4.2 Gravity-Driven Flow Regime Characterization
191
4.2.4.2 The Analysis Method
Fractal Analysis and Dimension
Fractal analysis is employed here to assess the fractal characteristics of data. As well
known, the fractal dimension is a statistical quantity that gives a global description
of how complex a geometry is. The dimension is the critical quantity to study fractal
objects. A fractal dimension is an index for characterizing fractal patterns or sets by
quantifying their complexity as a ratio of the change in detail to the change in scale
[50]. Perfect fractal sets can be decomposed into N similar copies of itself. Each
copy is scaled down by a factor s. Then, the quantities N and s are correlated by a
power law, i.e., N (s) ∝ s
−D , where D is the fractal dimension.
Multiplicative Cascade Method (MCM)
The Multiplicative Cascade Method (MCM) is used to characterize the self-similarity
and scale invariance. In general, a signal at coarser scales could be reconstructed from
the raw signal by processing the window-averaged measurement at the selected scale
(λ; [x + 1; x + λ]) =
1
λ
x+λ
x 0 =x+1
(1; x 0 ),
(4.19)
where (1; x 0 ) is the raw signal. x = kλ, and k = 0, 1, . . . ,
L
λ
, where L is the total
length of the signal. The window size λ = 2
0
, 2
1
, · · · and the ensemble average of
q
th order moments of the window-averaged field at scale λ is defined as
x)
q
=
1
L/λ
x 0 )
q
x 0 =kλ+1
.
(4.20)
A power-law relationship exists between the ensemble average of moments and the
order q for multi-fractals
x)
q
∼ λ
−K(q)
(4.21)
The K(q) versus q contains the necessary information for characterizing the intermittency of the velocity-time series. The generalized dimension of the multi-fractal
signals is D(q) = 1 −
K(q)
(q−1)
, where
K(q)
(q−1)
is the co-dimension [51]. The multi-fractal
analysis is a quantitative measure of the degree of the intermittency for that: (1). A
constant function is not intermittent with D(q) = 1; (2). An impulse function is deeply
intermittent with D(q) = 0. As a result, C 2 = 1 − D(2) is used here to characterize
the intermittency.
In this chapter, six cases of different fixed recirculation flow rates of 1, 10, 50,
100, 500, and 1000 particles per second (shortly named as Case F1, F2, F3, F4, F5,
and F6, respectively) are investigated for comparative study and analysis. The total
number of particles in the pebble bed is 6894.
191
4.2.4.2 The Analysis Method
Fractal Analysis and Dimension
Fractal analysis is employed here to assess the fractal characteristics of data. As well
known, the fractal dimension is a statistical quantity that gives a global description
of how complex a geometry is. The dimension is the critical quantity to study fractal
objects. A fractal dimension is an index for characterizing fractal patterns or sets by
quantifying their complexity as a ratio of the change in detail to the change in scale
[50]. Perfect fractal sets can be decomposed into N similar copies of itself. Each
copy is scaled down by a factor s. Then, the quantities N and s are correlated by a
power law, i.e., N (s) ∝ s
−D , where D is the fractal dimension.
Multiplicative Cascade Method (MCM)
The Multiplicative Cascade Method (MCM) is used to characterize the self-similarity
and scale invariance. In general, a signal at coarser scales could be reconstructed from
the raw signal by processing the window-averaged measurement at the selected scale
(λ; [x + 1; x + λ]) =
1
λ
x+λ
x 0 =x+1
(1; x 0 ),
(4.19)
where (1; x 0 ) is the raw signal. x = kλ, and k = 0, 1, . . . ,
L
λ
, where L is the total
length of the signal. The window size λ = 2
0
, 2
1
, · · · and the ensemble average of
q
th order moments of the window-averaged field at scale λ is defined as
x)
q
=
1
L/λ
x 0 )
q
x 0 =kλ+1
.
(4.20)
A power-law relationship exists between the ensemble average of moments and the
order q for multi-fractals
x)
q
∼ λ
−K(q)
(4.21)
The K(q) versus q contains the necessary information for characterizing the intermittency of the velocity-time series. The generalized dimension of the multi-fractal
signals is D(q) = 1 −
K(q)
(q−1)
, where
K(q)
(q−1)
is the co-dimension [51]. The multi-fractal
analysis is a quantitative measure of the degree of the intermittency for that: (1). A
constant function is not intermittent with D(q) = 1; (2). An impulse function is deeply
intermittent with D(q) = 0. As a result, C 2 = 1 − D(2) is used here to characterize
the intermittency.
In this chapter, six cases of different fixed recirculation flow rates of 1, 10, 50,
100, 500, and 1000 particles per second (shortly named as Case F1, F2, F3, F4, F5,
and F6, respectively) are investigated for comparative study and analysis. The total
number of particles in the pebble bed is 6894.