4th International Symposium on Particle Image Velocimetry
Göttingen, Germany, September 17-19, 2001
PIV’01 Paper 1116
A. Lecuona, U. Ruiz-Rivas, J. Nogueira
Abstract The trajectories of heavy particles (
ρ particle
/
ρ fluid
1) are simulated in a 2D free vortex flow. The results shown that heavy particles, even with small diameters, cannot properly trace the fluid and develop a centrifugal motion. This behavior leads to a quick depletion of particles in the vortex core, which, in seed-dependent measurements as PIV or LDV, produce a marked increase of measurement errors. Some examples are giving and the evolution of the particle density is simulated.
1
Introduction
The dynamics of a particle imbedded in a fluid flow has gathered constant attention in the last two decades.
Basically, this is due to the existence of a wide range of particle-laden flows in nature and of considerable engineering applications. On the other hand, the study of the particle motion is relevant to the implementation of flow measurement techniques such as LDV and PIV. These techniques rely in the motion of seeded particles, and therefore their liability is based on the capability of the particles to accurately follow the flow.
In this paper, we will analyze the dynamics of a tracer particle in a vortical flow. Flows with strong vortices are also widely present in nature and engineering applications (i.e. the Karman Vortex Street, aircraft wing-tip vortices, the mixing layer between parallel streams, etc.).
We will consider a small spherical particle imbedded in a two dimensional, isolated vortex. The particle diameters will vary around 1 micron and the density ratio between fluid and particle is of order 10
-3
. This is typical for tracers such as oil drops or solid particles in airflows.
Throughout this paper we will consider that the particle motion does not affect the fluid flow, an assumption based on the small particle dimensions and the moderate number of particles imbedded in the flow. This is, of course, a general assumption taken in all the seed-dependent measurement techniques. Also, the separation between particles is commonly several orders of magnitude larger than the particle diameter, so that each particle can be considered as isolated. This means that the diffusion of particles, collisions between particles and particle breaking processes will not be considered. Finally, for the case of liquid seeds, the vapor pressure of oils used for seeding is so low at usual working conditions that evaporation is negligible.
2
Equation of Motion
The general form of the equation of motion for a small rigid sphere was proposed by Maxey and Riley (1982) and is generally accepted. It considered the effect of five forces acting over the particle: the steady state drag force, the gravitational force, the added (or virtual) mass effect, the fluid acceleration at the particle location (pressure gradient) and the Basset force (considering the time history of the particle). This equation of motion is then given by the formula: d v
= d t
18
D
2
µ
ρ c p u
− v
+
1
24
D
2 ∇ 2 u
+
9
D
ρ
ρ c p
ν
π
0 t d d t
+
ρ
ρ c p
D u
+
D t g


1
− u
− v
+
1
24
D
2 ∇ 2 u
( t
ρ
ρ c p


+
1
2
ρ
ρ c p
− τ ) −
0 .
5 d
τ d d t u
− v
+
1
40
D
2 ∇ 2 u
+
(1) where D is the particle diameter;
µ
,
ν
, and
ρ
are respectively the dynamic and kinematic viscosity and the mass
A. Lecuona, U. Ruiz-Rivas, J. Nogueira, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Spain
Correspondence to:
Dr. Ulpiano Ruiz-Rivas, , Escuela Politécnica Superior, Universidad Carlos III de Madrid, Leganés, Spain
Avda de la Universidad 30, 28911 Leganés, Spain, E-mail: ulpiano@ing.uc3m.es
1

PIV’01 Paper nnnn density (denoted p or c to distinguish between particle and continuum-fluid), u is the velocity vector of the flow at the particle location and v is the particle velocity vector. The operator d/dt denotes time derivatives following the particle and the operator D/Dt a time derivative following the fluid. The
∇ 2
u terms are the Faxen correction for the non-uniformity of the flow. The bar over a symbol indicates that it is a dimensional quantity, while its nondimensional form will be written without a bar
The vortex characteristic radius R o
and velocity U o
define respectively the length and velocity scales of the flow.
Therefore, we can introduce the non-dimensional variables: x
= x
R o u
= u
U o t
= t · U o
R o
(2)
In dimensionless form, equation (1) becomes
1
ε
2 d dt v
=
+
1
St
( u
−
3 v
+
2
π
ε
· St
0
T
1
24
δ
2 ∇ 2 u )
+ ε d dT
U
−
V
+
D u
Dt
+
G
(
1
− ε )
+
1
24
δ 2 ∇ 2
U
1
2
ε d dt
(
T
− τ
) −
0 .
5 d
τ u
− v
+
1
40
δ
2 ∇ 2 u
+
(3)
Four non-dimensional parameters appear in this equation: a Stokes number, a density ratio, a particle-vortex length scale ratio and a gravity coefficient:
St
=
p
U o
D
2
18
R o
=
c
p
=
D
R o
G
=
R o g
U o
2 (4) where the Stokes number (St) is the viscous time divided by the characteristic time scale of the vortex flow, R o
/U o
.
For a tracer particle in an airflow
δ ∼
O(10
-5
). Therefore the Faxen corrections can be neglected.
Lasheras and Tio (1994) have carried out the asymptotic study of the resulting equation, using a Modified Rankine
Vortex to model the flow field. They modification of the Rankine vortex is done to avoid discontinuity in the derivative. In terms of a cylindrical coordinate system, the stream function, the fluid velocity and the vorticity for this axisymmetric vortex are given by:
Ψ = −
U o
R o ln




1
+

 r
R o


2



 u
=
2 U
1
+

 o

 r r
R
R o o


2

 e
=
R o

 1
+
4 U

 r o
R o


2


2 e z
(5)
In Figure 1 we show the velocity and the vorticity fields
1.2
5
1
0.8
0.6
0.4
0.2
4
3
2
1
0
0 1 2 3
Radial Coordinate, r/Ro
4 5
0
Figure 1. The velocity and vorticity fields generated by a modified Rankine Vortex
The asymptotic analysis of Lasheras and Tio was carried out for three different regions of the flow field, defined by the order of magnitude of the position: x
∼
O(St), x
∼
O(1) and x
∼
O(St
-1
). We will show that, for the dynamics of a
2

PIV’01 Paper nnnn tracer particle in a large scale flow, the only region of interest is x
∼
O(1), where, according to the asymptotic analysis, the particle velocity is given by: v
= u
+
St
3
2
ε −
1


( u ·
∇ u
−
G
)
+
O
( )
(6)
Let us examine the orders of the different parameters involved in the dynamics of a tracing particle in a large-scale airflow. The particle diameters for a good tracer are D
∼
O(10 particles (Al and
∼
O(10
-1
2
-6
). Air tracers are commonly oil drops or solid
O
3
or similar), for which
ρ p
∼
O(10
3
). The vortex parameters (U o
, R o
) can be assumed to be
∼
O(10)
) respectively. With such conditions, the non-dimensional parameters of equation 3 are:
St
O ( 10
−
3
)
ε
O ( 10
−
3
)
δ
O ( 10
−
5
) G
O ( 10
−
2
)
With this values it seems reasonable to disregard the asymptotic analysis for x
∼
O(St) and x
∼
O(St
-1
) in a practical evaluation of seeding dynamics and to consider the error of equation 6 negligible. Now, let us separate the two components of the particle velocity in cylindrical coordinates: v r
( r ,
θ
)
=
St
3
2
ε −
1



− u
2
θ
( r ) r
−
G r
(
θ
)


+
O St v
θ
( r ,
θ
)
= u
θ
( r )
−
G
θ
(
θ
) St
(7)
3
2
ε −
1


+
O
( )
Being u θ
∼
O(1), the correction is negligible in the azimuthal term and the tracer particle can be considered to follow accurately. Nevertheless, considering the radial term it follows that, being u r
= 0, the perturbation cannot be neglected. Also, for the vortex parameters mentioned earlier, the gravity effect can be neglected and the particle velocity can be calculated as a function only of the radial coordinate: v r
( r )
=
St
3
2
ε −
1



− u 2
θ r
( r )


+
O v
θ
( r )
= u
θ
( r )
+
O
( )
−
4
(8)
In figure 2 we plot the two components field
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0 0
0 0.5
1 1.5
2 2.5
Radial Coordinate, r/Ro
3 3.5
4
Figure 2. Components of the velocity of a solid particle imbedded in a modified Rankine Vortex. The error bars (almost indistinguishable for the azimuthal component) show the error margin allowed in the asymptotic study.
3

PIV’01 Paper nnnn
3
Results
The analysis yet mentioned seems to be relevant to the implementation and normal use of the measurement techniques that lay on the visualization of a tracer, such as PIV or LDV. The relevant fact is not that the velocity obtained by looking at the tracer does not coincide with the fluid velocity. This error is small in the azimuthal velocity, which is the important magnitude. Also, the vorticity calculation is not affected by the particle inability of tracing the flow, as the radial velocity is function of the radial coordinate and thus it is irrotational. The problem is the possibility of finding large regions in the vortex core where no particle remains, thus rendering the measurement impossible. Some experimental results (for example Kompenhans and Stanislas (2000)) have shown that the inner zones of strong vortices get depleted of tracers, making the measurement very difficult or unaffordable. On the following we will try to state the major characteristics of this phenomenon.
According to equation 8, a good tracer imbedded in a Rankine Vortex will deviate from the flow motion in the radial direction. Therefore, the particle will follow a spiral path instead of following the circular streamlines of the flow.
This behavior can be observed in figure 3.
0.006
3
5.741337 10
0.004
0.002
3
0
0.002
0.004
3
5.622299 10
0.006
Figure 3. Trajectory of a solid particle imbedded in a modified Rankine Vortex
From equation 8 we can also obtain the radial position of the particle versus time. This has to be implemented avoiding the singularity in the vortex center (a particle located in the center of the vortex will not move from its equilibrium position). We have considered an initial position (t = 0) in the limit of the asymptotic study, r = St
-1
.
Figure 4 shows a plot of the non-dimensional quantities. Logarithmic scale is used to show the behavior for small radii.
10
4
3.5
t (r = 2Ro)
1 3
0.1
0.01
0.001
0.1
1 10
Time · Stokes Number, (t·St)/(Ro/Uo)
Figure 4. Radial motion of the particle versus time
2.5
2
1.5
1 t (r = Ro) t (r = 0.5Ro)
100
0.5
t (r = 0.25Ro)
0
0.001
0.01
0.1
Initial Radial Position, ro/Ro
1
Figure 5. Time for a particle to escape from the vortex core, t d
We can thus calculate the time that a tracer needs to leave a vortex core, t d
. Figure 5 shows, with non-dimensional variables, the time that a particle located in a certain initial position needs to arrive to certain characteristic radial positions (0.25R o
, 0.5R o
, R o
and 2R o
). For example, a vortex of U o
= 25m/s and R o
= 0.05m seeded with oil drops of
1.5
µ m of diameter (St = 3.4·10
-3
) will have its core (r < R o
) practically depleted of particles in approximately 0.3s, which seems a small time lapse. These values are typical for an Airbus Model tested in an wind tunnel, flying at
60m/s. The time of depletion will represent a traveling distance of the order of 20 wingspans.
4

PIV’01 Paper nnnn
More detailed information can be given. The evolution in time and space of the concentration of particles can be obtained solving the continuity equation for the particle concentration C:
∂
C
∂ t
+ ∇
( C · v )
=
∂
C
∂ t
+
∂
( C · v
∂ r r
)
=
0
(9)
Equation 9 can be easily integrated. Figure 6 shows the concentration evolution for tracers of 0.5
µ m and 1.5
µ m of diameter in the vortex already mentioned. The smaller particles show a slower tendency to be depleted from the vortex core. As the residence time goes inversely with the Stokes number and this goes with D
2
, the time behavior of the two types of particles differ by a factor 9.
1.6
1.6
1.4
1.4
1.2
1.2
1 1
0.8
0.6
0.4
0.2
0.07
0.17
0.27
0.37
0.47
0.57
0.77
1.00
0.8
0.6
0.4
0.2
0.01
0.03
0.04
0.06
0.13
0.24
0.38
0.58
1.00
0 0
0 0.5
1 1.5
2 2.5
Radial Coordinate, r/Ro
3 3.5
4 0 0.5
1 1.5
2 2.5
Radial Coordinate, r/Ro
3 3.5
a b
Figure 6. Concentration of particles as a function of radial location and for different times (in s) in a Rankine vortex with Uo =
25m/s and Ro = 0.05m for particles with diameters: a) 0.5
µ m; b)1.5
µ m.
4
Note that, in addition to the diminishment of particle concentration in the vortex core there is an increment at higher radii: This effect can also damage the acquisition of good measurements in such zones.
All this information has been implemented in a computer program that shows the evolution of seeding particles in a
PIV recording as time increases. Figure 7 shows several images obtained for a Rankine vortex seeded with 1.5
µ m particles. Note that even for small times the vortex core is strongly depleted of particles, thus rendering impossible the measurement using any seed-dependent technique. The accumulation of particles in an annular region outside the vortex core can be observed in the images for larger times.
The parameters involved in this behavior are, as have been shown, the vortex parameters, the fluid viscosity and the particle diameter and density. The first three are given by the experiment and generally they cannot be changed, while the particle density cannot be changed considerably. Therefore, to avoid this problem, the use of smaller particles seems customary. Nevertheless, the use of small particles has its drawbacks, as the ratio between the wavelength of the illuminating light and the particle diameter decreases. Another problem is that the usual equipments for the production of seeding particles have problems for diameters smaller than 1 micron.
t = 0s t = 0.1s t = 0.3s
5

PIV’01 Paper nnnn
t = 0.7s
t = 1.0s
Figure 7. Evolution of tracer particles of 1
µ m of diameter in a Rankine vortex (indicated by the black circle) with Uo = 25m/s and Ro = 0.05m.
Nevertheless, usually the seeding is not mono-dispersed and particles smaller than 1 micron are seeded in the flow.
To take this effect into account, we have considered a particle generator that produces particles with the distribution shown in figure 8, obtained using a modified Laskin nozzle (Kompenhans (2001)). The distribution can be modeled using a Log-Normal, Rossin-Rambler or Gamma Distribution. Here we have chosen a Gamma distribution because it provides better fits for both the smaller and the larger particles, as can be seen in the graph (Note that the experimental measurement for the smaller diameter might be over-dimensioned, carrying information of smaller particles, due to the problems existing in measuring such tiny particles).
1.E+07
Experimental Data
Rossin Rambler
Gamma
Log Normal
1.E+06
1.E+05
1.E+04
1.E-07 1.E-06
Particle Size (m)
Figure 8. Particle size distribution of a typical generator.
1.E-05
The process to obtain the images in figure 7 can be repeated with the information of the particle size distribution. In figure 9 we have plotted the results.
t = 0s t = 0.1s
6
t = 0.3s

PIV’01 Paper nnnn
t = 0.7s
t = 1.0s
Figure 9. Evolution of tracer particles from the data shown in figure 8 (mean diameter 1
µ m) in a Rankine vortex with Uo =
25m/s and Ro = 0.05m.
The comparison between the images of figure 7 and 9 show, for the larger times, a less dramatic depletion of the core in the second case. But for smaller times the effect is the opposite. The generator distribution for figure 9 is centered in the particle diameter used for figure 7. Then, for small times, when the particles of figure 7 have not gotten the time to leave the vortex, the effect of the comparison is set on the higher diameter particles of the distribution, while for larger times, we observe the smaller particles of the generator in the zone where the particles of figure 7 have already been depleted.
Nevertheless, the use of the smaller particles to soften the depletion effect is questionable. The ratio (particle diameter) / (light wavelength) increases and in this zone reaches order 1, so optical problems arise.
4
Conclusions
The evolution of heavy particles in a flow has been studied and applied to seeding particles for LDV or PIV measurements in airflows. The results show that strong vortices are depleted of seeding particles, while an increment of particles occur simultaneously in an annular zone outside the vortex core. The effect is less rigorous for smaller particle diameters (t d
∼
D
2
). Particles around a micron will be depleted from the core of a strong vortex (for example, an aircraft wing tip vortex) in less than one second.
The effect is studied and its main characteristics are pointed. The use of poli-dispersed seeding seems to mitigate the problem, as a considerable amount of particles have diameters smaller than 1 micron. Nevertheless, particles with diameters smaller than a micron have other well-known drawbacks, and relying on them for PIV measurements seems problematic.
The basic equations that run these processes have been implemented in a program that synthesize images able for
PIV measurements.
References
Kompenhans J, Stanislas M.(2000) Application of the PIV Measurement Technique for Aerodynamic Research in
EC Projects Workshop on European Research on Aerodynamic Engine/Airframe Integration for Transport
Aircraft, September 2000, Braunschweig, Germany
Kompenhans (2001) Private Communication
Lasheras J. Tio K-K (1994) Dynamics of a small spherical particle in steady two-dimensional vortex flows, Appl
Mech Rev 47, 6, 2, S61-S69.
Maxey M; Riley J (1982) Equation of motion for a small rigid sphere in a nonuniform flow. Phys Fluids 26 (4) p.
883-889.
7