2.5 Particle Velocimetry Measurements
73
Fractal Analysis and Dimension
Fractal analysis is used to assess fractal characteristics. In detail, a set of perfect
fractal data, either irregular or regular, appears similarly at all scales of magnification.
The fractal dimension is a statistical quantity that gives a global description of how
complex or irregular the geometry is. The dimension is the key quantity of the fractal
objects. A perfect fractal set can be decomposed into N similar copies of itself, each
scaled down by a factor s. The quantity N and s have the relationship
N (s) ∝ s
−D
(2.7)
where D is the fractal dimension. The fractal dimension is an index for characterizing
the complexity of geometries as a ratio of change in length/area to the change in the
measurement scale [26].
2.5.3.3 Velocity Profile
As shown in Fig. 2.16, the velocity profiles in the horizontal direction on four heights
are obtained by averaging the velocities of particles’ overall times at each radial
coordinate r . To get suitable statistical averages, several runs of the experiments
have been carried out on each height at a circulating rate of 150 particles per minute.
At the same time, the same number of particles is reloaded into the experiment vessel
through the inlet tubes.
In Fig. 2.16, the averages are plotted by blue dots which indicate the Gaussian
distributions. The average vertical velocities of the pebbles in the middle zone are
getting greater gradually and neatly as the height decreases. The results are similar to
the experiment carried out by Maza et al. [19, 27]. The cloud of small gray dots stands
for the velocities of individual particles. A bar is provided, indicating the standard
deviation of the measured values. In general, the deviations of the velocities of
individual pebbles from the mean are quite significant, as illustrated by the standard
deviation. The standard deviation near the center zone of the pebble bed is negatively
related to the pebble-bed height h (Fig. 2.16), namely when the height is lower, the
vertical velocities fluctuate more drastically.
2.5.3.4 Time Evolution of Individual Particle Velocity
The deviations of velocities of individual particles are quite large. To obtain more
information on the movement of individual particles, more details are shown by tracking the individual particles in successive frames. Several rectangular zones (4d × 5d)
centered at (h, x) or (h, r ) are used to study the path and instantaneous velocity of
each particle.
The mean vertical velocity of particles inside the rectangular zones centered
at height h and x (or r ) at time t is named as V hx,t . Figure 2.17 illustrates the
73
Fractal Analysis and Dimension
Fractal analysis is used to assess fractal characteristics. In detail, a set of perfect
fractal data, either irregular or regular, appears similarly at all scales of magnification.
The fractal dimension is a statistical quantity that gives a global description of how
complex or irregular the geometry is. The dimension is the key quantity of the fractal
objects. A perfect fractal set can be decomposed into N similar copies of itself, each
scaled down by a factor s. The quantity N and s have the relationship
N (s) ∝ s
−D
(2.7)
where D is the fractal dimension. The fractal dimension is an index for characterizing
the complexity of geometries as a ratio of change in length/area to the change in the
measurement scale [26].
2.5.3.3 Velocity Profile
As shown in Fig. 2.16, the velocity profiles in the horizontal direction on four heights
are obtained by averaging the velocities of particles’ overall times at each radial
coordinate r . To get suitable statistical averages, several runs of the experiments
have been carried out on each height at a circulating rate of 150 particles per minute.
At the same time, the same number of particles is reloaded into the experiment vessel
through the inlet tubes.
In Fig. 2.16, the averages are plotted by blue dots which indicate the Gaussian
distributions. The average vertical velocities of the pebbles in the middle zone are
getting greater gradually and neatly as the height decreases. The results are similar to
the experiment carried out by Maza et al. [19, 27]. The cloud of small gray dots stands
for the velocities of individual particles. A bar is provided, indicating the standard
deviation of the measured values. In general, the deviations of the velocities of
individual pebbles from the mean are quite significant, as illustrated by the standard
deviation. The standard deviation near the center zone of the pebble bed is negatively
related to the pebble-bed height h (Fig. 2.16), namely when the height is lower, the
vertical velocities fluctuate more drastically.
2.5.3.4 Time Evolution of Individual Particle Velocity
The deviations of velocities of individual particles are quite large. To obtain more
information on the movement of individual particles, more details are shown by tracking the individual particles in successive frames. Several rectangular zones (4d × 5d)
centered at (h, x) or (h, r ) are used to study the path and instantaneous velocity of
each particle.
The mean vertical velocity of particles inside the rectangular zones centered
at height h and x (or r ) at time t is named as V hx,t . Figure 2.17 illustrates the