Particle-Turbulence Interaction
in a Homogeneous, Isotropic
Turbulent Suspension
Christian Poelma
Gijs Ooms1
e-mail: [email protected]
Laboratory for Aero- and Hydrodynamics,
J.M. Bergerscentrum,
Delft University of Technology,
Mekelweg 2,
2628 CD Delft, The Netherlands
1
A review is given of numerical, analytical, and experimental research regarding the
two-way coupling effect between particles and fluid turbulence in a homogeneous, isotropic turbulent suspension. The emphasis of this review is on the effect of the suspended
particles on the spectrum of the carrier fluid, in order to explain the physical mechanisms
that are involved. An important result of numerical simulations and analytical models
(neglecting the effect of gravity) is that, for a homogeneous and isotropic suspension with
particles with a response time much larger than the Kolmogorov time scale, the main
effect of the particles is suppression of the energy of eddies of all sizes. However for a
suspension with particles with a response time comparable to or smaller than the Kolmogorov time, the Kolmogorov length scale will decrease and the turbulence energy of
(nearly) all eddy sizes increases. For a suspension with particles with a response time in
between the two limiting cases mentioned above the energy of the larger eddies is suppressed, whereas the energy of the smaller ones is enhanced. Attention is paid to several
physical mechanisms that were suggested in the literature to explain this influence of the
particles on the turbulence. In some of the experimental studies, certain results from
simulations and models have, indeed, been confirmed. However, in other experiments
these results were not found. This is attributed to the role of gravity, which leads to
turbulence production by the particles. Additional research effort is needed to fully understand the physical mechanisms causing the two-way coupling effect in a homogeneous, isotropic, and turbulently flowing suspension. This review contains 47
references. 关DOI: 10.1115/1.2130361兴
Introduction
The occurrence of dispersed two-phase flows in nature and industrial applications is abundant. Despite a significant research
interest, there are still several open questions in this topic, limiting
a good quantitative prediction of these flows. In this publication, a
review is given of recent progress relating to one of the more
fundamental problems: the so-called two-way coupling between
the dispersed phase and turbulence. Two-way coupling refers to
the effects of the fluid on the particles and vice versa. The review
is limited to rigid particles in homogeneous and isotropic turbulence, in order to avoid further complications. For the same reason, no work on shear flows is incorporated. Even though inclusion of these publications would greatly increase the amount of
available data, it would also significantly complicate the analysis,
since, among other phenomena, shear-induced turbulence production and particle migration might obscure other effects.
Because of the growth of computer power, detailed direct numerical simulations 共DNS兲 of the behavior of particles in the turbulent fluid velocity field and of the two-way coupling effect became possible. Also, theoretical models were developed that try to
capture the main physical mechanisms occurring in turbulent suspensions. Moreover, because of the progress in experimental techniques, new and accurate measurement results became available.
We will give a review of publications about the two-way coupling effect in turbulent suspensions that appeared in the literature
during the last decade. It is, of course, impossible to discuss all
important publications. However, we hope that the review will
give a clear picture of the progress in understanding of turbulent
suspensions. Attention will be paid to numerical simulations, theoretical models, and experiments. Emphasis will be placed on the
influence of the particles on the turbulent kinetic energy spectrum
1
To whom correspondence should be addressed.
Transmitted by Assoc. Editor A. Tsinober.
78 / Vol. 59, MARCH 2006
of the carrier fluid. We know that turbulence spectra are not sufficient to characterize turbulence completely. However, in the literature particular attention was given to the influence of particles
on the turbulence spectrum in order to derive a physical understanding of the two-way coupling effect.
In the first part of this review we will integrate the knowledge
from numerical simulations with respect to the dependence of the
turbulence spectrum of the carrier fluid on the presence of the
particles in the fluid. In the second part such an integration will be
given with respect to theoretical explanations given in the literature on the two-way coupling effect, in general, and the dependence of the turbulence spectrum on the particles, in particular.
The third part is devoted to a comparison between experimental
data and results from numerical simulations and theoretical
models.
Several good reviews about particle-laden turbulent flows were
published during the past decade, see for instance Hetsroni 关1兴,
Elghobashi 关2兴, Crowe et al. 关3兴, and Mashayek and Pandya 关4兴.
Our review is different, in our opinion, in the sense that we pay
particular attention to the physical mechanisms that play an important role in the interaction between particles and fluid turbulence in a turbulently flowing suspension. In recent years some
interesting publications have been written in which detailed information is given about these mechanisms.
2
Relevant Parameters
We define a number of parameters that are of importance for
turbulently flowing suspensions and that will be used in this review. The volume fraction of the dispersed phase 共the particles兲 is
defined as
Copyright © 2006 by ASME
Transactions of the ASME
␦Vd
␦V
␺=
共1兲
where ␦Vd is the volume of the dispersed phase in the volume ␦V
of the suspension 共assumed large enough to ensure averaging兲.
The ratio of the average particle spacing 共dspacing兲 and the particle
diameter 共d p兲 is related to the volume fraction by
dspacing
dp
冉 冊
␲
=
6␺
1/3
共2兲
The particle mass loading is given by
␾=␺
␳p
␳f
共3兲
in which ␳ p and ␳ f are the densities of the particles and the carrier
fluid, respectively. The particle response time to changes in the
surrounding fluid is expressed by the Stokes time
␶p =
2␳ pd2p
9␳ f ␯
共4兲
where ␯ is fluid kinematic viscosity. For the fluid phase the Kolmogorov time is defined by
␶ k = ␩ 2/ ␯
共5兲
with ␩ the Kolmogorov length scale. The integral time scale is
given by
␶⌳ = ⌳/u⬘
共6兲
in which ⌳ is the integral length scale of turbulence and u⬘ the
root-mean-square value of the turbulent velocity fluctuations. The
Stokes number is defined in this review as the ratio of the particle
time scale and the Kolmogorov time scale
St = ␶ p/␶k
共7兲
For St→ 0, the particle behaves as a fluid tracer. For St→ ⬁, the
particle is unresponsive to the fluctuations of the flow. Obviously,
the physically most interesting situation occurs when it approaches unity: particles follow the large scale fluid motions.
Finally, the flow regime around the particles can be characterized by means of a Reynolds number
Rep =
uTVd p
␯
共8兲
In this equation, uTV represents the terminal velocity 共the velocity a particle attains falling in a quiescent medium兲. Alternatively, it can be defined using, e.g., the root-mean-square value of
the free-stream turbulence. In has been postulated by Hetsroni 关1兴
that Rep can be used to determine whether particles attenuate
共Rep ⬍ 100兲 or enhance 共Rep ⬎ 400兲 the fluid turbulence level.
3
Direct Numerical Simulations
3.1 Effect of Particles on Turbulence Spectrum. An early
indication about the influence of the particles on the turbulence
spectrum was given by Squires and Eaton 关5,6兴. They used direct
numerical simulation to study statistically stationary, homogeneous, isotropic turbulence. They considered particle motion in
the Stokes regime. Gravitational settling was neglected. Computations were performed using both 323 and 643 grid points. The
particles were treated as point particles. To achieve the stationary
flow, a steady nonuniform body force was added to the governing
equations. Particle sample sizes up to 106 were used in the simulations. Mass loadings of 0.1, 0.5, and 1.0 were considered. They
used the following values for the ratio of the particle response
time 共␶ p兲 and the Kolmogorov time scale of turbulence
Applied Mechanics Reviews
Fig. 1 Effect of mass loading on spatial energy spectra for
„␶p / ␶k = 1.5…. With increasing mass loading the energy at large
wave numbers increases relative to the energy at small wave
numbers. From Ref. †5‡.
共␶k兲 : ␶ p / ␶k = 0.75, 1.4, 1.5, 5.2, 7.5, and 15.0. Squires and Eaton
calculated the effect of the mass loading on the spatial turbulent
kinetic energy spectrum of the carrier fluid. The 共dimensionless兲
spatial spectrum for ␶ p / ␶k = 1.5 as function of the 共dimensionless兲
wave number for different mass loadings is shown in Fig. 1. E共␬兲
is the turbulent energy spectrum as function of wave number. As
mentioned earlier ␩ is the Kolmogorov length and ␾ the mass
loading. The spectra are normalized using q2, the total turbulent
kinetic energy for each particular case. It can be seen that with
increasing mass loading, the energy at large wave numbers increases relative to the energy at small wave numbers. As Squires
and Eaton found that the total turbulent energy decreases with
increasing mass loading, it can be concluded that at small wave
numbers 共where most of the energy is located兲 the energy decreases with increasing mass loading compared to the particle-free
case. Whether the energy at large wave numbers decreases or
increases with respect to the particle-free case cannot directly be
concluded from the results of Squires and Eaton. For that it would
be necessary to multiply the spectra of Fig. 1 with the total energy
q2. The relative increase of the distribution of energy at large
wave numbers with respect to the energy at small wave numbers
was found for all particle response times used in the simulations.
More details about the influence of particles on the spatial turbulence spectrum of the fluid became available via the work of
Elghobashi and Truesdell 关7兴. In contrast to Squires and Eaton,
they examined turbulence modulation by particles in decaying isotropic turbulence. A point-particle approximation was again made.
They used the particle equation of motion derived by Maxey and
Riley 关8兴. Like Squires and Eaton they found that the coupling
between the particles and fluid results in an increase in the turbulent energy at the large wave numbers relative to the energy at
small wave numbers. Moreover, they concluded from their calculations that the large wave number energy for a flow with particles
共with a sufficient small response time兲 is even larger than the large
wave number energy for the particle-free case at the same time in
the decay process.
Like Squires and Eaton, also Boivin et al. 关9兴 made a detailed
DNS study of the modulation of statistically stationary, homogeneous, and isotropic turbulence by particles. Gravitational settling
was neglected and the particle motion was assumed to be governed by drag. The ratio of the particle response time to the Kolmogorov time scale had the following values: 1.26, 4.49, and
11.38, and the particle mass loading was equal to: 0.2, 0.5, and
MARCH 2006, Vol. 59 / 79
Fig. 2 Turbulence kinetic energy spectrum for ␶p / ␶k = 1.26: ␾
= 0 „solid line…, 0.2 „dotted…, 0.5 „dashed…, and 1.0 „dasheddotted…. With increasing mass loading the energy at large wave
numbers is enhanced relative to the energy at small wave numbers. Data taken from Ref. †9‡.
1.0. The velocity field was made statistically stationary by forcing
the small wave numbers of the flow. Again the effect of particles
on the turbulence was included by using a point-force approximation. Fluid turbulence spatial energy spectra, derived by Boivin et
al. 关9兴 for different mass loadings are shown in Fig. 2 for ␶ p / ␶k
= 1.26 and in Fig. 3 for ␶ p / ␶k = 11.38. Note that the figures as they
are represented here are replotted with double-logarithmic axes, to
facilitate comparison to the other figures. In their simulations the
dimensionless maximum value of k is about equal to the number
of grid points in each direction of their cubic simulation domain.
So the region around k = 1 represents the energy-containing eddies
and we can say that the wave number along the horizontal axis is
made dimensionless by means of the integral length scale. In general, there is a similar damping effect on the smaller wave numbers of the fluid turbulence by both particles with a large response
time and particles with a small response time. At the larger wave
numbers the turbulence kinetic energy is attenuated by particles
Fig. 3 Turbulence kinetic energy spectrum for ␶p / ␶k = 11.38: ␾
= 0 „solid line…, 0.2 „dotted…, 0.5 „dashed…, and 1.0 „dasheddotted…. With increasing mass loading the energy at all wave
numbers decreases. Data taken Ref. †9‡.
80 / Vol. 59, MARCH 2006
Fig. 4 Kinetic energy spectrum of the carrier fluid at t = 5.0.
The cross-over wave number increases with increasing particle
response time. From Ref. †11‡.
with a large response time, but increased by particles with a small
response time. It should be noted that in Figs. 1–3 the highest
wave numbers appear to be contaminated with noise, since they
show a distinct “plateau” 共Fig. 1兲 or even increase 共Figs. 2 and 3兲
at these wave numbers. Nevertheless, even if this part of the graph
is ignored, the trends mentioned above remain.
A next step in achieving information about the dependence of
the turbulence spectrum on the presence of particles in a turbulent
suspension was provided by Sundaram and Collins 关10兴. Like Elghobashi and Truesdell they performed DNS simulations of
particle-laden isotropic decaying turbulence. The particle response
time was in the range 1.6⬍ ␶ p / ␶k ⬍ 6.4. The ratio of the particle
density and fluid density was of the order 103. The drag force on
the particles was described by Stokes law, and the influence of the
gravitational force was neglected. The DNS results showed again
that the turbulent energy spectrum of the fluid is reduced at small
wave numbers and increased at large wave numbers 共compared to
the particle-free case兲 by the two-way coupling effect. They also
concluded that the location of the cross-over point 共the wave number where the influence of the particles changes from a
turbulence-damping effect to a turbulence-enhancing one兲 is
shifted toward larger wave numbers for larger values of the particle response time ␶ p.
In their DNS calculations Ferrante and Elghobashi 关11兴 fixed
both the volume fraction 共␺ = 10−3兲 and mass fraction 共␾ = 1兲 for
four different types of particles, classified by their ratio of the
particle response time and the Kolmogorov time scale of turbulence. The ratio ␶ p / ␶k had the values 0.1, 0.25, 1.0, and 5.0. The
ratio of the particle density 共␳ p兲 and the fluid density 共␳ f 兲 is 103.
In Fig. 4 the spatial turbulent energy spectrum E共t , ␬兲 for the
carrier fluid in the suspension is given 共for the case without gravity兲 at a certain moment during the decay process. ␬ is the wave
number made dimensionless by means of the the integral length
scale L. In the figure the result indicated by case A is for the
particle-free flow, the results indicated by cases B, C, D, and E are
for the carrier fluid in the suspension with particles of increasing
response time 共␶ p / ␶k = 0.1, 0.25, 1.0, and 5.0兲, respectively. Microparticles 共case B兲 increase E共t , ␬兲 relative to the particle-free flow
共case A兲 at wave numbers ␬ 艌 12 and reduce E共t , ␬兲 relative to
case A for ␬ ⬍ 12, such that E共t兲 = 兰E共t , ␬兲d␬ in case B is larger
than in case A 共the particle-free case兲. Also for the cases C, D, and
E, the particles dampen the turbulence at small wave numbers
compared to the particle-free flow and enhance the turbulence at
Transactions of the ASME
large wave numbers. However the cross-over wave number increases with increasing particle response time. As can be seen
from Fig. 4 large particles 共case E兲 contribute to a faster decay of
the turbulent kinetic energy by reducing the energy content at
almost all wave numbers, except for ␬ ⬎ 87, where a slight increase of E共t , ␬兲 occurs.
3.2 Physical Mechanisms. In the publication of Squires and
Eaton 关6兴, a first proposal is made to explain the physical mechanism responsible for the nonuniform distortion of the turbulence
energy spectrum by particles. In their opinion this nonuniform
distortion is due to the preferential concentration of particles in
the turbulent flow field. They showed that particles with a small
response time 共␶ p / ␶⌳ Ⰶ 1兲 exhibit significant effects of preferential
concentration in regions of low vorticity and high strain rate. The
effect of high concentrations of particles in these regions leads to
an increase in small-scale turbulent velocity fluctuations. This production of small-scale fluctuations subsequently causes the viscous dissipation rate in the carrier fluid to be increased 共for particles with a small response time兲. Squires and Eaton also showed
that preferential concentration causes a significant disruption of
the balance between production and destruction of dissipation,
again leading to a selective modification of the turbulence spectrum. More research is, in our opinion, needed to fully understand
the contribution of preferential concentration to the two-way coupling effect.
Boivin et al. 关9兴 found that in a turbulently flowing suspension
the cascade process 共energy transport from the large to the small
eddies兲 is influenced by the particles. They calculated the spectrum of the fluid-particle energy exchange rate. In the small wave
number part of this spectrum the turbulent fluid motion transfers
energy to the particles, i.e., the particles act as a sink of kinetic
energy. At larger wave numbers of the spectrum the energy exchange rate is positive, indicating that particles are capable of
adding kinetic energy to the turbulence. This energy, “released”
by the particles, is not immediately dissipated by viscous effects
but is, in fact, responsible for the relative increase of small-scale
energy compared to the particle-free case observed in the energy
spectra for particles with a small response time.
A different type of physical mechanism for the two-way coupling effect was proposed in the publication by Ferrante and Elghobashi 关11兴. They provided explanations for the behavior of microparticles and large particles. Because of their fast response to
the turbulent velocity fluctuations of the carrier fluid, the microparticles are not ejected from the vortical structures of their initial
surrounding fluid. The inertia of the microparticles causes their
velocity autocorrelation to be larger than that of the surrounding
fluid. Since the microparticles’ trajectories are almost aligned with
fluid points’ trajectories, and their kinetic energy is larger than that
of the surrounding fluid, the particles will transfer part of their
own energy to the fluid. 共The idea is that the microparticles accelerate quickly in an eddy to the velocity of the surrounding fluid.
After that acceleration period the small but finite inertia of the
particles causes, on average, a larger kinetic energy for the particles than the surrounding fluid, as the particles tend to retain
their velocity for a longer time. Much more quantitative details,
based on their DNS calculations, are given in the publication by
Ferrante and Elghobashi.兲 On the other hand the microparticles
increase the viscous dissipation rate relative to that of the particlefree flow. The reason is that the microparticles remain in their
initially surrounding vortices, causing these vortical structures to
retain their initial vorticity and strain rates longer than for the
particle-free flow. 共Again, it is the small but finite inertia of the
particles that causes this effect.兲 The net effect is positive for the
turbulent kinetic energy of the carrier fluid, as the gain in energy
due to the transfer of energy from the particles is larger than the
increase in viscous dissipation. The DNS calculations also show
that the microparticles directly interact with the small scales of
motion, augmenting their energy content. The triadic interaction
Applied Mechanics Reviews
of wave numbers then alters the energy content of the other scales
of motion, such that after few integral time scales the energy
spectrum is modified at all the wave numbers as compared to the
particle-free case.
For large particles the explanation is different: because of their
significant response time, large particles do not respond to the
velocity fluctuations of the surrounding fluid as quickly as microparticles do, but rather escape from their initial surrounding fluid
共crossing the trajectories of fluid points兲. Large particles retain
their kinetic energy longer than the surrounding fluid. However,
because of the “crossing trajectories” effect, the fluid velocity autocorrelation is larger than the correlation between the particle
velocity and the fluid velocity, causing a transfer of energy from
the fluid to the particles. On the other hand, large particles reduce
the lifetime of eddies, causing a viscous dissipation rate 共which is
smaller than for the particle-free flow兲. 共It is shown that large
particles interacting with a clockwise vortex create a counterclockwise torque on the fluid, which, in turn, reduces the vorticity.兲 The net result of the two opposing effects is a reduction of
turbulent kinetic energy for a suspension with large particles at
nearly all wave numbers relative to the kinetic energy for the
particle-free turbulent flow. Again for more quantitative details the
publication of Ferrante and Elghobashi should be consulted. We
think that their analysis is a very interesting and important step in
the direction of a physical understanding of the two-way coupling
effect.
3.3 Effect of Finite Particle Size. Numerical work in which
the particles are fully resolved is slowly becoming available in the
literature. An example of this type is the work by ten Cate 关12,13兴,
who carried out numerical simulations of a 共homogeneous and
isotropic兲 turbulent suspension taking into account the finite size
of the particles 共by satisfying the no-slip boundary condition at the
particle-fluid interface兲. Nevertheless, the short-range interactions
between the particles had to be added explicitly, since they could
not be resolved on the grid. For the generation of sustained turbulent conditions a spectral forcing scheme was implemented using the lattice-Boltzmann technique. In these simulations the particle volume concentration is varied between 2% and 10% 共which
is probably no longer in the two-way coupling regime兲 and the
particle to fluid density ratio was between 1.15 and 1.73. The
Taylor-scale Reynolds number was 61. Results were presented
concerning the influence of the particle phase on the turbulent
energy spectrum. Fluid motion was generated at length scales in
the range of the particle size, which resulted in a strong increase
in the rate of energy dissipation at the small length scales. With
respect to the turbulent energy spectrum, little difference was
found between the spectra with and without particles at the lowest
wave numbers. At the intermediate wave numbers the particles
reduced the fluid kinetic energy 共see Fig. 5兲. At the larger wave
numbers the spectra crossed at a clear cross-over or pivot point,
and the particles increased the kinetic energy with respect to the
particle-free case.
In contrast to suspensions studies, an alternative is the very
fundamental approach of studying a single particle. Examples of
this type of work, in which a fully resolved particle in a turbulent
flow is studied, can be found in, e.g., publications by Mittal 关14兴
and Bagchi and Balachandar 关15兴. In the former work, the production of turbulence due to vortex shedding is studied. It is found
that this can contribute to turbulence production when Rep ⬎ 300,
yet only when the free-stream turbulence level is sufficiently low.
In the work by Bagchi and Balachandar, various forces 共inertial,
viscous, history兲 are studied, in detail, in homogeneous and straining flow. This type of work may also clarify to what level the
point-force approximation, that is used in most numerical and
theoretical work, is valid.
3.4 Discussion of DNS Results. Some general conclusions
can be drawn from the literature on numerical simulations of turbulently flowing suspensions. In the numerical simulations, the
MARCH 2006, Vol. 59 / 81
force fp共t , r兲 caused by the friction of the fluid with the particles,
which is often approximated by
fp共t,r兲 =
␾␳ f
␶p
关v共t,r兲 − u共t,r兲兴
共10兲
Here v共t , r兲 is the velocity field of the particles, considered as a
continuous medium with density m p / l3 = ␳ f ␾, where m p is the
mass of a particle, l3 suspension volume per particle, and ␾ the
mass loading parameter
␾ = m p/ ␳ f ᐉ 3
Fig. 5 Scaled energy spectra „E = E„k… / „⑀2/3␩5/3…… of the singlephase simulations „dotted… and two-phase simulations „continuous…. The wavenumber is made dimensionless with the particle wave number: kp = 2␲ / dp. ␾ = 0.005, ␳p / ␳f = 1.414, StK
= 0.207. Data taken from Ref. †13‡.
effect of the turbulence generation by the particle wakes and by
the vortices shed by the particles was not taken into account. It
would also have been difficult to include this effect of turbulence
generation, as the particles in these publications were treated as
point-particles 共apart from the work by ten Cate兲. This is a significant simplification, and the results should thus be interpreted
with care. From the simulations it can be concluded that for a
suspension with particles with a response time much larger than
the Kolmogorov time scale, the main effect of the particles is
suppression of the energy of eddies of all sizes 共at the same energy
input into the suspension as for the particle-free case兲. So for such
a suspension the total turbulent energy of the carrier fluid will be
smaller than the total turbulent energy of the fluid for the particlefree case 共at the same energy input兲. However, for a suspension
with particles with a response time comparable to or smaller than
the Kolmogorov time, the Kolmogorov length scale will decrease
and the turbulence energy of 共nearly兲 all sizes increases. In that
case the total turbulent energy of the carrier fluid can be larger
than the total turbulent energy of the fluid for the particle-free
case. For a suspension with particles with a response time in between the two limiting cases mentioned above, the energy of the
larger eddies is suppressed, whereas the energy of the smaller
ones is enhanced. Several physical mechanisms have been proposed to explain these effects. It is not possible, at the moment, to
decide which mechanism is the most important one, or whether
they all contribute to explain the influence of particles on the
turbulence of the carrier fluid in a suspension. More research is
needed on this subject. It is also desirable to extend the DNS work
with finite-size particles to investigate, in detail, the important
influence of turbulence generation by particle wakes and vortices.
4
4.1 Introduction. The starting point for analytical models described in the literature is often the Navier-Stokes 共NS兲 equation
for the velocity of the carrier fluid u共t , r兲 with external forces
␳f
冋
册
⳵
+ 共u · ⵜ兲 − ␯ⵜ2 u + ⵜp = f p + f
⳵t
共9兲
p共t , r兲 is the pressure and ␳ f is the fluid density. The random
vector field f共t , r兲 represents the stirring force responsible for the
maintenance of the turbulent flow. The equation includes also the
82 / Vol. 59, MARCH 2006
The validity to represent f p共t , r兲 in the form of Eqs. 共10兲 and 共11兲
is based on the assumption of space homogeneity of the particle
distribution. If the particles are not homogeneously distributed,
one can add additional equations, e.g., the number density. It is
assumed that the particles are small enough for the Stokes drag
law to be valid. For the simple case of monosize particles moving
under the influence of the Stokes drag, the equation of motion for
the particles 共considered as a continuous phase兲 has the following
form 共see, for instance, 关16兴兲
冉 冊冋
mp
l3
4.2
册
⳵
+ 共v · ⵜ兲 v = − f p
⳵t
共12兲
Physical Mechanisms.
4.2.1 Two-Fluid Models. The treatment mentioned above of
the particle phase as a continuum, i.e., the assumption of two
interpenetrating fluids, is the basis for the so-called two-fluid
models. Many publications have been written about the two-way
coupling effect using a version of the two-fluid model, and good
reviews about this type of research are available 共see, for instance,
关4兴兲. As mentioned before, we will concentrate on publications in
which special attention is given to the influence of the particles on
the turbulence in a homogeneous, isotropic suspension and in
which an attempt is made to understand the underlying physical
mechanism.
The equations given above were used by Baw and Peskin 关17兴
to derive a set of “energy-balance” equations for the following
functions:
E f f 共k兲—energy spectrum of the fluid turbulence 共E共k兲 in the
nomenclature used in other publications兲
• E f f,p共k兲—energy spectrum of the fluid turbulence along a
particle trajectory
• E fp共k兲—fluid-particle covariance spectrum
• E pp共k兲—particle energy spectrum
•
In the balance equations, the following energy transfer functions
occur:
•
•
•
•
Theoretical Models
共11兲
•
T f f,f 共k兲—energy transfer in fluid turbulence
T fp,f 共k兲—transfer of fluid-particle correlated motion by the
fluid turbulence along the particle path
T fp,p共k兲—transfer of fluid-particle correlated motion by the
particles
T pp共k兲—transfer of particle-particle correlated motion by the
particle motion
⌸q,f 共k兲—fluid-particle energy exchange rate.
Baw and Peskin made a set of simplifying assumptions in order to
be able to analyze the balance equations. First, they assumed that
the particles do not respond to the fluid velocity fluctuations due
to their 共very large兲 inertia. Therefore
E f f,p共k兲 = E f f 共k兲
共13兲
and
Transactions of the ASME
T fp,f 共k兲 = T fp,p共k兲 = T pp,p共k兲 = 0
共14兲
This assumption is, of course, not realistic for particles satisfying
the Stokes’ approximation. Their next assumption
⌸q,f = ␾关E fp共k兲 − E f f,p共k兲兴/␶ p
共15兲
may be understood as a statement that the fluid-particle exchange
rate is statistically the same for all scales characterized by a
k-independent frequency ␥ p = ␾ / ␶ p. This is reasonable for particles with very large inertia, but then Stokes law is not valid. For
particles satisfying Stokes law, assumption Eq. 共15兲 has to be
replaced with a more realistic, k-dependent frequency ␥ p共k兲.
A serious difficulty in the derivation of the energy-balance
equations is how to find a closure expression for third-order velocity correlation functions, responsible for the various energy
transfer functions. Baw and Peskin assumed that T f f,f 共k兲 can be
expressed similarly as in the case of a pure 共single phase兲 flow
T f f,f 共k兲 = −
5/3
d ⑀1/3
f k E f f 共k兲
␣
dk
共16兲
where ⑀ f is the viscous dissipation in the pure fluid 共without particles兲 and ␣ is the so-called Kolmogorov constant. This assumption is questionable. According to the spirit of the RichardsonKolmogorov cascade picture of turbulence, one may express
inertial range objects, like T f f,f 共k兲 in terms of again inertial range
quantities, like k, E f f 共k兲, and ⑀共k兲 共the energy flux in k space兲. In
a single-phase flow, indeed ⑀共k兲 = ⑀ f . However, this is not the case
for a turbulent suspension due to the fluid-particle energy exchange, given by Eq. 共15兲. With this simplified model, Baw and
Peskin predicted the following influences on the energy spectrum
of the fluid turbulence due to the particles:
•
•
•
a decrease of the energy in the energy-containing range of
the spectrum
an increase in the inertial range of the spectrum
a decrease in the viscous dissipation range.
Boivin et al. 关9兴 used the same model as Baw and Peskin 关17兴.
They also applied assumptions similar to Eqs. 共15兲 and 共16兲. Fortunately, they took into account the response of the particles to the
turbulent velocity fluctuations by relaxing assumptions of Eqs.
共13兲 and 共14兲 and also accounted for the very important physical
effect of the energy dissipation due to the drag around the particles. For that reason they approximated T f f,f 共k兲 and T fp,f 共k兲 as
follows:
T f f,f 共k兲 = −
d
关⑀ f − ⌸q,f 共k兲兴1/3k5/3E f f 共k兲
␣dk
共17兲
T fp,f 共k兲 = −
d
关⑀ f − ⌸q,f 共k兲兴1/3k5/3E fp共k兲
␣dk
共18兲
and
Note that this closure has the same weakness as Eq. 共16兲, involving the dissipation range value ⑀ f . With the above-described
changes with respect to the model as developed by Baw and Peskin, Boivin et al. found an increase in the viscous dissipation
range of the fluid turbulence spectrum for small values of the
particle response time ␶ p.
4.2.2 Single-Fluid Models. Some analytical models were developed, in which the turbulent suspension was treated as a single
fluid with effective 共frequency- and wave-number-dependent兲
physical properties. Felderhof and Ooms 关18兴 共see also the
follow-up publications Ooms and Jansen 关19兴 and Ooms et al.
关20兴兲 developed an analytical model for the dynamics of a suspension of solid spherical particles in an incompressible fluid based
on the linearized version of the Navier-Stokes equation. In particular, the effect of the particles-fluid interaction on the effective
Applied Mechanics Reviews
Fig. 6 Turbulent kinetic energy spectrum of the carrier fluid
„␶ É 1.5…. The cross-over wave number increases with increasing particle response time in accordance with numerical simulations. From Ref. †22‡.
transport coefficients and on the turbulent energy spectrum of the
suspension was studied. Also the hydrodynamic interaction between the particles and the influence of the finite size of the particles were incorporated. However, it is needless to say that the
nonlinearity of the Navier-Stokes equation is of crucial importance in the problem of turbulence. The authors were well aware
of this problem, but 共as mentioned兲 wanted to study, in particular,
the influences of the particle-particle hydrodynamic interaction
and of the finite particle size at a relatively high particle volume
concentration. In order to improve the turbulence modeling, they
included in one of their publications a wave-number-dependent
turbulence viscosity. However, as the turbulence cascade process
was not properly accounted for, they never found an increase of
the turbulent kinetic energy at large wave numbers for particles
with a small response time 共as found by numerical simulations
and some experiments兲. The importance of this work lies in the
description of the particle-particle hydrodynamic interaction at
high particle volume fractions and of the finite particle size.
L’vov et al. 关21兴 develop a one-fluid theoretical model for a
stochastically stationary, homogeneous, isotropic turbulently
flowing suspension. It is based on a modified Navier-Stokes equation with a wave-number-dependent effective density of suspension and an additional damping term representing the fluidparticle friction 共described by Stokes’ law兲. It can be considered as
an improvement of the work of Felderhof and Ooms because in
their model, L’vov et al. incorporated a description of the cascade
process of turbulence. Ooms and Poelma 关22兴 extended the theoretical model in such a way that it can be applied to a decaying,
homogeneous, and isotropic turbulent suspension and can be compared to the DNS data of Ferrante and Elghobashi 关11兴. They
calculated, for instance, the energy spectra E共␬兲 for the carrier
fluid in the suspension for the five cases 共A, B, C, D, and E兲
discussed by Ferrante and Elghobashi and compared the predictions with their DNS results. Here only the results are shown for
the particle-free flow 共case A兲, the microparticles 共case B兲, and the
large particles 共case E兲. The results for the other particles 共cases C
and D兲 are in between those for cases B and E. For a certain
moment in the decay process, the results are shown in Fig. 6.
There is a difference between model predictions 共Fig. 6兲 and the
DNS results 共Fig. 4兲 at small values of ␬ 共␬ ⬇ 1兲. This is due to a
difference in the boundary condition at the small wave-number
end between the two methods. It is clear from Fig. 6 that the
particles dampen the turbulence for small values of ␬ 共large eddies兲 and enhance the turbulence for large values of ␬ 共small
MARCH 2006, Vol. 59 / 83
eddies兲. However, there is a difference. The microparticles 共case
B兲 enhance the turbulence over a much larger range of ␬ values
than the large particles 共case E兲. For microparticles the enhancement is so strong that the total energy over all eddies is larger than
for the particle-free flow. That is not the case for the large particles. The cross-over wave number 共the wave number where the
influence of the particles changes from a turbulence-damping effect to a turbulence-enhancing one兲 increases with increasing particle response time. This result is the same as found in the DNS
calculations.
The following physical mechanism is proposed by L’vov et al.
关21兴 to explain the obtained results. According to them an important effect of the particles is, that they increase the effective density of the suspension. As the dynamic viscosity is not much influenced at low values of the particle volume fraction, the
kinematic viscosity of the suspension will decrease compared to
the kinematic viscosity for the particle-free case. This will decrease the Kolmogorov length scale and hence elongate the inertial subrange of the energy spectrum. There is a second effect that
is, in particular, important in the inertial subrange. There are two
competing effects in that subrange: an energy suppression due to
the fluid-particle friction and an energy enhancement during the
cascade process due to the decrease of the effective density of the
suspension with decreasing eddy size. Particles become less involved in the eddy motion with decreasing eddy size. A more
detailed investigation of this effect has been made by L’vov et al.,
and it is shown that this effect can lead to a significant enhancement of the turbulence in the inertial subrange dependent on conditions, such as the ratio of the particle response time the integral
time scale. It is the combination of the two effects mentioned
above, that explains the phenomena observed in Fig. 6 in terms of
the model.
Druzhinin 关23兴 共see also Druzhinin and Elghobashi 关24兴兲 studied the two-way coupling effect on the decay rate of isotropic
turbulence laden with solid spherical microparticles whose response time is much smaller than the Kolmogorov time scale. An
asymptotic analytical solution was obtained for the kinetic energy
spectrum of the carrier fluid, which indicated that the two-way
coupling increases the fluid inertia in the fluid momentum equation by the factor 共1 + ␾兲. 共␾ is the particle mass fraction.兲 The net
result is a reduction of the decay rate of turbulence energy as
compared to that of the particle-free turbulence. The analytical
solution was also extended up to the first order in 共␶ p / ␶k兲 and is
applicable for particles with small but finite inertia.
4.2.3 Phenomenological Models. Yuan and Michaelides 关25兴
presented a model for the turbulence modification in particle laden
flows based on the interaction between particles and turbulence
eddies. Two predominant mechanisms for the suppression and
production of turbulence were identified: 共i兲 the acceleration of
particles in eddies is the mechanism for the turbulence reduction
and 共ii兲 the flow velocity disturbance due to the wake of the particles or the vortices shed by the particles is taken to be the responsible mechanism for turbulence enhancement. The effect of
the two mechanisms were combined to yield the overall turbulence intensity modulation. The model exemplifies the effect of
several variables, such as particle size, relative velocity, Reynolds
number, ratio of densities, etc. A comparison to available experimental data confirmed that the model predicts rather well the observed changes in turbulence intensity. In the model no attention
is devoted to the influence of the particles on the turbulent energy
spectrum of the carrier fluid. Although the model shows fair
agreement with experimental data for the turbulence intensity, it
involves ad hoc modeling procedures leaving many questions regarding turbulence modulation unanswered. It has to be emphasized, however, that the publication by Yuan and Michaelides is
one of the few that pays attention to the generation of turbulence
in the wake behind the particles for large values of the particle
84 / Vol. 59, MARCH 2006
Reynolds number. Similar models that also considers the turbulence generation have been published by Yarin and Hetsroni 关26兴
and Crowe 关27兴.
4.3 Discussion of Results From Theoretical Models. As a
general conclusion from the theoretical work, it can be stated that
the more recent analytical models for a homogeneous, isotropic,
turbulent suspension predict the same effect of the particles on the
turbulent energy spectrum of the carrier fluid as the effect predicted by direct numerical simulations. For a suspension with particles with a response time much larger than the Kolmogorov time
scale, the main effect of the particles is suppression of the energy
of eddies of all sizes 共at the same energy input into the suspension
as for the particle-free case兲. Thus for such a suspension, the total
turbulent energy will be smaller than the total turbulent energy for
the particle-free case 共at the same energy input兲. However, for a
suspension with particles with a response time comparable to or
smaller than the Kolmogorov time, the Kolmogorov length scale
will decrease and the turbulence energy of 共nearly兲 all sizes increases. In that case the total turbulent energy can be larger than
the total turbulent energy for the particle-free case. For a suspension with particles with a response time in between the two limiting cases mentioned above, the energy of the larger eddies is
suppressed, whereas the energy of the smaller ones is enhanced.
An interesting point is that it seems possible to give different
physical explanations for the influence of the particles on a 共decaying兲 homogeneous, isotropic turbulent suspension. One explanation 共given by Ferrante and Elghobashi兲 is based on a microscopic picture about the interaction between individual particles
and their local fluid flow environment. The other one 共given by
关21兴兲 uses a macroscopic picture with eddy-size-dependent suspension properties, such as effective density. Both pictures give a
satisfactory explanation, not only in words but also mathematically 共for details, see the relevant publications兲.
It is important to point out that only in a few analytical models
the turbulence generation due to the particle wakes or vortices
shed by the particles is taken into account. In this respect it is
interesting to mention briefly the work of Parthasarathy and Faeth
关28兴. It will also be discussed in the section on experimental work.
They investigated theoretically and experimentally the continuous
phase properties for the case of nearly monodisperse glass particles falling in a stagnant water bath. This yielded a stationary,
homogeneous flow in which all turbulence properties were due to
the effects of turbulence modulation by the particles. Their theoretical model is based on a linear superposition of undistorted
particle wakes in a nonturbulent environment. The model predicts
many properties of the flow reasonably well. However, it yields
poor estimates of the integral length scale and streamwise spatial
correlations. This model seems a good starting point for further
research on the turbulence generation by particle wakes.
As a final remark we stress the point that more attention may
also be given to the theoretical 共and numerical兲 investigation of
the influence of preferential concentration 共clustering兲 of particles
in the turbulent flow field on the two-way coupling effect. During
the last ten years several publications concerning theoretical and
numerical studies of particles clustering in a turbulent flow field
have appeared in the literature. Some of them are summarized
below. We first emphasize, however, that in these publications the
effect of turbulence on particle clustering is studied. The influence
of this clustering on the turbulence of the fluid velocity field 共the
two-way coupling effect that is the subject of this review兲 is not
considered. Elperin et al. 关29兴 proposed a theory in which particle
clusters are caused by the combined influence of particle inertia
共leading to a compressibility of the particle velocity field兲 and a
finite velocity correlation time of the fluid flow field. Particles
inside turbulent eddies are carried to the boundary regions between them by inertial forces. This clustering mechanism acts on
all scales of turbulence and increases toward small scales. The
turbulent diffusion of particles decreases toward smaller scales.
Therefore, the clustering instability dominates at the Kolmogorov
Transactions of the ASME
scale. An exponential growth of the number of particles in the
clusters is inhibited by collisions between the particles. The end
result can be a strong clustering whereby a finite fraction of particles is accumulated in the clusters, or a weak clustering when a
finite fraction of collisions occurs in the clusters. A crucial parameter for clustering is the particle radius, which has to be larger
than a certain critical value. Also Balkovsky et al. 关30兴 considered
clustering of inertial particles suspended in a turbulent flow and
developed a statistical theory of this phenomenon based on a Lagrangian description of turbulence. The initial growth of concentration fluctuations from a uniform state is studied, as is its saturation due to finite-size effects, imposed either by the Brownian
motion or by a finite distance between the particles. The statistics
of these fluctuations is independent of the details of the velocity
statistics, which allows the authors to predict that the particles
cluster at the Kolmogorov scale of turbulence. Also the probability distribution of the concentration fluctuations is calculated.
They discuss the possible role of the particle clustering in the
physics of atmospheric aerosols, in particular, cloud formation.
5
Experimental Work
5.1 Introduction. In this section, an overview is given of the
experimental studies of dispersed turbulent flows found in the
literature. Experimental work on the interaction of particles and
turbulence in homogeneous, isotropic flows is mostly done in
grid-generated turbulence. This is the closest approximation to
true homogeneous, isotropic turbulence, while still being experimentally feasible. Most experiments use a static grid through
which a fluid moves with a constant mean velocity. Alternatively,
an oscillating grid in a enclosed tank can be used 关31–33兴. The
advantage of these is that the turbulence level that can be attained
is relatively high. Additionally, the flow does not have a mean
flow component. This can be beneficial if Lagrangian measurements are required, such as tracking particles over a longer time to
study, e.g., dispersion properties. The biggest drawback of these
experiments is the fact that there often is a strong gradient in the
particle concentration. To overcome this drawback, experiments
are being done in microgravity 关34兴. The focus of this review will
be on the classic approach: by placing a grid in a wind tunnel or
water channel.
5.2 Grid-Generated Turbulence. The main concept of gridgenerated turbulence is simple: a fluid passes a grid with a certain
solidity. Strong gradients in the axial direction are generated 共i.e.,
“jets” emerging from the openings in the grid兲, which break up to
form nearly isotropic turbulence. The macroscopic length scale of
the turbulence is determined by the mesh spacing M. This length
scale, as any other length scale of the flow, grows proportional to
the square root of the downstream position: L ⬀ x1/2. Usually, it is
found that at a downstream distance of 20 mesh spacings the flow
is reasonably isotropic 共⬍10% difference between axial and transversal components兲. Because of the mixing behavior of a turbulent
flow, no influence of the separate mesh bars can be observed from
this distance on; the flow is homogeneous in the plane perpendicular to the mean flow direction. Typically, the turbulence level,
defined as the ratio of the root-mean-square of the fluid fluctuations 共u⬘兲 to the mean flow velocity U, is of the order of a few
percent. It is mainly determined by the mean fluid velocity and the
solidity of the grid. The turbulence obviously decays, therefore,
strictly speaking, the flow is not homogeneous in the axial direction. With respect to a typical particle time scale 共e.g., the Stokes
time兲, however, this decay is slow. Therefore, the flow can be
considered to be homogeneous in the context of particleturbulence interaction. The decay of the turbulent kinetic energy is
usually assumed to decay proportional to the inverse of the distance 共u⬘2 ⬀ x−1兲. The conventional way of representing the results
is therefore by plotting the reciprocal value of the turbulent kinetic
energy versus the distance to the grid. For an extensive treatment
Applied Mechanics Reviews
of grid-generated flow, one is referred to the classic experiments
by, e.g., Comte-Bellot and Corrsin 关35兴 or Batchelor 关36兴.
5.3 Overview of Experiments. The first thing that becomes
clear when reviewing the available literature is the fact that each
of the experiments has a relatively narrow scope. Even though
most papers claim to study the two-way interactions between the
phases, in general, the authors are often limited to the study of
only one or two related phenomena. Two-phase flows are notoriously difficult to study experimentally 关37兴, so each of the experiments has to be more or less tailored to study a certain aspect of
the interaction. A broad classification of the experiments can be
made:
•
•
•
•
changes in carrier-phase properties
particle-induced turbulence
clustering, preferential concentrations
particle-phase properties
The focus of the numerical and theoretical sections was mainly
on the first class: the influence of particles on the carrier-phase
spectrum. Only a few experiments have reported this because of
the difficulties in obtaining good measurements in such flows.
Additionally, the decay of the total turbulent kinetic energy
共i.e., the integral of the spectrum兲 is discussed. A full discussion of
all related phenomena is clearly unfeasible. For example, the work
on the effective settling velocity of particles in a turbulent flow is
an important parameter in many engineering models and the topic
of numerous publications. The settling velocity of a particle is a
direct result of particle-fluid interaction. But since settling obviously indicates a strong contribution from gravitational forces, it
is therefore inherently anisotropic. Since most theories and numerical works exclude gravity, this topic is not considered here.
5.4 Influence of the Particles on the Carrier Phase
Spectrum. Only three experiments have reported measurements
of the influence of particles on the carrier phase turbulence in
homogeneous isotropic turbulence: Schreck and Kleis 关38兴, Hussainov et al. 关39兴 and Geiss et al. 关40兴. The first one used solid
particles in water, yielding a density ratio of the order of unity.
The latter two used solid particles in a vertical wind tunnel 共density ratio of order 103兲. Work in progress on the topic has recently
been reported by Nishino et al. 关41兴 and Poelma et al. 关42兴, which
will hopefully contribute to the data available in the near future.
Where needed, their preliminary results are discussed.
Schreck and Kleis 关38兴 used two types of particles: glass 共relative density 2.5兲 and neutrally buoyant plastic particles. Decaying
grid turbulence offers the chance to study the dynamics of turbulent suspensions. In Fig. 7, the 共reciprocal value of the兲 turbulent
kinetic energy is plotted for the single-phase and particle-laden
case. These data were obtained using laser Doppler anemometry
共LDA兲. As can be seen, the turbulence level is lower for all mass
loadings compared to the single-phase flow. This was also the case
for the neutrally buoyant particles. The slopes in Fig. 7 corresponding to the particle-laden cases are somewhat steeper than the
single-phase case, indicating an increased dissipation rate.
In the longitudinal spectrum a very small decrease in the energy
at large scales could be observed, as well as an increase in energy
at small scales 共Fig. 8兲. This was most evident for the neutrally
buoyant particles. The Stokes number of these particles was 1.9. It
should be noted that the Stokes number decreases as the turbulence decays because of the growth of the Kolmogorov scales. For
the transversal spectra the reverse effect was seen: the small scales
have less energy for the particle-laden flows. The overall effect of
these phenomena resulted in almost identical shapes of the onedimensional spectrum for the particle-laden and particle-free
flows. Therefore, these results seem in contradiction to the outcome of the numerical work, which predicts changes in the onedimensional spectrum at this Stokes number and mass loading
MARCH 2006, Vol. 59 / 85
Fig. 7 Influence of mass load on decay of turbulent kinetic
energy; 0.65 mm glass particles in water. Reproduced from Ref.
†38‡.
共␾ = 3.8% 兲. Obviously, there is a significant difference in volume
load 共e.g., Ferrante and Elghobashi: ␺ = 0.1%, Schreck and Kleiss:
␺ = 1.5%兲, yet one would expect more of an effect with higher
loads.
Hussainov et al. 关39兴 used particles that were similar to those
used by Schreck and Kleis 关38兴, yet instead of a water channel
they used a wind tunnel. This led to a mass fraction that was
significantly larger 共␾ = 10% 兲, but more importantly also provided
very large Stokes numbers 共i.e., of the order 103兲. Measurements
are again done using LDA. Surprisingly, the effects that Hussainov and co-workers measure are less than those obtained by
Schreck and Kleiss despite the higher mass load: the decay rate is
somewhat larger, but very similar to the single-phase decay rate
共see Fig. 9兲. In the far-downstream region they find that the equilibrium turbulence level 共i.e., fully developed pipe/channel flow兲
is lower than for the particle-free case.
In the reported transversal spectra 共see Fig. 10兲 an increase of
energy is observed at higher frequencies 共viz. above 1 kHz兲. Even
though this agrees with Schreck and Kleis, it can be debated that
this is well within their measurement uncertainty and there is no
Fig. 8 Longitudinal spectrum for unladen and laden flow;
0.65 mm neutrally buoyant particles in water. Reproduced from
Ref. †38‡.
86 / Vol. 59, MARCH 2006
Fig. 9 Decay of turbulent kinetic energy for particle-free and
particle-laden flow. Data taken from Ref. †39‡.
difference between the single-phase and particle-laden fluid spectra. Similar results, yet less pronounced, were found in experiments with a grid with mesh spacing twice as large. A possible
explanation for the absence of a clear cross-over in the spectrum
might be the large Stokes number of the particles. Since they are
unresponsive to most fluid fluctuations, true two-way coupling
effects cannot be expected; there is only influence of the particle
on the fluid and not vice versa.
In a recent publication, Geiss et al. 关40兴 reported very similar
experiments as Hussainov’s. The main difference is the used particle size 共120, 240, and 480 ␮m兲, which is significantly smaller
than the experiments mentioned earlier. Still, the Stokes times of
the glass particles are very much larger than unity 共the Kolmogorov scales are not reported, only Stokes numbers based on the
integral time scale are reported. Obviously these are smaller than
the Kolmogorov scale-based values兲. For their measurements,
they used a phase Doppler anemometry. This enabled them to
measure both the fluid and the particle velocity simultaneously. At
a mass fraction of up to 0.077, they do not find any influence on
the 共normalized兲 carrier phase spectrum within the accuracy of the
measurements. On the other hand, there was an influence on the
total kinetic energy of the carrier phase and also on the decay rate.
One of their findings was the fact that there seems to be a thresh-
Fig. 10 Smoothened energy spectra for single-phase „␾ = 0…
and particle-laden „␾ = 0.005… flows. Data taken from Ref. †39‡.
Transactions of the ASME
Fig. 11 Decay of the axial „streamwise… and transversal kinetic energy for single-phase and
particle laden case: left: ␾ = 0.37%, right: ␾ = 2.67%. Data replotted from Ref. †40‡.
old in the mass load above which turbulence dampening occurs.
However, this effect seems rather weak considering their results
given in Figs. 11 and 12, which show the decay of the carrier
phase turbulent kinetic energy for various mass loads. A remarkable result of their measurement is the fact that the flow becomes
anisotropic while it decays. This was not observed by e.g.,
Schreck and Kleis, but has been confirmed by recent work of
Poelma et al. 关42兴 and Poelma 关43兴. An example of their decaying
axial and transversal kinetic energy components is shown in Fig.
13. Poelma et al. added ceramic particles 共␺ = 0.1%, d p = 280 ␮m,
␳ p / ␳ f = 3.8, Rep = 18兲 to their decaying grid-generated turbulence
in a water channel. It is clear that in the initial stages after the
grid, there is less turbulence compared to the single-phase flow at
the same centerline velocity. The axial component appears to decay slower than the transversal component, a result very similar to
that reported in Figs. 11 and 12. Note that in the work of Poelma,
the mass fraction is significantly lower 共because the experiments
were done in a water channel兲. The reason for the slower decay is
probably particle-induced turbulence; the overal decay rate appears lower because the particles generate turbulence, predominantly in the axial 共gravitational兲 direction. This behavior is in
contradiction to the often-cited rule of thumb that particles start
producing turbulence when Rep ⬎ 400.
The detailed mechanisms of the generation of turbulence by
particles has been studied by an number of authors. Parthasarathy
and Faeth 关28兴 performed studies using LDA of turbulence generation due to falling glass particles in stagnant water. An impor-
tant conclusion was the fact that the turbulence generated by a
falling particle is anisotropic; the mean streamwise 共i.e., in the
direction of gravity兲 component is roughly a factor 2 higher than
the cross-stream component. Furthermore, it was found that the
spectrum of the fluid phase showed energy at a large range of
frequencies, even though the particle Reynolds numbers were
small. This indicates that energy is generated at 共or transported to兲
scales larger than the particle size. Chen et al. 关44兴 and Lee et al.
关45兴 did similar experiments in a counterflow wind tunnel. They
found that even at low volume loads 共0.003%兲, the particles can
generate a turbulence level of up to 5%.
In a recent study in grid-generated turbulence, Nishino et al.
关41兴 also found strong anisotropic particle-induced turbulence.
Again, grid-generated turbulence in a water channel was studied.
The particles they used were 1.0 and 1.25 mm glass particles,
their size being in between the Kolmogorov and integral length
scales 共Rep = 138, 208兲. Instead of having a constant mass load
they studied a decaying mass load using a innovative idea: the
共upward兲 mean flow was chosen equal to the mean settling velocity of the particles. The particles were trapped in the test section
with only a small hole at the top. This led to a slowly decreasing
mass/volume load, which was measured using image processing
techniques. The turbulence statistics of the fluid, as measured by a
particle image velocimetry 共PIV兲 system, could thus be measured
as a function of the volume load. Figure 14 shows the influence of
the mass load on the turbulence level. The most important feature
Fig. 12 Decay of the axial „streamwise… and transversal kinetic
energy for single-phase and particle laden case. ␾ = 4.99%.
Data replotted from Ref. †40‡.
Fig. 13 Decay of axial and transversal kinetic energy for
single-phase and particle-laden case „␺ = 0.1%, ␾ = 0.38%, dp
= 280 ␮m, ␳p / ␳f = 3.8…. Data taken from Ref. †42‡.
Applied Mechanics Reviews
MARCH 2006, Vol. 59 / 87
Fig. 14 Turbulence level modification by 1.0 mm glass particles in water as function of volume concentration, Rep = 140.
Data taken from Ref. †41‡.
of this graph is the fact that while the axial component increases
significantly 共and more or less linearly兲 with volume load, the
transversal component increases only slowly. In total, the turbulence level is roughly three times larger than for the particle-free
flow. These results are in contradiction with the experiments of
Schreck and Kleiss 关38兴, who found a slight damping 共with somewhat smaller particles of the same density兲. The explanation given
by Nishino et al. 关41兴 is the fact that the potential energy of the
particles is transferred to kinetic energy. This is the same effect as
was studied by Parthasarathy and Faeth 关28兴, apart from the presence of ambient turbulence in Nishino’s work.
In addition to the particle-induced anisotropy, they observed
large fluctuations in the concentration of particles. More specific,
they observed what they called “columnar particle accumulation,”
i.e., vertical bands of high concentration. This could also be observed in the transversal autocorrelation function, which changed
drastically. Similar phenomena have also been observed by
Poelma et al. 关42兴, also using PIV. These large density fluctuations
may also be the cause of the increased turbulence level. Obviously, these effects could only be identified due to the whole-field
character of the PIV measurements, in contrast to the earlier
single-point LDA work.
The inhomogeneities in particle distribution observed by
Nishino et al. 关41兴 and others have been investigated by a number
of authors, and the effect is often referred to as “clustering” or
“preferential concentration.” Eaton and Fessler 关46兴 collected a
significant amount of experiments and simulations done in this
field. The main conclusion that can be drawn from this overview
is that these effects occur when the Stokes number of the particles
is close to unity. Work by Fallon and Rogers 关34兴, based on imaging of particles in microgravity, indicated that the presence of
gravity 共or any other body force兲 reduces preferential concentration phenomena. Preferential concentration can have significant
effects. For instance, Aliseda et al. 关47兴 studied the effect of preferential concentration on the settling velocity of particles. The
system was studied using imaging-based techniques. They observed a quasi-linear relationship between the effective settling
velocity and the local concentration. The settling velocity of particles is an important parameter in, e.g., many engineering models,
but a discussion of this parameter is beyond the scope of this
review.
5.5 Discussion of Experimental Work. No significant
changes in the shape of the carrier-phase turbulence spectrum
共normalized by the total turbulent kinetic energy兲 are observed in
the experiments. However, the total turbulent kinetic energy of the
88 / Vol. 59, MARCH 2006
fluid phase is lower for most experiments, which indicates that
there is some way of coupling between the phases. The fact that
the Stokes number of the particles used in most experiments is an
order of magnitude larger than unity, might explain the absence of
bigger changes in the spectrum. It seems to us that a damping
effect due to the particles 共with large Stokes number兲 takes place
at the largest scales of turbulence, and that because of the cascade
process of turbulence all other smaller eddies also receive less
energy than in the particle-free case. If energy would be added or
taken at any other position in the spectrum, this would become
evident in the shape of the spectrum. On the other hand, particles
are able to generate turbulence. The particle-induced turbulence is
anisotropic, and energy is generated at a large range of scales in
the spectrum. The two effects, 共i兲 damping of the fluid motions at
large scale and cascade transport to smaller scales and 共ii兲
particle-induced turbulence production, are competing processes
in dispersed two-phase flows. A more quantitative description using the presently available experimental data is impossible. The
governing parameters such as density ratio, ratio of particle size
and fluid length scale, and volume load, are too different to be
able to compare the present results directly. More research is
therefore needed, preferably varying one single parameter 共as was
done in, e.g., Nishino’s work with the volume fraction兲.
6
Conclusion
The numerical work from different researchers for the two-way
coupling effect in a homogeneous, isotropic, turbulently flowing
suspension agrees reasonably well with respect to the effect of
particles on the turbulence spectrum of the carrier phase: low
wave numbers are suppressed, while energy is gained at higher
wave numbers 共dependent on the Stokes number兲. The cross-over
point—the wave number above which the energy is larger compared to the single-phase case—shifts to larger wave numbers for
larger Stokes numbers. The overall effect can be either a damping
of the turbulence level or an increase, depending on the particle
Stokes number and the volume load. Several physical explanations for this phenomenon have been given in the literature.
Analytical theories for the two-way coupling have been developed based on the physical mechanisms that are thought to govern
the system. For instance, the pivoting of the carrier-phase spectrum is reasonably well predicted, even though the results are still
somewhat qualitative. At the moment, no comprehensive theory
that integrates all phenomena exists, however. In particular, preferential concentration effects are not included. This effect can play
an important role.
The available experimental data for a homogeneous, isotropic
turbulent suspension are scarce. The pivoting of the spectrum was
not observed in the obtained spectra, neither in solid/liquid nor
solid/gas flows, even though the mass loads were comparable to
those used in the numerical simulations. The fact that the Stokes
number of the used particles was rather high in most experiments
may partly account for the discrepancy. According to the physical
explanations mentioned above, a large Stokes number implies that
the cross-over 共pivoting兲 point would move to very high wave
numbers; thus, effectively damping on all scales occurs. This is in
agreement with what is observed in most experiments.
Another important conclusion from the experiments is the effect of gravity, which generates a strongly anisotropic system.
This effect is not included in most numerical simulations and
analytical theories; therefore, direct comparison is not trivial. It is
important to extend the simulations and theories so that the effect
of gravity is incorporated.
Acknowledgment
Parts of this work was supported by the Technology Foundation
STW, Applied Science Division of NWO, and the technology programme of the Ministry of Economic Affairs. The authors would
like to thank Dr. Dreizler and Dipl.-Ing. Chrigui for supplying the
original data used in Figs. 11 and 12.
Transactions of the ASME
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Christian Poelma received his M.Sc. degree in Chemical Engineering in 1999 from the Delft University of
Technology (TUD), The Netherlands. At the TUD, he started his Ph.D. research in the Laboratory for Aero
and Hydrodynamics. His main research topics are experimental fluid mechanics, turbulence, and dispersed
two-phase flows. In 2004, he obtained his Ph.D. on experimental work on turbulence modification by
particles. This review paper was prepared as part of his thesis. In the summer of 2004, he started as a
postdoc (funded by a NWO Talent Fellowship) at the California Institute of Technology in Pasadena to study
the unsteady aerodynamics of insect flight.
Applied Mechanics Reviews
MARCH 2006, Vol. 59 / 89
Gijs Ooms studied applied physics at the Delft University of Technology (TUD). He received his doctoral
degree in 1971 on a topic from the field of fluid mechanics. Thereafter he worked for Shell (in Amsterdam,
Houston, and Rijswijk) and the TUD. At Shell he was involved in research and technology development.
Although he gradually developed into a manager of research, he always continued carrying out his own
research projects. At the TUD he is professor of fluid mechanics at the Laboratory for Aero- and Hydrodynamics. He is also scientific director of the J.M. Burgers Centre (the national research school for fluid
mechanics). His scientific interest is on two-phase flow, turbulence, and the influence of high-frequency
acoustic waves on the flow through a porous material.
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