2.5 Particle Velocimetry Measurements
111
cally. In summary, the idea that the angle distribution is a vital clue and feature for
the spatial rearrangement of the arches is reinforced.
Furthermore, the relative change of positions of arching particles and δ P (α ≈60
◦ )
during an avalanche are considered here. The difference between the resultant movement of all particles in the arches and the individual particle can be calculated as
follows:
δV =
(v 1 − v t ) 2 + (v 2 − v t ) 2 + · · · + (v s − v t ) 2
(2.26)
v i is the velocity vector of particle i in the arch (i=1, 2,..., s) and (v t ) =
v 1 +v 2 +···+v s
s
.
Arches, including s particles, retain a stable structure when all particles move along
almost in same motion trajectory. Thus, there is little difference between the vectorial
average of all particle velocities between any individual particle velocity. When the
arch experiences an avalanche, the particle members will show irregular dispersion
and disordered motion. The δV is more likely to become significantly large. The
avalanche sees the significant changing of both the δV and PDF (α ≈60
◦ ). The
rectangular region centered in H = 30d, X = 24d is focused on to explore more,
and the data, including δV and δ P (α ≈60
◦ ) in each avalanche for a given arch size
(s=2, 3, 4, respectively) in sequence, is collected. Herein, 250 avalanche events are
chosen for each arch size.
Figure 2.45 illustrates the δV and δ P (α ≈60
◦ ) evolution in the 250 avalanche
events for the arch sizes from two to four. The correlation of δV and δ P (α ≈60
◦ )
will be studied. The cross-correlation of (X t , Y t ) is calculated by
ρ(X t , Y t ) =
E[(X t − μ x )(Y t − μ y )]
σ X σ Y
(2.27)
where μ x and σ X are, respectively, the mean and standard deviation of the process
(X t ), which are constant over time due to stationarity. Quantitative values show
that the correlation coefficient (ρ(δV ,δ P)=0.5055) for the four-particle arches and
(ρ(δV ,δ P)=0.5132) for the three-particle arches are higher than that (ρ(δV ,δ P)=
0.4112) for the two-particle arches. The condition of 0.5< ρ <0.8 is considered to be
significantly correlated, while the condition of 0.3< ρ <0.5 is considered as lowly
correlated. Therefore, it can be concluded that δV and δ P are significantly correlated
for the larger arches, whereas it is slowly correlated for the smaller arches.
In addition, the relationship between the internal topological structure changing
and velocity variation during the avalanche is evidenced. In a broader sense, the
arch breaking keeps its particle members move in different directions and depart
from their original orbits. Meanwhile, the relative positions between particles are
changed with a significant decrease of the successive angles near 60
◦ between the
neighbors of the particle. It could be predicted that the arch breaking and rebuilding
have something to do with the velocity fluctuation in the particle flow. More will be
discussed in the next section.
111
cally. In summary, the idea that the angle distribution is a vital clue and feature for
the spatial rearrangement of the arches is reinforced.
Furthermore, the relative change of positions of arching particles and δ P (α ≈60
◦ )
during an avalanche are considered here. The difference between the resultant movement of all particles in the arches and the individual particle can be calculated as
follows:
δV =
(v 1 − v t ) 2 + (v 2 − v t ) 2 + · · · + (v s − v t ) 2
(2.26)
v i is the velocity vector of particle i in the arch (i=1, 2,..., s) and (v t ) =
v 1 +v 2 +···+v s
s
.
Arches, including s particles, retain a stable structure when all particles move along
almost in same motion trajectory. Thus, there is little difference between the vectorial
average of all particle velocities between any individual particle velocity. When the
arch experiences an avalanche, the particle members will show irregular dispersion
and disordered motion. The δV is more likely to become significantly large. The
avalanche sees the significant changing of both the δV and PDF (α ≈60
◦ ). The
rectangular region centered in H = 30d, X = 24d is focused on to explore more,
and the data, including δV and δ P (α ≈60
◦ ) in each avalanche for a given arch size
(s=2, 3, 4, respectively) in sequence, is collected. Herein, 250 avalanche events are
chosen for each arch size.
Figure 2.45 illustrates the δV and δ P (α ≈60
◦ ) evolution in the 250 avalanche
events for the arch sizes from two to four. The correlation of δV and δ P (α ≈60
◦ )
will be studied. The cross-correlation of (X t , Y t ) is calculated by
ρ(X t , Y t ) =
E[(X t − μ x )(Y t − μ y )]
σ X σ Y
(2.27)
where μ x and σ X are, respectively, the mean and standard deviation of the process
(X t ), which are constant over time due to stationarity. Quantitative values show
that the correlation coefficient (ρ(δV ,δ P)=0.5055) for the four-particle arches and
(ρ(δV ,δ P)=0.5132) for the three-particle arches are higher than that (ρ(δV ,δ P)=
0.4112) for the two-particle arches. The condition of 0.5< ρ <0.8 is considered to be
significantly correlated, while the condition of 0.3< ρ <0.5 is considered as lowly
correlated. Therefore, it can be concluded that δV and δ P are significantly correlated
for the larger arches, whereas it is slowly correlated for the smaller arches.
In addition, the relationship between the internal topological structure changing
and velocity variation during the avalanche is evidenced. In a broader sense, the
arch breaking keeps its particle members move in different directions and depart
from their original orbits. Meanwhile, the relative positions between particles are
changed with a significant decrease of the successive angles near 60
◦ between the
neighbors of the particle. It could be predicted that the arch breaking and rebuilding
have something to do with the velocity fluctuation in the particle flow. More will be
discussed in the next section.