Multiscale aspects of multiphase flows, exemplified by the flow of bubbles in an inclined channel, according to the classification introduced by E and Enquist. 8
Results from a simulation of several hundred bubbles of different sizes in turbulent up flow in a vertical channel at an early time when only a few of the small bubbles have moved to the wall. In the left frame only the bubbles are shown but in the frame on the right the vortical structures, visualized by the λ − 2 method 26 are shown also. The physical and computational parameters are listed in the text.
The temperature in a turbulent bubbly upflow. In the middle frame the view is parallel to the walls and the side frames show the wall temperature and the bubbles next to each wall. The red fluid is hot and the blue fluid is cold. The physical and computational parameters are listed in the text.
The Nusselt number versus time for the simulation shown in Figure 3 .
The scalar field around a few freely rising bubbles at one time. In the frame on the left the mass transfer is computed on the same grid as the fluid flow and the mass transfer is significantly over-predicted. On the right the mass transfer is computed using the multiscale approach described in Ref. 29 . The physical and computational parameters for the simulations are listed in the text.
The total scalar released by the bubble versus time, for the simulations shown in Figure 5 , computed in the same way as the fluid flow (dashed line) and using the multiscale approach described in the text (solid line).
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