172
4 Numerical Methods and Simulation for Pebble Flows
The condition of 0.5< ρ( ˙
F, V ) < 0.8 is considered to be significantly correlated, while the condition of 0.3< ρ( ˙
F, V ) < 0.5 is considered as poorly correlated. Therefore, it can be concluded that ˙
F and V are significantly correlated for
the high discharging flow rate, whereas it is poorly correlated for the slow discharging
flow rate.
The following analyses provide further explanations of the correlation between
˙
F and V . From Eq. (4.2),
F ∼ kx + ηV ,
˙
F ∼ k˙ x + η ˙
V ∼ kV + η ˙
V ,
(4.14)
where x is the mean-deformation of particles. When ˙
F and V are well correlated,
it is appropriate to assume
˙
F ∼ ρ( ˙
F, ˙
V )V
(4.15)
where ρ( ˙
F, ˙
V ) can be considered as a constant. Thus, the correlation coefficient
ρ( ˙
F, ˙
V ) and stiffness factor k are found to play a similar role, provided the η ˙
V
term is negligible. On the other hand, the condition
η ˙
V ≈ 0
(4.16)
indicates either the negligible time derivative of mean velocity or the small relative
value of η(i.e.,
η
k
≈ 3 × 10
−5
1). If the mean velocity variation is steady, the
velocity derivative is not very large, and
˙
V )
V )
should be small. Together with
η
k
1),
it leads to
η ˙
V )
kV )
1. Thus,
˙
F
kV
and the significant correlation between ˙
F and V
will exist.
In addition, the correlation coefficient between ˙
F and V is found to be
fairly small for all flows (see Table 4.2, ρ( ˙
F, ˙
V ) < 0.2, where ρ < 0.3 is usually considered as weakly correlated or even independent). In other words, with
η ˙
V kV and ˙
V independent of ˙
F, it guarantees the dominant role of V
in determining ˙
F. Thus, they are correlated more or less.
On the other hand, ρ( ˙
F, V ) is large in rapid particle flow (significant correlation) and small in slow or slow particle flow (low correlation, Table 4.2). Therefore,
the correlation between ˙
F and V may provide a useful indicator for categorizing the rapid and slow particle flow regimes. When ˙
F and V are significantly
correlated, it is considered as a steady rapid dense flow. Otherwise, it is a slow flow.
4.2.2.2 Autocorrelation
To further elucidate the characteristics or mechanisms between the rapid and slow
particle flows, the autocorrelation function is introduced to evaluate the time variation
of the mean force and velocity. As the variables are almost steady in time when
t > 100s (Fig. 4.5), the autocorrelation function of the stationary time can be obtained
4 Numerical Methods and Simulation for Pebble Flows
The condition of 0.5< ρ( ˙
F, V ) < 0.8 is considered to be significantly correlated, while the condition of 0.3< ρ( ˙
F, V ) < 0.5 is considered as poorly correlated. Therefore, it can be concluded that ˙
F and V are significantly correlated for
the high discharging flow rate, whereas it is poorly correlated for the slow discharging
flow rate.
The following analyses provide further explanations of the correlation between
˙
F and V . From Eq. (4.2),
F ∼ kx + ηV ,
˙
F ∼ k˙ x + η ˙
V ∼ kV + η ˙
V ,
(4.14)
where x is the mean-deformation of particles. When ˙
F and V are well correlated,
it is appropriate to assume
˙
F ∼ ρ( ˙
F, ˙
V )V
(4.15)
where ρ( ˙
F, ˙
V ) can be considered as a constant. Thus, the correlation coefficient
ρ( ˙
F, ˙
V ) and stiffness factor k are found to play a similar role, provided the η ˙
V
term is negligible. On the other hand, the condition
η ˙
V ≈ 0
(4.16)
indicates either the negligible time derivative of mean velocity or the small relative
value of η(i.e.,
η
k
≈ 3 × 10
−5
1). If the mean velocity variation is steady, the
velocity derivative is not very large, and
˙
V )
V )
should be small. Together with
η
k
1),
it leads to
η ˙
V )
kV )
1. Thus,
˙
F
kV
and the significant correlation between ˙
F and V
will exist.
In addition, the correlation coefficient between ˙
F and V is found to be
fairly small for all flows (see Table 4.2, ρ( ˙
F, ˙
V ) < 0.2, where ρ < 0.3 is usually considered as weakly correlated or even independent). In other words, with
η ˙
V kV and ˙
V independent of ˙
F, it guarantees the dominant role of V
in determining ˙
F. Thus, they are correlated more or less.
On the other hand, ρ( ˙
F, V ) is large in rapid particle flow (significant correlation) and small in slow or slow particle flow (low correlation, Table 4.2). Therefore,
the correlation between ˙
F and V may provide a useful indicator for categorizing the rapid and slow particle flow regimes. When ˙
F and V are significantly
correlated, it is considered as a steady rapid dense flow. Otherwise, it is a slow flow.
4.2.2.2 Autocorrelation
To further elucidate the characteristics or mechanisms between the rapid and slow
particle flows, the autocorrelation function is introduced to evaluate the time variation
of the mean force and velocity. As the variables are almost steady in time when
t > 100s (Fig. 4.5), the autocorrelation function of the stationary time can be obtained