182
4 Numerical Methods and Simulation for Pebble Flows
4.2.3.4 A General Rule for Gravity-Driven Dense Particle Flow
A series of numerical experiments with a wide range of circulating rates ranging
from 1 to 2500 (nearly free outflow) pebbles per second have been carried out.
Different circulating rates lead to particle flows with different mean velocities and
more importantly, a transformation of flow behavior.
The transformation of the flow behavior can be noted from E − t profiles of these
cases in Fig. 4.13. The total kinetic energy of the bulk pebble flow within the control
volume is recorded for every time step (5 × 10
−5 s) during the simulation. Data
points with a time interval of 1 × 10
−3 s are extracted from a period of time to plot
the E − t profile. The time interval of 1 × 10
−3 s is believed to be short enough to
capture momentary energy impulses whose time scale usually ranges from 0.01 to
0.1s. So, the variation of the kinetic energy is plotted against time for cases with
different discharging rates. At low circulating rates, the impulses are discrete, and
the intermittence is evident (5 particles per second and 20 particles per second). As
the circulating rate increases, the frequency of impulses increases rapidly, leading to
the intermittence being covered up gradually (80 particles per second). However, the
E − t profile still shows a base level of the kinetic energy, which means that most
irregular impulses present one-direction above the base level. Nonetheless, when it
turns to the circulating rates higher than 320 particles per second, the intermittence
is hardly recognizable, and the profile turns to an intense two-directional fluctuation.
The change of the profile pattern indicates the change of flow behavior. Therefore,
it is believed there is a need for subdividing these dense particle flows, which used
to be roughly treated as a single elastic–quasi-static flow.
The data of E − t obtained from the numerical experiments are further processed
to study the statistical distribution of the instantaneous kinetic energy (E(t)). By
setting the order of the magnitude of the instantaneous energy (log 10 E(t)) as the
horizontal axis, similar statistical distributions of E(t) are obtained for a wide range
of gravity-driven dense particle flows (refer to Fig. 4.14). These distribution profiles have a similar pattern of one “peak” of high probability density and two-side
“tails” of low probability density. Moreover, all these distributions show high similarity with Gaussian distribution with quite large R-square values. For the sake of an
easy expression, these distribution patterns with self-similarity are informally called
“single-peak” distribution. As a result, the general rule on E(t) distribution is that
at different mean velocities (or circulating rates), during a long enough period, the
order of the magnitude of the instantaneous kinetic energy (log 10 E(t)) should follow
the “single-peak” distribution.
Figure 4.14 demonstrates that the universal distribution pattern of log 10 E(t) is
applicable to general gravity-driven dense particle flows from very slow flow (1 particle per second) to basically free outflow (around 2500 particles per second). It is
noted that the order of the magnitude of E(t) rises as an increase of the circulating
rate, because a high circulating rate makes overall particle material flow faster and
get larger kinetic energy. Additionally, the peak probability density of a low circulating rate is much larger than that of the high circulating rate. This is because a
slowand intermittent flow (low circulating rate) has a wider range of log 10 E(t), and
4 Numerical Methods and Simulation for Pebble Flows
4.2.3.4 A General Rule for Gravity-Driven Dense Particle Flow
A series of numerical experiments with a wide range of circulating rates ranging
from 1 to 2500 (nearly free outflow) pebbles per second have been carried out.
Different circulating rates lead to particle flows with different mean velocities and
more importantly, a transformation of flow behavior.
The transformation of the flow behavior can be noted from E − t profiles of these
cases in Fig. 4.13. The total kinetic energy of the bulk pebble flow within the control
volume is recorded for every time step (5 × 10
−5 s) during the simulation. Data
points with a time interval of 1 × 10
−3 s are extracted from a period of time to plot
the E − t profile. The time interval of 1 × 10
−3 s is believed to be short enough to
capture momentary energy impulses whose time scale usually ranges from 0.01 to
0.1s. So, the variation of the kinetic energy is plotted against time for cases with
different discharging rates. At low circulating rates, the impulses are discrete, and
the intermittence is evident (5 particles per second and 20 particles per second). As
the circulating rate increases, the frequency of impulses increases rapidly, leading to
the intermittence being covered up gradually (80 particles per second). However, the
E − t profile still shows a base level of the kinetic energy, which means that most
irregular impulses present one-direction above the base level. Nonetheless, when it
turns to the circulating rates higher than 320 particles per second, the intermittence
is hardly recognizable, and the profile turns to an intense two-directional fluctuation.
The change of the profile pattern indicates the change of flow behavior. Therefore,
it is believed there is a need for subdividing these dense particle flows, which used
to be roughly treated as a single elastic–quasi-static flow.
The data of E − t obtained from the numerical experiments are further processed
to study the statistical distribution of the instantaneous kinetic energy (E(t)). By
setting the order of the magnitude of the instantaneous energy (log 10 E(t)) as the
horizontal axis, similar statistical distributions of E(t) are obtained for a wide range
of gravity-driven dense particle flows (refer to Fig. 4.14). These distribution profiles have a similar pattern of one “peak” of high probability density and two-side
“tails” of low probability density. Moreover, all these distributions show high similarity with Gaussian distribution with quite large R-square values. For the sake of an
easy expression, these distribution patterns with self-similarity are informally called
“single-peak” distribution. As a result, the general rule on E(t) distribution is that
at different mean velocities (or circulating rates), during a long enough period, the
order of the magnitude of the instantaneous kinetic energy (log 10 E(t)) should follow
the “single-peak” distribution.
Figure 4.14 demonstrates that the universal distribution pattern of log 10 E(t) is
applicable to general gravity-driven dense particle flows from very slow flow (1 particle per second) to basically free outflow (around 2500 particles per second). It is
noted that the order of the magnitude of E(t) rises as an increase of the circulating
rate, because a high circulating rate makes overall particle material flow faster and
get larger kinetic energy. Additionally, the peak probability density of a low circulating rate is much larger than that of the high circulating rate. This is because a
slowand intermittent flow (low circulating rate) has a wider range of log 10 E(t), and