78
2 Experiments in Pebble Flows
• The autocorrelation time τ c seems to follow the power-law relation (τ c ∼ H
α )
with height on different streamlines. The index α equals 2.23, 1.69, 1.51, and 0.93
for streamlines No.1–No.4, respectively. It can be seen that the streamline with a
larger radial coordinate value indicates a smaller power-law index.
2.5.3.6 Intermittent Characteristics of Particle Velocities
The intermittent behavior is usually accompanied by the presence of rare bursts of
enhanced activity. Thus, the single impulse function in the constant signal suggests
a highly intermittent pattern, while a constant or uniformly varying signal is not
intermittent at all [32]. In light of the observation of the intermittency of particle
motions in the aforementioned section, it is necessary to define an intermittency
measure for quantitative analysis, since few investigations on the intermittency of
the slow dense granular flows in the silo bed have been reported.
As aforementioned, a Multiplicative Cascade Method (MCM) is introduced [22,
33] to characterize the self-similarity and scale invariance, i.e., the typical property
of fractals. In detail, the signals at coarser scales could be reconstructed from the raw
signal by processing the window average of the measurement at a selected scale:
(λ; [x + 1; x + λ]) =
1
λ
x+λ
x 0 =x+1
(1; x 0 ),
(2.8)
where (1; x 0 ) is the raw signal; X = kλ, and k = 0, 1, · · · ,
L
λ
; L is the total length
of the signal. The window size λ = 2
0
, 2
1
, · · · , and the ensemble average of the q
th
order moments of the window-averaged field at scale λ is defined as
x)
q
=
1
L/λ
L
x 0 =kλ+1
x 0 )
q
(2.9)
The velocities from the experiment have been analyzed by the multiplicative cascade method (Fig. 2.21). The velocity–time series exhibits a type of statistical scale
invariance and self-similarity. A power-law relationship exists between the ensemble
average of moments and the order q for multifractals:
x)
q
∼ λ
−K (q)
(2.10)
The K (q) contains the necessary information for characterizing the intermittency
of the velocity–time series. The generalized dimension is D(q) = 1 −
K (q)
(q−1)
for the
multifractal signals, where
K (q)
(q−1)
is the co-dimension [34]. The multifractal analysis
may be quantitatively related to the degree of intermittency; for that, a constant
function is extremely not intermittent with D(q)=1 and an impulse function is deeply
2 Experiments in Pebble Flows
• The autocorrelation time τ c seems to follow the power-law relation (τ c ∼ H
α )
with height on different streamlines. The index α equals 2.23, 1.69, 1.51, and 0.93
for streamlines No.1–No.4, respectively. It can be seen that the streamline with a
larger radial coordinate value indicates a smaller power-law index.
2.5.3.6 Intermittent Characteristics of Particle Velocities
The intermittent behavior is usually accompanied by the presence of rare bursts of
enhanced activity. Thus, the single impulse function in the constant signal suggests
a highly intermittent pattern, while a constant or uniformly varying signal is not
intermittent at all [32]. In light of the observation of the intermittency of particle
motions in the aforementioned section, it is necessary to define an intermittency
measure for quantitative analysis, since few investigations on the intermittency of
the slow dense granular flows in the silo bed have been reported.
As aforementioned, a Multiplicative Cascade Method (MCM) is introduced [22,
33] to characterize the self-similarity and scale invariance, i.e., the typical property
of fractals. In detail, the signals at coarser scales could be reconstructed from the raw
signal by processing the window average of the measurement at a selected scale:
(λ; [x + 1; x + λ]) =
1
λ
x+λ
x 0 =x+1
(1; x 0 ),
(2.8)
where (1; x 0 ) is the raw signal; X = kλ, and k = 0, 1, · · · ,
L
λ
; L is the total length
of the signal. The window size λ = 2
0
, 2
1
, · · · , and the ensemble average of the q
th
order moments of the window-averaged field at scale λ is defined as
x)
q
=
1
L/λ
L
x 0 =kλ+1
x 0 )
q
(2.9)
The velocities from the experiment have been analyzed by the multiplicative cascade method (Fig. 2.21). The velocity–time series exhibits a type of statistical scale
invariance and self-similarity. A power-law relationship exists between the ensemble
average of moments and the order q for multifractals:
x)
q
∼ λ
−K (q)
(2.10)
The K (q) contains the necessary information for characterizing the intermittency
of the velocity–time series. The generalized dimension is D(q) = 1 −
K (q)
(q−1)
for the
multifractal signals, where
K (q)
(q−1)
is the co-dimension [34]. The multifractal analysis
may be quantitatively related to the degree of intermittency; for that, a constant
function is extremely not intermittent with D(q)=1 and an impulse function is deeply