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Saltation movement of large spherical particles
Z. Chara, J. Dolansky, and B. Kysela
Citation: AIP Conference Proceedings 1863, 030038 (2017);
View online: https://doi.org/10.1063/1.4992191
View Table of Contents: http://aip.scitation.org/toc/apc/1863/1
Published by the American Institute of Physics
Saltation Movement of Large Spherical Particles
Z. Chara, J. Dolansky and B. Kysela
Institute of Hydrodynamics CAS, v. v. i., Prague, Czech Republic
Abstract. The paper presents experimental and numerical investigations of the saltation motion of a large spherical
particle in an open channel. The channel bottom was roughed by one layer of glass rods of diameter 6 mm. The plastic
spheres of diameter 25.7 mm and density 1160 kgm-3 were fed into the water channel and theirs positions were viewed by
a digital camera. Two light sheets were placed above and under the channel, so the flow was simultaneously lighted from
the top and the bottom. Only particles centers of which moved through the light sheets were recorded. Using a 2D PIV
method the trajectories of the spheres and the velocity maps of the channel flow were analyzed. The Lattice-Boldzmann
Method (LBM) was used to simulate the particle motion.
Keywords: Saltation, channel flow, PIV, LBM.
PACS: 47.11.Qr , 47.27.nd, 47.55.Kf, 47.80.Jk
INTRODUCTION
A motion of solid particles can be found in a wide spectrum of industrial and environmental applications sediment transport in rivers, hydraulic and pneumatic conveying in pipes, chemical reactors, heat exchangers. A lot
of studies have been focused on an investigation of a wake behind a sphere in uniform unbounded flow condition.
But the presence of the sphere close to a wall has received less attention. Rao et al.[1] studied the wake behind steel
spheres of diameters 4.7 and 6.5 rotated on an inclined surface. The particle Reynolds number was up to Re=300.
Ozgoren et al. [2] investigated the wake behind a motionless Plexiglas sphere of diameter 42.5 mm for different gap
heights between the sphere and a wall in an open channel. Dey et al. [3] used a Vectrino probe to measure wakes
behind spheres of diameters 20, 30 and 40 mm fixed on a rough wall.
This paper focuses on the investigation of the velocity field around the sphere of diameter 25.7 mm saltating over
a rough surface in an open channel. 2D PIV method was used to measure the velocity field experimentally and the
LBM method was used to simulate the sphere motion numerically.
EXPERIMENTAL ARRANGEMENT
The experiments were performed in the horizontal open channel of a cross section 0.25x0.25 m and length 6 m.
Both the side walls and the bottom are made from glass plates. The bottom was covered by one layer of the glass
rods of diameter 6 mm. The rods were placed perpendicularly on the flow direction. As the particles the plastic
spheres of the diameter 25.7 mm and density 1160 kgm-3 were used. By the help of a special device the spheres were
fed into the channel flow parallelly with the side walls. The 2D PIV system consisted of a camera NanoSense MK
III with 4 GB inner memory and frame rate 1000 fps in full resolution 1280x1024 pixels. The capacity of the camera
inner memory enabled to pick up the frames in a total time 3.3 sec. The seeding particles used were hollow glass
10 m particles (Dantec Inc. DK). As the sources of illumination were used Leica KL2500LCD equipped with
lightline with adjustable focusing optics. This light source worked in a continuous mode. The lights were placed
above the water level and under the channel bottom. The flow depth in the measuring section was h=64 mm and the
flow rate was 8.6 l/s. The flow Reynolds number based on the hydraulic diameter was Re=Umean H/=22000. The
local positions of the particle were determined by a cross correlation method.
NUMERICAL SIMULATIONS
The Lattice Boltzmann Method (LBM) is recently used as an alternative of CFD methods for numerical
modeling of a wide class of problems of fluid dynamics, e.g., multiphase systems, dissipative particle dynamics,
non-Newtonian fluid flows, suspension or biomaterials flows, turbulence. In the LBM the fluid is composed of
fictive particles which interact in nodes and propagate along the lattice links with velocities ci. The fictive particles
are represented by particle distribution functions f(x,ci,t) which give probabilities of finding of a fictive particle in a
International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2016)
AIP Conf. Proc. 1863, 030038-1–030038-4; doi: 10.1063/1.4992191
Published by AIP Publishing. 978-0-7354-1538-6/$30.00
030038-1
node x with a certain discrete velocity ci in time t. The collision and propagation process follows from the lattice
Boltzmann equation
1
f x  c i t ,t  t   f i x ,t   f i eq  x ,t ,ux ,t   f i x ,t 

where the term on the right-hand side represents the mostly used Bhatnagar-Gross-Krook (BGK) collision operator
in which the change of the distribution function fi is proportional to its difference from equilibrium fieq. The
relaxation parameter τ then expresses rate of relaxation to local equilibrium. The equilibrium distribution function
fieq is given by the discretized Maxwell distribution of fictive particle velocities
 c u c u 2  c s2 u 2 
f i eq   wi 1  i 2  i

cs
2cc2


where wi are the weights normalized to unity, u is the velocity flow and cs is the lattice sound speed. The collision
operator fulfills the first law of thermodynamics, i.e., conservation of mass and momentum. Macroscopic quantities
(density ρ, momentum ρu, momentum flux Π) of the flow are then obtained as moments over particle distributions fi
and discrete velocities ci. The LBM offers a unique computational frame to calculate mutual interaction of
macroscopic particles and the fluid (composed of the fictive particles represented by the particle distributions), e.g.,
Yu and Fan [4]. Influence of the particle on the fluid is evaluated as the (moving) boundary condition on the solid
particle surface while action of the fluid on the particle can be computed as the momentum transfer of the adjacent
fictive particles. The no-slip boundary condition is supposed to represent solid boundaries of the macroscopic
particles and the domain walls. In the developed simulation the Ansumali's proposal, (Ansumali et al. [5]) of the
hybrid diffusive bounce-back combined with the Grad's refilling algorithm was used for the moving particles. The
hybrid diffused bounce-back condition combines some advantages of the standard bounce-back and of the diffusive
boundary condition within the LBM. Namely, it retains the locality of the bounce-back scheme and it smooths
populations fi in the immediate surroundings of the moving boundary. These conditions are realized in three steps:
First the Ladd’s two-sided bounce-back is applied, then the modified fluid density ρ is computed at the boundary
nodes and this density is finally used as an argument of equilibrium fieq at these nodes. To refill populations in
uncovered nodes the Grad’s approximation is used

c u σ  u : u  u 2I   p  cs2 I
f i Grad  wi     i 2 
ci : ci  cS2I 
4
c
2
c
s
s


as it is able to reduce effectively fluctuations at these nodes (σ is the stress tensor and I is the identity matrix).
The interaction of open boundaries, i.e., inflow and outflow, with the fluid domain is not negligible and can
deteriorate solution as it is known that the LBM approach suffers from pressure wave reflections caused by
compressibility effects of the method, e.g., Izquierdo et al. [6]. Equilibrium distribution approach was chosen for the
specified velocity profile at the inlet, and the simple extrapolation scheme (Yu et al., [7]) for the outflow boundary.
Finally, to model the free surface of the flow the standard approach of Körner et al. [8] was employed.
The particle motion is determined by interaction between the fluid and the macroscopic particles. Thus force of
the flow is expressed by the difference Δp between momentum of fictive particles which incomes and leaves
(macroscopic) particle boundaries. Time rate of this momentum transfer Δp/Δt defines the hydrodynamic forces by
which the flow acts on the objects




w
F  2  f i  2i  c i v 
c
xb
i 
s

which are calculated as a sum over momentum contributions from all fictive particles incident to the boundary
nodes. The second term stands for the momentum contribution due to (macroscopic) particle motion.
The in-house simulation software developed for three-dimensional processes employs the D3Q15 lattice (mainly
to save computational time). The resolution of the computational domain was given by the number of nodes per the
particle diameter. The requirement of 25 nodes over the particle diameter 0.0254 m resulted in the resolution 286 ×
150 × 118 for the physical domain 0.144 × 0.076 × 0.06 m which corresponded to the part of the optical box where
the particle trajectories were observed. For this choice the lattice space step Δ
1 corresponded to Δx ~ 10-4 m in
physical units. Although there is no straightforward way to choose Δt with respect to Δx it was assumed that the
order of the time step and the space step is related as Δt / Δx = 0.1, which is enabled by the hyperbolic nature of
lattice Boltzmann equation (e.g., Succi [9]). Into the flow determined by the averaged velocity profile, the particles
were released from experimentally obtained initial positions with initial velocities.
030038-2
25
a)
vp [m/s]
0.1
y [mm]
20
0.0
15
-0.1
c)
40
0.50
30

[1/
s]
up [m/s]
0.45
20
0.40
10
b)
d)
0
20
40
60
80
100
0
20
40
60
80
x [mm]
x [mm]
FIGURE 1. Trajectories and velocities of the saltating spheres. The spheres moved from the left to the right.
40
0.6
y [mm]
30
0.5
0.4
20
0.3
0.2
10
0.1
0
0
a)
10
20
30
40
50
x [mm]
60
70
80
90
100
110
0
40
0.6
30
y [mm]
0.5
0.4
20
0.3
0.2
10
0.1
b)
0
0
10
20
30
40
50
x [mm]
60
70
80
90
100
110
0
FIGURE 2. Contours of the velocity magnitude on the central plane - a) measured b) simulated.
25
Experiment
LBM
20
y [mm]
0
15
10
0
20
40
60
80
x [mm]
FIGURE 3. Trajectories of the spherical particles
030038-3
100
100
RESULTS AND DISCUSSION
Examples of the measured trajectories of the saltating particles are shown in Fig.1a. Due to the limited size of the
image window only parts of the trajectories were detected. Since the time between two successive images was 0.001
sec it was possible to determine both the horizontal and the vertical velocity components. The velocities along the
trajectories are shown in Figs 1b and 1c. As can be seen in these figures the horizontal velocity components are
more or less constant and the vertical velocities linearly depend on the positions. It indicates a parabolic shape of the
trajectories. To estimate angular particle velocities each particle was colored by two perpendicular lines. The angular
velocities were detected automatically by an image analysis. Along individual trajectory the measured angular
velocities were nearly constant but for different trajectories they varied from 15 to 30 s-1, Fig. 1d. Even after impact
the considerable changes of the angular velocities were not observed. It means that the distribution of the velocity on
the sphere surface is mostly responsible for the particle rotation.
By the 2D PIV method maps of velocity vectors as well as contours of velocity magnitude were determined at a
plane going through the particle centers . The particle velocities smoothly follow the channel flow. A relatively large
volume of fluid is moving together with the particles. Vortical structures typical for flows around fixed solid objects
were not observed. The experimentally determined contours of the velocity magnitude when the position of the
sphere was just on the top of its trajectory are shown in Fig. 2a. The sphere is painted by the color corresponding to
the particle velocity. The initial parameters of the trajectories e.g. particle positions, particle velocities and mean
velocity profile were used as inputs for the LBM simulation. Interactions of the fluid and the particles were
evaluated within the unified computational frame of the LBM simulation. The action of the particles on the fluid was
solved by the so called hybrid diffusive BOUNCE-BACK scheme which suitably modifies particle distributions on
the moving boundaries. Mutual dependence of these two steps determined substantially outcomes of the simulation.
The resulting simulated contours of the velocity for the same flow conditions as in Fig. 2a are shown in Fig. 2b.
Upstream and downstream the sphere the simulated velocities are smaller than the experimental ones. It indicates
that the simulated flow stream is flowing around the sphere. Comparison of the simulated and the measured particle
trajectories are shown in Fig. 3. It seems that the simulated trajectories are a lit bit shorter than the trajectories
experimentally determined.
CONCLUSIONS
Using the image processing the trajectories and velocities of the large spherical particles as well as the flow maps
around the particles were experimentally determined. No wake structures typical for the pure flow around fixed solid
objects were observed. The LBM method was used to simulate the particle trajectories. The simulations slightly
undervalued the lengths of the particle trajectories.
ACKNOWLEDGMENTS
The supports under project No. 15-18870S of the Grant Agency of the Czech Republic and RVO: 67985874 are
gratefully acknowledged.
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M. Ozgoren , A. Okbaz , S. Dogan , B Sahin and H. Akilli, EPJ Web of Conferences 25, 01066 (2012).
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Z. Yu and L.S. Fan, Particuology, 8, 539-543 (2010).
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S. Izquierdo, P. Martínez-Lera and N. Fueyo, Comp. & Math. App., 58, 914–921 (2009).
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