4.3 Three-Dimensional Pebble Flow
225
4.3.2.3 Effects of Recirculation Modes on Pebble Streamlines
In this section, the concept of pebble streamline is utilized. The pebble streamline is
computed differently from the concept of pebble spindles as mentioned earlier. The
difference between the pebble spindle and the pebble streamline mainly includes:
(1). In the pebble spindle concept, it is mainly based on the Lagrangian description
of the pebble movements. A pebble spindle is composed of all the pebbles starts to
move or pass over the same initial positions. (2). In the pebble streamline concept,
it is mainly based on the Eulerian description. As defined just like the streamline
in fluid flow, the time-averaged mean velocity is on the tangential direction on each
point of the streamline. Therefore, the streamline can be computed from the mean
velocity of pebbles at each location, which can be governed by
dx
¯
v x
=
dy
¯
v y
=
dz
¯
v z
(4.40)
where x, y, z are the coordinates, and ¯
v x , ¯
v y , and ¯
v z are the time-averaged mean velocities in the x, y, z directions, respectively.
Fig. 4.39 The averaged velocity fields (a) and the 14 streamlines of pebble obtained from the
averaged velocity fields (b–d) for the “Fa” (b), “Me” (c), and “Sl” (d) recirculation rates, respectively