**3.4 Future research**

We still have many issues to resolve in order to clarify the heat transfer characteristics of turbulent bubby upflows. Firstly, the size of the computational domain is possibly too small to obtain the correct statistics of the turbulent bubbly flow since we utilized a minimal channel. In addition, statistical errors may be large since the number of bubbles is small and the computational time is relatively short to obtain the turbulence statistics. Larger and longer simulations are required to resolve these problems. Secondly, we set the density ratio at 0.1 and the viscosity ratio at 1.0, which are much higher than the corresponding values in an air-water system. The bubble motions and the generation of vortices due to the bubbles should be examined by changing the values of these two ratios. The continuous-phase Prandtl number of 2.0 employed in the present study is low compared with that of about 7 at room temperature. Simulations at higher Prandtl numbers are desirable. Thirdly, only two flow patterns (the bubbly and the droplet flows) have been simulated in the present study. As was shown in Lu & Tryggvason (2008), the bubble distribution consists of a core where the flow is essentially homogeneous, and a wall layer with a larger number of bubbles sliding along the wall in the case where the buoyancy effect is dominant. This interesting situation should be examined in the future study. The heat transfer characteristics of bubbly drag-reducing flows is also an interesting topic to explore.
