198
4 Numerical Methods and Simulation for Pebble Flows
toward the region of higher velocities (falling more rapidly) than toward the region
of lower velocities driven by gravity. The right tails stand for extreme motion events
with larger magnitude. For the distribution of F1, the velocities of some pebbles can
deviate from the average value more than sixteen times the standard deviation. With
the increasing flow rate, the number of pebbles with the extreme motion events is
getting smaller, and the deviations of particles from the average values are getting
diminished. The flow rate of F6 is close to the rate of free discharge flow under
gravity and presents the near symmetry distribution. Nearly all velocities of the
pebbles concentrate around the mean velocity within fourfold standard deviation.
On the other hand, the slowing down of the recirculation flow rates is accompanied
by increasing sharpness in the velocity distributions. As aforementioned, the kurtosis
is a measure of the sharpness or the “tailing” of the probability distribution, and
indicates how high the distribution is around the mean value. For F1, it is more
than 90% of the pebble velocities staying around the mean value within one standard
deviation. As a consequence, the kurtosis reaches the immense value of 381.2, which
indicates the largest sharpness of this distribution profile. In other words, the velocity
of the pebble flow system keeps lower for the most time. Recalling that the kurtosis is
regarded as an intermittency measure to evaluate how much time the system remains
quiescent, larger kurtosis indicates greater intermittency. Moreover, it shows the
decreasing intermittency for the Cases F2, F3, and F4 with the kurtosis = 32.2, 8.25,
and 5.74, respectively. It should be noted that the kurtosis is 3 for normal Gaussian
distribution which is larger than the kurtosis for the distributions of F5 and F6. The
proportion of velocities close to the mean value remains much smaller for these two
distributions which present less intermittency (if the intermittency exists).
4.2.4.4 Fluctuation Analysis
The velocity fluctuation has been observed in Fig. 4.20, and needed to be analyzed
with the quantitative descriptions. The coefficient of variation (CV ) is a statistical
measure of spread that describes the amount of variability relative to the mean value,
i.e., CV =
standard deviation(σ )
mean value of datasets
. Because the CV is unitless, it can be adopted instead
of the standard deviation to compare the spread of the data sets that have different
average values. Thus, the CV measures the dispersion of the data points around the
mean. It is helpful for comparing the degree of variation from one data series to
another, even if the means are drastically different.
The CV s presented in Table 4.8 and Fig. 4.22, indicate a rapid drop tendency with
the rise of the recirculation flow rate. The standard deviation σ is approximately eight
times larger than the mean velocity at the lowest flow rate (Case F1) while σ only
accounts for thirty percent of the average velocity at the higher flow rate (Case F6).
It indicates that the relatively burst motion events with extreme velocities are more
frequent in the slower pebble flow than the more rapid flow.
Furthermore, concerning the fluctuation of kinetic energy, the fluctuation velocity
v
(t) = v(t) − −v of the chosen rectangular zone is calculated by subtracting the
time-average velocity from the velocity at time t. The total fluctuating kinetic energy
4 Numerical Methods and Simulation for Pebble Flows
toward the region of higher velocities (falling more rapidly) than toward the region
of lower velocities driven by gravity. The right tails stand for extreme motion events
with larger magnitude. For the distribution of F1, the velocities of some pebbles can
deviate from the average value more than sixteen times the standard deviation. With
the increasing flow rate, the number of pebbles with the extreme motion events is
getting smaller, and the deviations of particles from the average values are getting
diminished. The flow rate of F6 is close to the rate of free discharge flow under
gravity and presents the near symmetry distribution. Nearly all velocities of the
pebbles concentrate around the mean velocity within fourfold standard deviation.
On the other hand, the slowing down of the recirculation flow rates is accompanied
by increasing sharpness in the velocity distributions. As aforementioned, the kurtosis
is a measure of the sharpness or the “tailing” of the probability distribution, and
indicates how high the distribution is around the mean value. For F1, it is more
than 90% of the pebble velocities staying around the mean value within one standard
deviation. As a consequence, the kurtosis reaches the immense value of 381.2, which
indicates the largest sharpness of this distribution profile. In other words, the velocity
of the pebble flow system keeps lower for the most time. Recalling that the kurtosis is
regarded as an intermittency measure to evaluate how much time the system remains
quiescent, larger kurtosis indicates greater intermittency. Moreover, it shows the
decreasing intermittency for the Cases F2, F3, and F4 with the kurtosis = 32.2, 8.25,
and 5.74, respectively. It should be noted that the kurtosis is 3 for normal Gaussian
distribution which is larger than the kurtosis for the distributions of F5 and F6. The
proportion of velocities close to the mean value remains much smaller for these two
distributions which present less intermittency (if the intermittency exists).
4.2.4.4 Fluctuation Analysis
The velocity fluctuation has been observed in Fig. 4.20, and needed to be analyzed
with the quantitative descriptions. The coefficient of variation (CV ) is a statistical
measure of spread that describes the amount of variability relative to the mean value,
i.e., CV =
standard deviation(σ )
mean value of datasets
. Because the CV is unitless, it can be adopted instead
of the standard deviation to compare the spread of the data sets that have different
average values. Thus, the CV measures the dispersion of the data points around the
mean. It is helpful for comparing the degree of variation from one data series to
another, even if the means are drastically different.
The CV s presented in Table 4.8 and Fig. 4.22, indicate a rapid drop tendency with
the rise of the recirculation flow rate. The standard deviation σ is approximately eight
times larger than the mean velocity at the lowest flow rate (Case F1) while σ only
accounts for thirty percent of the average velocity at the higher flow rate (Case F6).
It indicates that the relatively burst motion events with extreme velocities are more
frequent in the slower pebble flow than the more rapid flow.
Furthermore, concerning the fluctuation of kinetic energy, the fluctuation velocity
v
(t) = v(t) − −v of the chosen rectangular zone is calculated by subtracting the
time-average velocity from the velocity at time t. The total fluctuating kinetic energy