 0.607 and 0.654 respectively for confinement ratios of 2, 3 and 4. It can be deduced from these results that the rise velocities are comparable for both Bo = 50 and Bo = 100.

(a) (b)

(c) (d)

Figure 5.13 Aspect ratio for confinement ratios of 2, 3 and 4, and M o = 0.01: a) Bo = 1, b) Bo = 10, c)

Figure5.13shows the transient aspect ratio of the rising bubble for all confinement ratios, all Bond numbers and M o = 0.01. Similar to low Morton number study presented in section5.3.1the aspect ratio increases with increasing Bond number and confinement ratio. For Bo = 1 the aspect ratios are close to 1. For Bo = 10 (Fig. 5.13b) the aspect ratios in the steady state are 1.28, 1.51 and 1.57 respectively. Both for Bo = 50 and 100, the bottom surface of the bubble goes through oscillations reflecting in aspect ratio oscillations. The aspect ratio for Bo = 50 reaches a steady state at t = 6.5 with aspect ratios of 2.36, 3.12 and 3.37. In the case of low Morton number (Fig. 5.7c) only a quasi-steady state was achieved. This clearly indicates the higher viscous damping at the higher Morton number. For Bo = 100 (Fig. 5.7d) the aspect ratio reaches quasi steady state at t = 6.50 with the steady state aspect ratios being 2.65, 3.75 and 4.12 respectively. It is also interesting to mention that the highest aspect ratio for M o = 0.001 was approximately 27 whereas highest aspect ratio for M o = 0.01 is approximately 7.6.

Table 5.2Terminal Reynolds number

Bond Number

(Bo)

Confinement Ratio (CR)

2 3 4

Morton Number (M o) Morton Number (M o) Morton Number (M o)

0.001 0.01 0.001 0.01 0.001 0.01

1 1.18 0.42 1.81 0.63 2.11 0.75

10 19.34 7.89 21.44 10.42 23.10 11.35

50 55.65 30.27 64.42 35.97 69.87 38.91

100 93.83 51.69 109.12 61.25 116.29 66.04

Table 5.2 summarizes the terminal Reynolds number for all the cases. It can be seen that the terminal Reynolds number increases with Bond number and confinement ratio but decreases with

Morton number. The increase in terminal Reynolds number with Bond number is attributed to higher bubble velocity due to larger buoyancy force for larger bubbles. The viscosity is higher for higher Morton number cases hence the drag force on the bubble will be larger and subsequently the terminal Reynolds number will be smaller. Similarly effects of wall are higher for low confine- ment ratio cases therefore with increase in confinement ratio terminal rise velocity decreases. It is also worthwhile mentioning that the increase in terminal Reynolds number is not linear with the confinement ratio.