dropping small particles into a turbulent fluid, we know from daily
experience that the particles will be swept up in the swirling eddies
and vortices of the fluid motion. But how will the particles be
arranged in the flow, and does the addition of these particles smooth
out the flow or make it more turbulent? These questions not only have
industrial and technological implications, but are at the heart of our
understanding of turbulence. It is also a puzzle that has resisted
conventional fluid dynamics analysis. Now, in a paper published in Physical Review Letters, Tanaka and Eaton [1]
of Stanford University have scrutinized a set of experimental
measurements of particle motion in turbulent flows and find that a new
dimensionless parameter, the particle momentum number Pa, can be used to assess whether particles will enhance or attenuate the turbulence.

turbulence has been a longstanding challenge. The press release
accompanying the 1982 Physics Nobel Prize awarded to Kenneth G. Wilson
for his theory of critical phenomena [2]
cited “fully developed turbulence” as a prime example of an important
and yet unsolved problem in classical physics. Mathematically,
turbulence is described by the Navier-Stokes equations, and in 2000,
the Clay Mathematics Institute called unlocking the secrets of these
equations one of seven “Millenium Problems” and offered a $1 million
prize for their solution [3].
Turbulence is such a difficult problem because of its multiscale
character: Although the large length scales (of the order of some
maximal outer length scale L, for example, the size of the container) can be many orders of magnitude larger than the inner length scales of order η
(where the smoothing effects of viscosity become important), they are
nevertheless strongly coupled to each other. The problem becomes worse
the larger the Reynolds number Re (the ratio of inertial forces to
viscous forces, for which high or low values characterize turbulent vs
laminar flow, respectively) because L/η~Re3/4.
While the transition to turbulence occurs at a Reynolds number of
several thousands (depending on the geometry), typical turbulent flow
in the lab would have Re~106 , and in the atmosphere, flows with Re~109 can easily occur.

problem obviously becomes further complicated when the turbulent flow
transports particles—a situation that is omnipresent in nature and
technology. Examples include aerosols, rain drops, snow flakes or dust
particles in the atmosphere (and especially in clouds), plankton in the
ocean, or catalytic particles and bubbles in process technology. For
these situations it is a priori
not clear how the particles distribute in the turbulent flow, which
consists of vortices of various sizes—clearly, the distribution will be
inhomogeneous (Fig. 1). Particles heavier than the carrier fluid are
thrown out of the vortices due to centrifugal forces, whereas light
particles accumulate close to the vortex cores—an effect that everybody
can easily observe when stirring a glass of bubbly water. Neither is it
a priori clear whether the particles enhance or attenuate the turbulence (see Ref. [4]
for a classical review article). Take heavy particles in turbulent
water: On one hand, one could argue that the particles thrown into
still water will sink and thus excite some flow and therefore the flow
should also be enhanced when starting with a turbulent flow situation.
On the other hand, putting heavy particles in motion and rotation in
turbulent flow costs energy and therefore the turbulence intensity
should decrease.

One would
hope to be able to predict the enhancement or attenuation of turbulence
in the way common to fluid dynamics—by looking at the appropriate
dimensionless numbers, such as the Reynolds number mentioned before.
The classical dimensionless parameters for this problem would be the
large scale Reynolds number Re
of the turbulent flow, the density ratio of the dispersed particles and
the carrier fluid, the volume concentration of particles, and the
Stokes number St, which is the ratio of the particle relaxation time and the intrinsic timescale of the turbulent flow.

Tanaka and Eaton [1] have now mapped out 30 experimental data sets taken from literature with different combinations of Re and St,
arguing that the particle concentration and the density ratio should
only lead to a quantitative effect with regard to turbulence
enhancement or attenuation. However, they did not find any systematic
trend in the turbulence modification in this ReSt
plane. Data sets showing either turbulent kinetic-energy attenuation or
augmentation were seemingly randomly scattered over the ReSt plane, suggesting that the Stokes number is not the correct control parameter for turbulence modification.

finding and their further dimensional analysis of the underlying
Navier-Stokes equations with an extra forcing term due to the dispersed
particles, led Tanaka and Eaton to introduce a new type of
dimensionless parameter, which they call the particle momentum number Pa. One version of this parameter can be written as Pa=Re2St(η/L)3, which one would expect should scale as ~Re1/4St, given the aforementioned expression for L/η. Now, in the RePa plane the 30 analyzed data sets do fall into different groups: For Pa<103 the turbulence is augmented, for 103<Pa<105 it is attenuated, and for Pa>105
it is augmented again. This finding is surprising as (i) the dependence
of the turbulent kinetic energy is nonmonotonic as a function of Pa, and (ii) one would assume that a simple rescaling of St with Re1/4 would not all of a sudden lead to a grouping of the data sets.

any doubt this paper will trigger much further analysis. Presently,
only data sets that show at least 5% attenuation or augmentation have
been included in the study, in order to overcome experimental
inaccuracies. I would expect the relative inaccuracies of the turbulent
kinetic-energy modification to be smaller in numerical simulations of
two-way-coupled point-particles in Navier-Stokes turbulence, such as
those done in Refs. [5, 6, 7, 8],
where the particles act back on the flow, supplying an additional
driving mechanism. Although the Reynolds numbers achieved in such
simulations are considerably smaller than those in the data analyzed by
Tanaka and Eaton [1], they would allow calculation of three-dimensional plots of the turbulent kinetic-energy modification as a function of Re and Pa,
from which systematic trends could be derived. The gap between
numerical simulations and experiment could be narrowed by extending
Tanaka and Eaton’s analysis of experimental data towards smaller
Reynolds numbers. Another extension of the parameter space would
involve particles lighter than the carrier fluid, such as bubbles in
turbulent flow, for which a wealth of numerical [9, 10] and experimental [11, 12]
data on the energy modification exist. Tanaka and Eaton’s work thus
gives hope that we can finally obtain order from the mist of turbulent
data points on dispersed multiphase flow.


thank Enrico Calzavarini (Ecole Normale Supérieure, Lyon, France) for
providing Fig. 1 and for many stimulating discussions over the years.
Moreover, I would like to thank the Fundamenteel Onderzoek der Materie
(FOM) for continuous support.