76
2 Experiments in Pebble Flows
2.5.3.5 Correlation of Velocity in Time
In the above section, the velocity fluctuation with time has been observed. It indicates
the autocorrelation between the fluctuations of vertical velocities on different heights.
The autocorrelation functions (Eq. 2.28) are shown in Fig. 2.19, which demonstrate
the following:
• The velocities become uncorrelated and asymptotically approach zero until after
about 15s at all heights. After such a period, the relative displacement in the vertical
coordinate is about 0.15d–0.25d. Thus, the particles maintain some fluctuations,
and these fluctuations seem not to be zero even as long as 100s after the initial
decaying. Furthermore, these fluctuations are apparently different from the rapid
dilute granular flows with the velocity autocorrelation functions decaying monotonically and exponentially down to constant zero finally [29–31]. As a result,
these fluctuations may indicate some coordinated interactions in the dense phase
movement with ordered microstructures.
• The relaxation time decreases continuously. With a lower height, the decaying rate
goes faster (see the inset). It needs less time to reach the same level of correlation
for the particles in the low regions, or inversely, it becomes more correlated for
the particles in the higher zones. It is possibly caused by the speeding up of bulk
motion events in lower regions with higher average velocities and more significant
strength bulk motion events (as shown in Fig. 2.17).
Furthermore, an autocorrelation time τ c can be defined as the time for C(t) to fall
to (1/e)C hx (t = 0). It can be regarded as the characteristic period of autocorrelation.
Then, the particles remain less correlated with their previous velocities after some
time τ c . The rectangular zones centered on the streamlines (No.1, 2, 3, 4) are investigated to study the variation of autocorrelation time characteristics on the streamlines.
The following are the details (Fig. 2.20):
Fig. 2.19 Normalized
autocorrelation of vertical
velocity of particles in the
rectangular zones. The gray
dashed line is C =
1
e , where
e is the Napierian base. The
inset is the autocorrelation
curves with the time from 0
to 20s for clarity
2 Experiments in Pebble Flows
2.5.3.5 Correlation of Velocity in Time
In the above section, the velocity fluctuation with time has been observed. It indicates
the autocorrelation between the fluctuations of vertical velocities on different heights.
The autocorrelation functions (Eq. 2.28) are shown in Fig. 2.19, which demonstrate
the following:
• The velocities become uncorrelated and asymptotically approach zero until after
about 15s at all heights. After such a period, the relative displacement in the vertical
coordinate is about 0.15d–0.25d. Thus, the particles maintain some fluctuations,
and these fluctuations seem not to be zero even as long as 100s after the initial
decaying. Furthermore, these fluctuations are apparently different from the rapid
dilute granular flows with the velocity autocorrelation functions decaying monotonically and exponentially down to constant zero finally [29–31]. As a result,
these fluctuations may indicate some coordinated interactions in the dense phase
movement with ordered microstructures.
• The relaxation time decreases continuously. With a lower height, the decaying rate
goes faster (see the inset). It needs less time to reach the same level of correlation
for the particles in the low regions, or inversely, it becomes more correlated for
the particles in the higher zones. It is possibly caused by the speeding up of bulk
motion events in lower regions with higher average velocities and more significant
strength bulk motion events (as shown in Fig. 2.17).
Furthermore, an autocorrelation time τ c can be defined as the time for C(t) to fall
to (1/e)C hx (t = 0). It can be regarded as the characteristic period of autocorrelation.
Then, the particles remain less correlated with their previous velocities after some
time τ c . The rectangular zones centered on the streamlines (No.1, 2, 3, 4) are investigated to study the variation of autocorrelation time characteristics on the streamlines.
The following are the details (Fig. 2.20):
Fig. 2.19 Normalized
autocorrelation of vertical
velocity of particles in the
rectangular zones. The gray
dashed line is C =
1
e , where
e is the Napierian base. The
inset is the autocorrelation
curves with the time from 0
to 20s for clarity