**2.6 Computational conditions**

Since many grid points are needed to resolve the fluid motions around and inside the bubbles, simulations are limited to low Reynolds numbers with small computational domains. We utilize so-called 'minimal turbulent channel' in the present study. The simulations are performed on the domain of )2/(2 as in Lu & Tryggvason (2008). In the simulation of Lu & Tryggvason (2008), the constant pressure gradient to drive the flow was set so that the friction Reynolds number Re was 127.2. Therefore, the size of the computational domain was 2004.254400 in wall units, and the domain was sufficiently large to sustain turbulence, as was shown in Jimenez & Moin (1991). The resulting channel Reynolds number was 3786 in their simulation of the single-phase turbulent flow. In their flow laden with nearly spherical bubbles, the channel Reynolds number was reduced to less than 2000, which may be too low to examine turbulence statistics of a bubbly flow. In the present study, the channel Reynolds number is set at 3786, and the volume flow rate is kept

Numerical Study on Flow Structures and

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 127

Number of grid points 256 256 128

Density ratio 0.1 Viscosity ratio 1.0 Diameter of bubbles 0.4 32.6

Number of bubbles 12 Eötvös number 0.36 Morton number <sup>10</sup> 2.91 10 Archimedes number <sup>4</sup> 1.27 10 Void fraction 0.04 Buoyancy parameter *Bu* <sup>2</sup> 8.17 10

smallest resolved capillary wave, and is described by 1/2 <sup>3</sup> ( )( ) *T x*

 

> *t t T*

*y*

 

Case B1 B2 B3 B4

Prandtl number (liquid) 2.0 1.0 2.0 2.0 Prandtl number (bubble) 2.0 1.0 20.0 20.0 Ratio of specific heat 1.0 1.0 1.0 10.0 Ratio of thermal conductivity 1.0 1.0 0.1 1.0

Number of grid points 96192192

Diameter of bubbles 0.4 24.4

As mentioned above, we conduct a simulation for neutrally buoyant droplets in order to assess the importance of the buoyancy effects. The density ratio of the dispersed-phase fluid is changed to 1.0 from 0.1 in the bubbly flow. Computational conditions are summarized in

Table 4. Computational conditions for the simulation with lower grid resolution.

Time increment 073.0

 

*x z*

 

2.00

represents the timescale of the

 

*c d* .

1089.7 <sup>3</sup>

79.251.0 64.2

*x*

 

*Tt* 

*t*

*y zx*

2 ( /2)

<sup>3</sup> 4.54 10 0.063

0.38 2.17

*x*

Channel Reynolds number 3786

Domain size

Time increment

Grid resolution

Table 2. Computational conditions for the bubbly flow. *T*

Table 3. Thermal properties for the bubbly flow.

**2.6.2 Droplet flow** 

Grid resolution

constant. Notice that the friction velocity (and the friction Reynolds number) is generally changed by the effects of the bubbles.

Non-slip boundary conditions are imposed in the wall-normal direction for the velocity components. Periodic boundary conditions are imposed in the x and z directions for the velocity, the pressure variance, *p P* , and the temperature variance, . As mentioned above, we assume a constant temperature gradient in the vertical (streamwise) direction. We impose a uniform heat flux from both walls. In the present study, the energy (enthalpy) of the system is kept constant, so that the instantaneous wall heat flux *qW* is given by

$$q\_{\mathcal{W}} = \frac{1}{2} \mathcal{G} \int\_0^{2\mathcal{S}} \overline{\rho \mathcal{C}\_P \mu} \left( y \right) dy. \tag{22}$$

Here, represents the spatial average in the *x* and *z* directions.

#### **2.6.1 Bubbly flow**

The simulations are performed with 256 256 128 rectangular grid cells. We set the fluid density inside the bubbles (density of the dispersed-phase fluid) to be one-tenth of that of the liquid (continuous-phase fluid) *d c* 0.1 , and we set the viscosities to be equal *d c* 1.0 to reduce the computational cost as in Lu & Tryggvason (2008). Air bubbles with a diameter of 1-2mm in water are considered in the present study. Eötvös number, Morton number, and Archimedes number of the bubbles are 0.36, <sup>10</sup> 2.91 10 , and 12700, respectively. These parameters correspond to a 1.64mm air bubble in the fluid whose viscosity is 1.84 times higher than that of the water at room temperature. Twelve bubbles with a diameter of 0.4 are introduced randomly into the turbulent singlephase flow in the channel. Computational conditions are summarized in Table 2.

Although most of the parameters employed here are quite close to those in Lu & Tryggvason (2008), the buoyancy parameter, *Bu*, is considerably higher than their value of 0.018. This indicates that the buoyancy effects are less important in our simulation. In order to assess the importance of the buoyancy effects, we conduct a simulation for neutrally buoyant droplets, where *Bu* is infinite , as will be explained in 2.6.2.

The thermal properties employed in the present simulation are summarized in Table. 3. The Prandtl number for the liquid (continuous-phase fluid) is set at a low value of 2.0 ( Pr 2.0 *<sup>c</sup>* ) to maintain high numerical accuracy. The Prandtl number for the gas (dispersed-phase fluid) is also set at 2.0 ( Pr 2.0 *<sup>d</sup>* ). In this case, the ratio of heat capacities per unit volume, *d Pd c Pc C C* , is 0.1 and the ratio of thermal conductivity, *d c k k* , is 1.0, respectively. Hereafter, we call this run Case B1. For comparison, three cases (Cases B2-B4) with different thermal porperties are simulated. In Case B2, we change the Prandtl number of the continuous-phase fluid to 1.0 to examine Prandtl number dependence of the heat transfer characteristics of the turbulent bubbly flow. In Cases B3 and B4, we change the thermal properties inside the bubbles. In Case B3, the thermal conductivity inside the bubbles is set at 1/10 of that in the surrounding liquid in order to examine the heat insulating effect due to the bubbles. In Case B4, the specific heat of the gas inside the bubble is set to be 10 times larger ( 1.0 *d Pd c Pc C C* ) to clarify the effect of lower heat capacity inside the bubbles.

In order to check the accuracy of the simulation, we have conducted a simulation under the same physical conditions with a lower grid resolution. The parameters for this simulation are summarized in Table 4.


Table 2. Computational conditions for the bubbly flow. *T* represents the timescale of the smallest resolved capillary wave, and is described by 1/2 <sup>3</sup> ( )( ) *T x c d* .


Table 3. Thermal properties for the bubbly flow.


Table 4. Computational conditions for the simulation with lower grid resolution.

## **2.6.2 Droplet flow**

126 Computational Simulations and Applications

constant. Notice that the friction velocity (and the friction Reynolds number) is generally

Non-slip boundary conditions are imposed in the wall-normal direction for the velocity components. Periodic boundary conditions are imposed in the x and z directions for the velocity, the pressure variance, *p P* , and the temperature variance, . As mentioned above, we assume a constant temperature gradient in the vertical (streamwise) direction. We impose a uniform heat flux from both walls. In the present study, the energy (enthalpy) of

the system is kept constant, so that the instantaneous wall heat flux *qW* is given by

 

buoyant droplets, where *Bu* is infinite , as will be explained in 2.6.2.

phase flow in the channel. Computational conditions are summarized in Table 2.

Here, represents the spatial average in the *x* and *z* directions.

2 0 <sup>1</sup> () . <sup>2</sup> *W P q G C u y dy* 

The simulations are performed with 256 256 128 rectangular grid cells. We set the fluid density inside the bubbles (density of the dispersed-phase fluid) to be one-tenth of that of

 *d c* 1.0 to reduce the computational cost as in Lu & Tryggvason (2008). Air bubbles with a diameter of 1-2mm in water are considered in the present study. Eötvös number, Morton number, and Archimedes number of the bubbles are 0.36, <sup>10</sup> 2.91 10 , and 12700, respectively. These parameters correspond to a 1.64mm air bubble in the fluid whose viscosity is 1.84 times higher than that of the water at room temperature.

Although most of the parameters employed here are quite close to those in Lu & Tryggvason (2008), the buoyancy parameter, *Bu*, is considerably higher than their value of 0.018. This indicates that the buoyancy effects are less important in our simulation. In order to assess the importance of the buoyancy effects, we conduct a simulation for neutrally

The thermal properties employed in the present simulation are summarized in Table. 3. The Prandtl number for the liquid (continuous-phase fluid) is set at a low value of 2.0 ( Pr 2.0 *<sup>c</sup>* ) to maintain high numerical accuracy. The Prandtl number for the gas (dispersed-phase fluid) is also set at 2.0 ( Pr 2.0 *<sup>d</sup>* ). In this case, the ratio of heat capacities per unit

Hereafter, we call this run Case B1. For comparison, three cases (Cases B2-B4) with different thermal porperties are simulated. In Case B2, we change the Prandtl number of the continuous-phase fluid to 1.0 to examine Prandtl number dependence of the heat transfer characteristics of the turbulent bubbly flow. In Cases B3 and B4, we change the thermal properties inside the bubbles. In Case B3, the thermal conductivity inside the bubbles is set at 1/10 of that in the surrounding liquid in order to examine the heat insulating effect due to the bubbles. In Case B4, the specific heat of the gas inside the bubble is set to be 10 times

 *d Pd c Pc C C* ) to clarify the effect of lower heat capacity inside the bubbles. In order to check the accuracy of the simulation, we have conducted a simulation under the same physical conditions with a lower grid resolution. The parameters for this simulation

, is 0.1 and the ratio of thermal conductivity, *d c k k* , is 1.0, respectively.

(22)

*d c* 0.1 , and we set the viscosities to be equal

are introduced randomly into the turbulent single-

changed by the effects of the bubbles.

the liquid (continuous-phase fluid)

Twelve bubbles with a diameter of 0.4

**2.6.1 Bubbly flow** 

 

volume,

*d Pd c Pc C C* 

larger ( 1.0 

 

are summarized in Table 4.

As mentioned above, we conduct a simulation for neutrally buoyant droplets in order to assess the importance of the buoyancy effects. The density ratio of the dispersed-phase fluid is changed to 1.0 from 0.1 in the bubbly flow. Computational conditions are summarized in

Numerical Study on Flow Structures and

**3. Results and discussion** 

(or 1400 *t*

the lower grid resolution.

flow are denoted by the superscript '+0'.

the bubbly (or droplet) flow.

**3.1.2 Bubble motions and vortices** 

as in the case of the bubbly flow.

**3.1 Flow structures of turbulent bubbly flow** 

quantities are related with the wall units as

**3.1.1 Wall shear stress and friction Reynolds number** 

In table 8, the relative magnitude of the wall shear stress *W W* <sup>0</sup>

represent those for the bubbly flow with the lower grid resolution.

 

quantities. The longer simulation for 2900 *t*

state about 1000

of 700 *t*

units, *utl* ,,, 

number Re Re 

when and 

Heat Transfer Characteristics of Turbulent Bubbly Upflow in a Vertical Channel 129

case with the QUICK scheme than that with the centered 2nd-order scheme. The Nusselt number for higher Prandtl number of 2Pr*c* is also slightly (about 1%) lower. It can be concluded that the effects of the numerical diffusion due to the QUICK scheme are small.

The turbulent bubbly (or droplet) flow and the temperature field reached a fully developed

reached the fully developed state, the simulation has further been conducted for the period

Since the simulations are performed under the condition of constant volume flow rate, the wall friction and therefore wall units are generally changed by the injection of the bubbles. Hereafter, the normalization of physical quantities is performed by the use of either the wall

depending on the situation. The quantities normalized by the wall units in the single-phase

0 0 0 00 0 Re Re , Re Re *W W*

*W W* 0 

Bubbly flow 1.65 (1.61) 1.28 (1.27) Droplet flow 1.37 1.17 Table 8. Wall shear stress and friction Reynolds number. The values in the parentheses

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

*lu t lu t*

    

 

2

 

 

> 

0

0 are shown for the bubbly and droplet flows. Note that these two

 

> 

*<sup>c</sup>* are fixed. The wall shear stress is increased by the factor of 1.65 (or 1.37) for

Re Re

, in each flow or those, 000 0 *utl* ,,,

*t* after the injection of the bubbles (or the droplets). After the turbulence

) for the bubbly (or droplet) turbulent flow to obtain statistical

has been performed for the bubbly flow with

, in the single-phase flow,

and the friction Reynolds

(23)

Table 5. Two cases with different Prandtl numbers are examined. Since the fluid density is uniform throughout the computation domain, the pressure Poisson equation is directly solved by the use of fast Fourier transform. The time increment and grid spacings are <sup>3</sup> 1056.7 *t* , 34.2 *zx* , 93.135.0 *y* .


Table 5. Computational conditions for the droplet flow.
