**3. Results and discussion**

128 Computational Simulations and Applications

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

> Prandtl number (continuous) 2.0 1.0 Prandtl number (dispersed) 2.0 1.0 Ratio of specific heat 1.0 1.0 Ratio of thermal conductivity 1.0 1.0

For comparison, a long simulation without bubbles was also conducted at the same channel Reynolds number of 3786 and Prandtl numbers of 2,1Pr*<sup>c</sup>* . The computational conditions are summarized in Table 6. Statistical quantities were obtained by taking averages over the period of about 30,000 in wall units. The time-averaged friction Reynolds number was 127.2

Number of grid points 9619296

Convection term QUICK Time increment <sup>2</sup> 1066.4 *t*

Grid spacing 42.230.0

Number of grid points 4812848

Grid type Staggered grid Convection term 2nd central

Time increment <sup>2</sup> 1066.4 *t*

Grid spacing 12.430.0

In order to estimate the effects of the numerical diffusion caused by the QUICK scheme, a simulation was conducted by using the centered 2nd-order scheme (consistent scheme) for the convection terms in a staggered grid system (Kawamura et al., 1998). It is found that the amplitude of the streamwise component of vorticity is slightly (about 1.5%) lower in the

Grid type Collocated grid

08.2,16.4

 

 

*x z*

*y*

(consistent scheme)

16.4,33.8

*x z*

*y*

Case D1 D2 Density ratio 1.0 1.0 Viscosity ratio 1.0 1.0

<sup>3</sup> 1056.7 *t* , 34.2 *zx* , 93.135.0 *y* .

Table 5. Computational conditions for the droplet flow.

Table 6. Computational conditions for the single-phase flow

Table 7. Computational conditions for the centered 2nd-order scheme.

**2.6.3 Single-phase flow** 

as in Lu and Tryggvason (2008).

The turbulent bubbly (or droplet) flow and the temperature field reached a fully developed state about 1000 *t* after the injection of the bubbles (or the droplets). After the turbulence reached the fully developed state, the simulation has further been conducted for the period of 700 *t* (or 1400 *t* ) for the bubbly (or droplet) turbulent flow to obtain statistical quantities. The longer simulation for 2900 *t* has been performed for the bubbly flow with the lower grid resolution.

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 units, *utl* ,,, , in each flow or those, 000 0 *utl* ,,, , in the single-phase flow, depending on the situation. The quantities normalized by the wall units in the single-phase flow are denoted by the superscript '+0'.
