202
4 Numerical Methods and Simulation for Pebble Flows
events and indicates a smaller value of C. Relative fluctuating kinetic energy (RFKE)
represents the magnitude of kinetic energy fluctuation over a period. Different from
C describing the uniform characteristics in time-domain, RFKE measures the ratio
between the total fluctuating kinetic energy (TFKE) to the total kinetic energy (TKE)
and indicates the fluctuation of the pebble flow in amplitude-domain. The consistent
pebble flow always shows smaller fluctuation around the mean velocity. For instance,
the pebble flow for case F5 and F6 present smaller fluctuation with the RFKE nearly
10%.
Besides, these two parameters are dimensionless and have the fixed range from 0
to 1. When the number of particles and time are enough to obtain steady statistics,
C and RFKE will not be affected by the particle number and recording time.
Based on the above analysis, a new criterion is suggested for categorizing the flow
regime of dense granular flow according to the values of C and RFKE. The specific
categorization principle for dense granular flow is suggested as follows:
• C ≥ 0.5: Intermittent pebble flow
• C <0.5, RFKE > 0.1: Transition Fluctuation pebble flow
• C <0.5, RFKE ≤ 0.1: Continuous pebble flow
Therefore, dense pebble flows are categorized into three sub-regimes characterized
mainly by their primary features. Obviously, intermittent pebble flow describes the
flow that keeps the steady quiescent state most of the time, and the burst motion events
with a larger magnitude of velocity happen at long, evident intervals. Continuous
flow is the fastest flow regime of dense pebble flows. It can flow steadily with little
fluctuation and smoothly like common liquid. The transition fluctuation pebble flow
covers the transition regime from the slow to rapid flows.
It seems that with only six cases it is hard to find the limit of flows. More flow
rates are analyzed, and the values of parameters C and RFKE are listed in Table 4.10.
The flow rate of 0.5 particles per second belongs to the intermittent flow. The flow
rates smaller than 0.5 particles per second are also analyzed. The intermittency index
C and RFKE are found to be larger with the reduction of the flow rates. However,
they cannot still reach the value of 1. In addition, the flow rates can reach the highest
value when the pebbles are discharged freely from the silo bed only under the effect
of gravity and without other manual controls. Herein, the higher flow rate of 1250
particles per second, which is almost equal to the free flow rate, is simulated to check
the parameters C and RFKE. Similarly, the two parameters become smaller with the
increase of the flow rates but cannot reach 0. To conclude, the two parameters can
describe the flow characteristics of all flow rates.
In this section, the effects of recirculation flow rates on the drained pebble flow
in the silo bed are mainly focused, and the effects can be entirely described by two
parameters C and RFKE. Thus, the suggested criterion for categorizing the flow
regimes of dense granular flow based on the values of C and RFKE is proposed to
describe the effects of the flow rates. Herein, the specific categorization principles
for dense granular flow are suggested for a better understanding of the effects of flow
rates.
4 Numerical Methods and Simulation for Pebble Flows
events and indicates a smaller value of C. Relative fluctuating kinetic energy (RFKE)
represents the magnitude of kinetic energy fluctuation over a period. Different from
C describing the uniform characteristics in time-domain, RFKE measures the ratio
between the total fluctuating kinetic energy (TFKE) to the total kinetic energy (TKE)
and indicates the fluctuation of the pebble flow in amplitude-domain. The consistent
pebble flow always shows smaller fluctuation around the mean velocity. For instance,
the pebble flow for case F5 and F6 present smaller fluctuation with the RFKE nearly
10%.
Besides, these two parameters are dimensionless and have the fixed range from 0
to 1. When the number of particles and time are enough to obtain steady statistics,
C and RFKE will not be affected by the particle number and recording time.
Based on the above analysis, a new criterion is suggested for categorizing the flow
regime of dense granular flow according to the values of C and RFKE. The specific
categorization principle for dense granular flow is suggested as follows:
• C ≥ 0.5: Intermittent pebble flow
• C <0.5, RFKE > 0.1: Transition Fluctuation pebble flow
• C <0.5, RFKE ≤ 0.1: Continuous pebble flow
Therefore, dense pebble flows are categorized into three sub-regimes characterized
mainly by their primary features. Obviously, intermittent pebble flow describes the
flow that keeps the steady quiescent state most of the time, and the burst motion events
with a larger magnitude of velocity happen at long, evident intervals. Continuous
flow is the fastest flow regime of dense pebble flows. It can flow steadily with little
fluctuation and smoothly like common liquid. The transition fluctuation pebble flow
covers the transition regime from the slow to rapid flows.
It seems that with only six cases it is hard to find the limit of flows. More flow
rates are analyzed, and the values of parameters C and RFKE are listed in Table 4.10.
The flow rate of 0.5 particles per second belongs to the intermittent flow. The flow
rates smaller than 0.5 particles per second are also analyzed. The intermittency index
C and RFKE are found to be larger with the reduction of the flow rates. However,
they cannot still reach the value of 1. In addition, the flow rates can reach the highest
value when the pebbles are discharged freely from the silo bed only under the effect
of gravity and without other manual controls. Herein, the higher flow rate of 1250
particles per second, which is almost equal to the free flow rate, is simulated to check
the parameters C and RFKE. Similarly, the two parameters become smaller with the
increase of the flow rates but cannot reach 0. To conclude, the two parameters can
describe the flow characteristics of all flow rates.
In this section, the effects of recirculation flow rates on the drained pebble flow
in the silo bed are mainly focused, and the effects can be entirely described by two
parameters C and RFKE. Thus, the suggested criterion for categorizing the flow
regimes of dense granular flow based on the values of C and RFKE is proposed to
describe the effects of the flow rates. Herein, the specific categorization principles
for dense granular flow are suggested for a better understanding of the effects of flow
rates.