**3.1.2 Bubble motions and vortices**

Fig. 3(a) shows a typical snapshot of the bubbles and the vortical structures visualized by the second invariant of velocity gradient tensor, 0125.0 *Q* . It is clearly seen that the bubbles tend to collect on the near-wall regions of the channel. The bubbles are slightly deformed form the spherical shape. As is shown in Fig. 3(b), the droplets are distributed rather uniformly throughout the channel though some droplets are located close to the walls as in the case of the bubbly flow.

Numerical Study on Flow Structures and

peaks at *y*

located at *y*

Figs. 3).

*u*

10

20

(a)

*+*

friction time for the single-phase flow, 0 2 ( )

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 131

Fig. 5(a) shows the profiles of the streamwise vorticity squared which is normalized by the

flows have high values compared with the single-phase flow, indicating that the generation of the streamwise vortices is enhanced by the injection of the bubbles or droplets. There are

corresponding peaks for the droplet flow are more distantly positioned from the walls, and the peaks in the single-phase flow are further away from the walls. In general, these peaks approach the walls as the friction Reynolds number is increased. We have conducted a

with that of the turbulent bubbly flow, to find that the peaks in the vorticity profile are

walls enhance the generation of quasi-streamwise vortices in the regions very close to the walls. This is verified by the visualization of the vortical structures around the bubbles (see

0.07 and 1.93 near the walls in the profile of the bubbly flow. The

0.19 and 1.81. This indicates that the bubbles (or droplets) rising along the

. The vorticities in the bubbly and droplet

, which is comparable

*y/*

(b)

*u*

10

20

*+*

the two peaks near the walls. As is shown in Fig. 6, these peaks and bumps correspond to

0.3 and 1.7 in addition to

0 12

 Single-phase Continuous-phase Dispersed-phase

*x*

simulation of the single-phase turbulent flow at a higher Re ( 160)

Fig. 6. A bubble and vortices. Light blue represents a wall.

*y/*

Fig. 7. The mean velocity profile for (a) bubbly flow and (b) droplet flow.

In the profile for the bubbly flow, there are small bumps at *y*

0 12

 Single-phase Continuous-phase Dispersed-phase

It is seen in the case of the bubbly flow that the vortices are locally activated by the bubbles in the near-wall regions. In the case of the droplet flow, strong vortices are observed even in the central region of the channel. Notice that the normalized values of *Q* are used for visualization in Figs. 3 and that the vortices are considerably strengthened due to the injection of the bubbles.

Figs. 4(a) and 4(b) show the time evolution of the locations of the bubbles and the droplets, respectively. The bubbles rise along the walls almost all the time. As is shown in Fig. 4(c), the void fraction has sharp peaks near the walls as a result of bubble accumulation toward the walls. Horizontal motions are much more noticeable for the droplets. Some droplets rise along the walls as the bubbles, however. This is also confirmed by the profile of the volume fraction of the droplet in Fig. 4(c).

Fig. 4. The path of (a) bubbles and (b) droplets. (c) Profiles of volume fraction.

Fig. 5. Profiles of streamwise vorticity squared that is normalized by (a) <sup>2</sup> 0 *t* and (b) <sup>2</sup> *t*.

It is seen in the case of the bubbly flow that the vortices are locally activated by the bubbles in the near-wall regions. In the case of the droplet flow, strong vortices are observed even in the central region of the channel. Notice that the normalized values of *Q* are used for visualization in Figs. 3 and that the vortices are considerably strengthened due to the

Figs. 4(a) and 4(b) show the time evolution of the locations of the bubbles and the droplets, respectively. The bubbles rise along the walls almost all the time. As is shown in Fig. 4(c), the void fraction has sharp peaks near the walls as a result of bubble accumulation toward the walls. Horizontal motions are much more noticeable for the droplets. Some droplets rise along the walls as the bubbles, however. This is also confirmed by the profile of the volume

> *y/*

> > Streamwise vorticity

squared

0.00

0.02

0.04

0.06

Fig. 4. The path of (a) bubbles and (b) droplets. (c) Profiles of volume fraction.

 Single-phase Bubbly flow Droplet flow

Fig. 5. Profiles of streamwise vorticity squared that is normalized by (a) <sup>2</sup>

*y/*

*y/*

0 *t*

and (b) <sup>2</sup> *t*

.

012

Volume Fraction

(b)

(c)

0.1

 Single-phase Bubbly flow Droplet flow

0 12

 Bubble Droplet

0 12

injection of the bubbles.

*t*

Streamwise vorticity

squared

0.00

0.02

0.04

0.06

200

400

(a)

600

*+*

fraction of the droplet in Fig. 4(c).

*y/*

(a)

*y/*

012

*t*

800

400

(b)

1200

*+*

0 12

Fig. 5(a) shows the profiles of the streamwise vorticity squared which is normalized by the friction time for the single-phase flow, 0 2 ( ) *x* . The vorticities in the bubbly and droplet flows have high values compared with the single-phase flow, indicating that the generation of the streamwise vortices is enhanced by the injection of the bubbles or droplets. There are peaks at *y* 0.07 and 1.93 near the walls in the profile of the bubbly flow. The corresponding peaks for the droplet flow are more distantly positioned from the walls, and the peaks in the single-phase flow are further away from the walls. In general, these peaks approach the walls as the friction Reynolds number is increased. We have conducted a simulation of the single-phase turbulent flow at a higher Re ( 160) , which is comparable with that of the turbulent bubbly flow, to find that the peaks in the vorticity profile are located at *y* 0.19 and 1.81. This indicates that the bubbles (or droplets) rising along the walls enhance the generation of quasi-streamwise vortices in the regions very close to the walls. This is verified by the visualization of the vortical structures around the bubbles (see Figs. 3).

Fig. 6. A bubble and vortices. Light blue represents a wall.

Fig. 7. The mean velocity profile for (a) bubbly flow and (b) droplet flow.

In the profile for the bubbly flow, there are small bumps at *y* 0.3 and 1.7 in addition to the two peaks near the walls. As is shown in Fig. 6, these peaks and bumps correspond to

Numerical Study on Flow Structures and

and spanwise components near the walls.

**3.1.4 Turbulence intensities** 

in 3.1.5.

Here, 

Shear stress profiles


0

1

(a)

Fig. 9. Budget for shear stress.

**3.1.5 Shear-stress profiles** 

respect to *y*, we obtain the relation

 

*y/*

012

 Reynolds stress Viscous stress Surface tension Buoyancy Sum

*du dy* denotes the viscous shear stress and

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 133

Figs. 8 show the profiles of velocity fluctuations of the liquid (continuous-phase fluid) for the single-phase, bubbly and droplet flows. In the case of the droplet flow, the streamwise component of the turbulence intensities decreases near the walls due to the droplets. The wall-normal and spanwise components, on the other hand, increase due to the presence of the droplets in the near-wall regions. The fluid motions normal to the interfaces of droplets are directly suppressed or induced by the presence of droplets. The droplets, on the other hand, indirectly affect the fluid motions. The turbulence is augmented by the droplets, and the redistribution mechanism of Reynolds stresses is enhanced. This may be one of the reasons for the decrease in the streamwise component and the increase in the wall-normal

The profiles of turbulence intensities in the bubbly flow resemble those in the droplet flow when they all are normalized by the friction velocity of the single-phase flow (figures not shown). When normalized by each value of the friction velocity, all components are considerably low in the turbulent bubbly flow compared with those in the droplet flow as is shown in Figs. 8. This is because the increase in the wall shear stress (and the friction velocity) is brought about by the factors other than turbulence augmentation, as is discussed

The wall-friction drag is increased due to the injection of the bubbles (Note that the volume flow rate is kept constant in the computation). Now, we examine the mechanism for the increase of the drag by considering the balance of forces in the channel. Taking the average of Eq.(2) over time and the *x* and *z* directions and integrating the averaged equation with

0 0 v ( ) ( ) ( ') ' ( )( ') ' 1 . *y y*

*<sup>y</sup> u y y f y dy g y dy*

third, and fourth terms on the right-hand side of Eq.(24) represent the Reynolds shear stress, the viscous shear stress, the surface-tension term, and the buoyancy term, respectively.

Shear stress profiles


0

1

*x W*

*y/*

(b)

 *<sup>W</sup>* 

(24)

 

> 

0 1 2

(0) (2 ) . The first, second,

 Reynolds stress Viscous stress Surface tension Buoyancy Sum

the trailing vortices around the bubbles. Relatively small trailing vortices are seen on the wall side of the bubble (e.g. a vortex pair marked by A), while relatively large vortices are also seen on the side of the channel center (e.g. vortices marked by B).

In Fig. 5(b), the profiles of the streamwise vorticity squared which is normalized by the friction time for each flow, <sup>2</sup> ( ) *x* , are shown. The vortices is relatively weak in the central region of the channel in the bubbly flow.

### **3.1.3 Mean velocity profiles**

Fig. 7(a) shows the mean velocity profiles for the bubbly flow. The black line represents the liquid velocity in the single-phase flow, and the red line and blue circles represent the liquid and gas velocities in the bubbly flow, respectively. Note that the gas velocity is not the rise velocity of the bubble centroids, but just the velocity of the gas inside the bubbles. Note also that the bubble diameter is 0.4 . Since the wall shear stress is increased with the flow rate fixed, the mean velocity normalized by the friction velocity is reduced compared with that for the single-phase flow.

The gas velocity is higher than the liquid velocity near the walls, while it is lower in the regions around *y* 0.25 and 1.75. The bubbles are exposed to high shear near the walls. The balance between this shear stress and the interfacial surface tension leads to the higher gas velocity near the walls and the lower gas velocity on the central side of the channel. In

fact, the streamwise velocity is homogenized by the circulating flow inside the bubble.

Fig. 7(b) shows the mean velocity profiles for the droplet flow. The velocity for the dispersed-phase fluid is remarkably lower than that for the continuous-phase fluid in the regions around *y* 0.25 and 1.75, indicating that the droplets are moving more slowly than the surrounding fluid.

Fig. 8. The rms velocity fluctuation profiles.

#### **3.1.4 Turbulence intensities**

132 Computational Simulations and Applications

the trailing vortices around the bubbles. Relatively small trailing vortices are seen on the wall side of the bubble (e.g. a vortex pair marked by A), while relatively large vortices are

In Fig. 5(b), the profiles of the streamwise vorticity squared which is normalized by the

Fig. 7(a) shows the mean velocity profiles for the bubbly flow. The black line represents the liquid velocity in the single-phase flow, and the red line and blue circles represent the liquid and gas velocities in the bubbly flow, respectively. Note that the gas velocity is not the rise velocity of the bubble centroids, but just the velocity of the gas inside the bubbles. Note also

fixed, the mean velocity normalized by the friction velocity is reduced compared with that

The gas velocity is higher than the liquid velocity near the walls, while it is lower in the

The balance between this shear stress and the interfacial surface tension leads to the higher gas velocity near the walls and the lower gas velocity on the central side of the channel. In fact, the streamwise velocity is homogenized by the circulating flow inside the bubble. Fig. 7(b) shows the mean velocity profiles for the droplet flow. The velocity for the dispersed-phase fluid is remarkably lower than that for the continuous-phase fluid in the

*v+*rms

1

0

 Single-phase Bubbly flow Droplet flow

*y/*

012

, are shown. The vortices is relatively weak in the central

Since the wall shear stress is increased with the flow rate

0.25 and 1.75. The bubbles are exposed to high shear near the walls.

0.25 and 1.75, indicating that the droplets are moving more slowly

*y/*

(b)

012

 Single-phase Bubbly flow Droplet flow

also seen on the side of the channel center (e.g. vortices marked by B).

*x*

friction time for each flow, <sup>2</sup> ( )

**3.1.3 Mean velocity profiles** 

that the bubble diameter is 0.4 .

*y/*

012

*w+*rms

Fig. 8. The rms velocity fluctuation profiles.

(c)

0

1

 Single-phase Bubbly flow Droplet flow

for the single-phase flow.

regions around *y*

regions around *y*

*u'+*rms

than the surrounding fluid.

(a)

0

1

2

3

region of the channel in the bubbly flow.

Figs. 8 show the profiles of velocity fluctuations of the liquid (continuous-phase fluid) for the single-phase, bubbly and droplet flows. In the case of the droplet flow, the streamwise component of the turbulence intensities decreases near the walls due to the droplets. The wall-normal and spanwise components, on the other hand, increase due to the presence of the droplets in the near-wall regions. The fluid motions normal to the interfaces of droplets are directly suppressed or induced by the presence of droplets. The droplets, on the other hand, indirectly affect the fluid motions. The turbulence is augmented by the droplets, and the redistribution mechanism of Reynolds stresses is enhanced. This may be one of the reasons for the decrease in the streamwise component and the increase in the wall-normal and spanwise components near the walls.

The profiles of turbulence intensities in the bubbly flow resemble those in the droplet flow when they all are normalized by the friction velocity of the single-phase flow (figures not shown). When normalized by each value of the friction velocity, all components are considerably low in the turbulent bubbly flow compared with those in the droplet flow as is shown in Figs. 8. This is because the increase in the wall shear stress (and the friction velocity) is brought about by the factors other than turbulence augmentation, as is discussed in 3.1.5.

#### **3.1.5 Shear-stress profiles**

The wall-friction drag is increased due to the injection of the bubbles (Note that the volume flow rate is kept constant in the computation). Now, we examine the mechanism for the increase of the drag by considering the balance of forces in the channel. Taking the average of Eq.(2) over time and the *x* and *z* directions and integrating the averaged equation with respect to *y*, we obtain the relation

$$-\overline{\rho u \mathbf{v}}(y) + \tau(y) + \int\_0^y \overline{f\_{\sigma x}}(y') dy' - \mathbf{g} \Big|\_0^y \overline{(\rho - \{\rho\})}(y') dy' = \tau\_W \left(1 - \frac{y}{\delta}\right). \tag{24}$$

Here, *du dy* denotes the viscous shear stress and *<sup>W</sup>* (0) (2 ) . The first, second, third, and fourth terms on the right-hand side of Eq.(24) represent the Reynolds shear stress, the viscous shear stress, the surface-tension term, and the buoyancy term, respectively.

Fig. 9. Budget for shear stress.

Numerical Study on Flow Structures and

 

**3.2.2 Mean temperature profiles** 

 <sup>0</sup> .

*Nu* 20.6 15.8 27.3

the bubbly flow with the lower grid resolution.

The profiles of the mean temperature variance, *<sup>W</sup>*

exceeds that of the friction Reynolds number (see Eq.(17)).

W

10

20

30

Table 9. The values of

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 135

higher than the corresponding values in the single-phase flow, respectively. In the droplet flow, the Nusselt numbers for Pr 1 *<sup>c</sup>* (Case D2) and Pr 2 *<sup>c</sup>* (Case D1) are respectively 19.8 and 27.1, which are very close to the corresponding ones in the bubbly flow in spite of the difference in the wall shear stress. By comparing Case B1 and Case B3, it is found that the reduction in the Nusselt number due to the insulating effect of the bubbles is very small. By comparing Case B1 and Case B4, we notice that the low heat capacity of the gas inside the

Case B1, B2, B3 B4 D1, D2

Case Single Phase B1 B2 B3 B4 D1 D2 Pr=2 Pr=1

Table 10. Time-averaged Nusselt numbers. The values in the parentheses represent those for

temperature variance is decreased in the whole region of the channel for the droplet flow. In the case of the bubbly flow, the temperature difference is decreased except in the core region of the channel. This increase in the core region indicates that the enhancement of fluid mixing due to the bubbles is rather confined to the near wall regions. The difference between the mean fluid temperature and the wall temperature is smaller in the multiphase flows than in the single-phase flow, which means that the increase of the Nusselt number

> *y/*

Fig. 11. The mean temperature profiles for the single-phase flow, Case B1 and Case D1.

0 12

19.5 (19.6)

27.1 (27.4)

 Single-phase Bubbly flow Droplet flow 29.5

, are drawn in Figs. 11. The

(29.3) 27.1 19.8

<sup>0</sup> 0.76 (0.76) 0.78(0.79) 0.85

bubbles leads to some amount of reduction in the Nusselt number.

(27.6)

The profiles of these four terms are drawn in Fig. 9. They are normalized by the wall shear stress. The sum of the four terms and the straight line of (1 ) *y* are also plotted in the figure. They agree well with each other, which indicates that the overall balance of forces is satisfied. The viscous shear stress is dominant in the near-wall regions as in the case of single-phase flows. The surface-tension term has large values in the regions of high void fractions (see Fig. 4). The bubbles are deformed by the mean shear in the near-wall regions and a restoring force due to the interfacial tension is acting to the liquid fluid. Since this term has large values near the walls, it makes a major contribution to the increase in the friction drag. The buoyancy term has relatively large values in the core region of the channel, which reduces the relative magnitude of the Reynolds shear stress there.

For the droplet flow, the buoyancy term is obviously zero. The surface-tension term has relatively large values near the walls since some droplets are located there. Its magnitude is smaller than that in the bubbly flow, however. Instead, the relative magnitude of the Reynolds shear stress is large in the droplet flow.

Fig. 10. Profiles of Reynolds shear stress normalized by (a) *<sup>W</sup>* <sup>0</sup> and (b) *<sup>W</sup>* .

Figs. 10 show the profiles of the Reynolds shear stress. They are normalized by the wall shear stress, *<sup>W</sup>* <sup>0</sup> , for the single-phase flow in Fig. 10(a). It is clearly seen that the magnitude of Reynolds shear stress is increased in the near-wall regions due to the presence of the bubbles (or droplets). This is because the momentum exchange is enhanced by the vortices generated around the bubbles (or droplets) near the walls. This increase of the Reynolds shear stress near the walls contributes to the increase in the wall friction. When normalized by *<sup>W</sup>* , the amplitude of the Reynolds stress is substantially reduced in the bubbly flow except in the close vicinity of the walls. This indicates that the relative role of the turbulence in the shear stress becomes diminished in the bubbly flow.

#### **3.2 Heat transfer characteristics of turbulent bubbly flow 3.2.1 Friction temperature and Nusselt number**

The friction temperature is also altered by the injection of the bubbles or droplets. The relative magnitude of the friction temperature is shown in Table 9. The time-averaged values of the Nusselt number in the single-phase flow are 15.8 and 20.6 for Pr 1 *<sup>c</sup>* and Pr 2 *<sup>c</sup>* , respectively (see Table 10). In the bubbly flow, the time-averaged Nusselt number for Pr 1 *<sup>c</sup>* (Case B2) and Pr 2 *<sup>c</sup>* (Case B1) are 19.5 and 27.3, which are 1.23 and 1.33 times higher than the corresponding values in the single-phase flow, respectively. In the droplet flow, the Nusselt numbers for Pr 1 *<sup>c</sup>* (Case D2) and Pr 2 *<sup>c</sup>* (Case D1) are respectively 19.8 and 27.1, which are very close to the corresponding ones in the bubbly flow in spite of the difference in the wall shear stress. By comparing Case B1 and Case B3, it is found that the reduction in the Nusselt number due to the insulating effect of the bubbles is very small. By comparing Case B1 and Case B4, we notice that the low heat capacity of the gas inside the bubbles leads to some amount of reduction in the Nusselt number.


Table 9. The values of <sup>0</sup> .

134 Computational Simulations and Applications

The profiles of these four terms are drawn in Fig. 9. They are normalized by the wall shear

figure. They agree well with each other, which indicates that the overall balance of forces is satisfied. The viscous shear stress is dominant in the near-wall regions as in the case of single-phase flows. The surface-tension term has large values in the regions of high void fractions (see Fig. 4). The bubbles are deformed by the mean shear in the near-wall regions and a restoring force due to the interfacial tension is acting to the liquid fluid. Since this term has large values near the walls, it makes a major contribution to the increase in the friction drag. The buoyancy term has relatively large values in the core region of the

For the droplet flow, the buoyancy term is obviously zero. The surface-tension term has relatively large values near the walls since some droplets are located there. Its magnitude is smaller than that in the bubbly flow, however. Instead, the relative magnitude of the

Reynolds shear stress

Figs. 10 show the profiles of the Reynolds shear stress. They are normalized by the wall

of Reynolds shear stress is increased in the near-wall regions due to the presence of the bubbles (or droplets). This is because the momentum exchange is enhanced by the vortices generated around the bubbles (or droplets) near the walls. This increase of the Reynolds shear stress near the walls contributes to the increase in the wall friction. When normalized

 *<sup>W</sup>* , the amplitude of the Reynolds stress is substantially reduced in the bubbly flow except in the close vicinity of the walls. This indicates that the relative role of the turbulence

The friction temperature is also altered by the injection of the bubbles or droplets. The relative magnitude of the friction temperature is shown in Table 9. The time-averaged values of the Nusselt number in the single-phase flow are 15.8 and 20.6 for Pr 1 *<sup>c</sup>* and Pr 2 *<sup>c</sup>* , respectively (see Table 10). In the bubbly flow, the time-averaged Nusselt number for Pr 1 *<sup>c</sup>* (Case B2) and Pr 2 *<sup>c</sup>* (Case B1) are 19.5 and 27.3, which are 1.23 and 1.33 times


, for the single-phase flow in Fig. 10(a). It is clearly seen that the magnitude

0

1

*y/*

(b)

012

*<sup>W</sup>* .

and (b)

channel, which reduces the relative magnitude of the Reynolds shear stress there.

 Single-phase Bubbly flow Droplet flow are also plotted in the

 Single-phase Bubbly flow Droplet flow

stress. The sum of the four terms and the straight line of (1 ) *y*

Reynolds shear stress is large in the droplet flow.

Reynolds shear stress

by 


0

1

(a)

shear stress, *<sup>W</sup>* <sup>0</sup>

*y/*

012

Fig. 10. Profiles of Reynolds shear stress normalized by (a) *<sup>W</sup>* <sup>0</sup>

in the shear stress becomes diminished in the bubbly flow.

**3.2 Heat transfer characteristics of turbulent bubbly flow** 

**3.2.1 Friction temperature and Nusselt number** 


Table 10. Time-averaged Nusselt numbers. The values in the parentheses represent those for the bubbly flow with the lower grid resolution.
