4.2 Gravity-Driven Flow Regime Characterization
199
(TFKE) for describing the degree of fluctuation in pebble flow for the analysis is
defined and obtained without considering the mass as follows:
TPKE =
1
2
(v
(t))
2
(4.23)
To compare the fluctuation kinetic energies for different flow rates, the relative fluctuating kinetic energy (RFKE) has been proposed as the ratio of TFKE to total kinetic
energy. The RFKE ranges from zero to one.
It is found in Fig. 4.22, that there are significant RFKE differences among various
flow rates. In detail, the RFKE nearly reaches around 95% for case F1 which reflects
the large magnitude of velocity under the burst motion events. By comparison, pebble
flow with higher flow rates in Cases F5 and F6 usually present the continuous flow
mode with slight fluctuation and the RFKE only stays approximately 10%.
Multi-Fractal Analysis
As aforementioned, to some extent, the kurtosis ranges from 1 to positive infinite
value, which could measure the intermittency of the pebble flow. However, this
parameter can’t provide detailed physical information of the dynamical system. Thus,
the multi-fractal analysis is adopted in this section to explore more details about the
pebble flow dynamics.
The aforementioned velocity-time series are processed by Eqs. (4.20–4.21), where
the non-dimensional analysis is adopted by dividing the velocity by the mean value.
The velocity signals at coarser scales have been reconstructed from the raw signal
by processing the window-average of the measurement at selected scale λ. The
ensemble average x)
q
of q
th -order moments of the window-averaged at the
scale λ is presented in Fig. 4.23. The different time-window-averaged scales λ =
2
0
, 2
1
, . . . , 2
7 are adopted here. The linear fitting of the data points is also presented
and the fitting parameters are displayed in Table 4.9.
From Fig. 4.23, the linear relation exists between the window size and the ensemble average in the logarithmic coordinate system, which indicates the power-law
relationship between the ensemble average of moments and the order q for multifractals x)
q
∼ λ
−K(q) . The velocity-time series exhibits a type of the statistical
scale invariance and self-similarity. The K(q) versus q contains the necessary information for characterizing the intermittency of the velocity-time series.
Recalling that the multi-fractal analysis is quantitatively related to the degree
of intermittency, a constant function is extremely not intermittent with D(q) = 1
and an impulse function is deeply intermittent with D(q) = 0. Herein the number
C = 1 − D(2) is chosen to characterize the intermittency. The higher value of C
means the larger intermittency and vice versa, where C ranges from 0 to 1.
The values of C of all the recirculation flow rates are shown in Fig. 4.24, and
these values correspond to the growing tendency of the kurtosis with the decreasing
recirculation flow rates. Specifically, the pebble flow for cases F1 and F2 presents
larger intermittency with the indices C = 0.7962 and 0.5027, respectively. Other
pebble flows are less intermittent with C slightly less than 0.5. In particular, the
199
(TFKE) for describing the degree of fluctuation in pebble flow for the analysis is
defined and obtained without considering the mass as follows:
TPKE =
1
2
(v
(t))
2
(4.23)
To compare the fluctuation kinetic energies for different flow rates, the relative fluctuating kinetic energy (RFKE) has been proposed as the ratio of TFKE to total kinetic
energy. The RFKE ranges from zero to one.
It is found in Fig. 4.22, that there are significant RFKE differences among various
flow rates. In detail, the RFKE nearly reaches around 95% for case F1 which reflects
the large magnitude of velocity under the burst motion events. By comparison, pebble
flow with higher flow rates in Cases F5 and F6 usually present the continuous flow
mode with slight fluctuation and the RFKE only stays approximately 10%.
Multi-Fractal Analysis
As aforementioned, to some extent, the kurtosis ranges from 1 to positive infinite
value, which could measure the intermittency of the pebble flow. However, this
parameter can’t provide detailed physical information of the dynamical system. Thus,
the multi-fractal analysis is adopted in this section to explore more details about the
pebble flow dynamics.
The aforementioned velocity-time series are processed by Eqs. (4.20–4.21), where
the non-dimensional analysis is adopted by dividing the velocity by the mean value.
The velocity signals at coarser scales have been reconstructed from the raw signal
by processing the window-average of the measurement at selected scale λ. The
ensemble average x)
q
of q
th -order moments of the window-averaged at the
scale λ is presented in Fig. 4.23. The different time-window-averaged scales λ =
2
0
, 2
1
, . . . , 2
7 are adopted here. The linear fitting of the data points is also presented
and the fitting parameters are displayed in Table 4.9.
From Fig. 4.23, the linear relation exists between the window size and the ensemble average in the logarithmic coordinate system, which indicates the power-law
relationship between the ensemble average of moments and the order q for multifractals x)
q
∼ λ
−K(q) . The velocity-time series exhibits a type of the statistical
scale invariance and self-similarity. The K(q) versus q contains the necessary information for characterizing the intermittency of the velocity-time series.
Recalling that the multi-fractal analysis is quantitatively related to the degree
of intermittency, a constant function is extremely not intermittent with D(q) = 1
and an impulse function is deeply intermittent with D(q) = 0. Herein the number
C = 1 − D(2) is chosen to characterize the intermittency. The higher value of C
means the larger intermittency and vice versa, where C ranges from 0 to 1.
The values of C of all the recirculation flow rates are shown in Fig. 4.24, and
these values correspond to the growing tendency of the kurtosis with the decreasing
recirculation flow rates. Specifically, the pebble flow for cases F1 and F2 presents
larger intermittency with the indices C = 0.7962 and 0.5027, respectively. Other
pebble flows are less intermittent with C slightly less than 0.5. In particular, the