2.5 Particle Velocimetry Measurements
113
Fig. 2.46 The time
evolution of vertical velocity
of the particles where
h = 15d, x = 0d
Table 2.3 Cross-correlation of velocities
Location Horizontal Direction H = 30d
Vertical Direction X = 0d
X = 0d
X = 8d
X = 16d X = 24d H = 57d H = 30d H = 15 H = 3d
ρ at
0.916
0.922
0.870
0.930
0.952
0.916
0.924
0.946
ρ nt
0.789
0.757
0.782
0.771
0.859
0.789
0.831
0.795
to move fast or to stay static. The process can be illustrated as follows: The arching particles are supported by mutually balanced forces. If one particle is removed,
the pebbles, which are supported by it, become unstable and tend to move. Then,
the particles in distant locations also lose their stability. As a result, an avalanche
of pebble displacements takes place, which results in the upward propagation of
instability. The activity stops when no more particles are unstable. More significant
displacement always means a higher degree of instability. Thus, arching particles
can fluctuate more drastically. It seems that larger velocities usually display greater
velocity fluctuations in a dense slow pebble flow, which is similar in the Couette flow
[60] and pebble flow in a driven heap [61].
Arch Lifetime and Fluctuation Velocity Autocorrelation Time
To explore more about the relationship between arch dynamics and the velocity
fluctuations, the analysis of the arch lifetime and fluctuation velocity autocorrelation
time should be the right choice. The characteristic lifetime L c of arches is determined
by a weighted average of lifetime probability distribution function. Besides, the
correlation of the fluctuation velocities at different spatial sections is investigated to
examine how closely the fluctuations are related with time going by. The normalized
autocorrelation of the vertical velocity at the rectangular zones with the coordinate
of center (h, x) is defined as
113
Fig. 2.46 The time
evolution of vertical velocity
of the particles where
h = 15d, x = 0d
Table 2.3 Cross-correlation of velocities
Location Horizontal Direction H = 30d
Vertical Direction X = 0d
X = 0d
X = 8d
X = 16d X = 24d H = 57d H = 30d H = 15 H = 3d
ρ at
0.916
0.922
0.870
0.930
0.952
0.916
0.924
0.946
ρ nt
0.789
0.757
0.782
0.771
0.859
0.789
0.831
0.795
to move fast or to stay static. The process can be illustrated as follows: The arching particles are supported by mutually balanced forces. If one particle is removed,
the pebbles, which are supported by it, become unstable and tend to move. Then,
the particles in distant locations also lose their stability. As a result, an avalanche
of pebble displacements takes place, which results in the upward propagation of
instability. The activity stops when no more particles are unstable. More significant
displacement always means a higher degree of instability. Thus, arching particles
can fluctuate more drastically. It seems that larger velocities usually display greater
velocity fluctuations in a dense slow pebble flow, which is similar in the Couette flow
[60] and pebble flow in a driven heap [61].
Arch Lifetime and Fluctuation Velocity Autocorrelation Time
To explore more about the relationship between arch dynamics and the velocity
fluctuations, the analysis of the arch lifetime and fluctuation velocity autocorrelation
time should be the right choice. The characteristic lifetime L c of arches is determined
by a weighted average of lifetime probability distribution function. Besides, the
correlation of the fluctuation velocities at different spatial sections is investigated to
examine how closely the fluctuations are related with time going by. The normalized
autocorrelation of the vertical velocity at the rectangular zones with the coordinate
of center (h, x) is defined as