Physics – Particles go with the flow

Physics 1, 18 (2008) DOI: 10.1103/Physics.1.18

# Particles go with the flow

*Physics
of Fluids Group, Faculty of Science and Technology, MESA+ and Impact
Institutes, University of Twente, 7500 AE Enschede, The Netherlands*

Published September 8, 2008

A

novel dimensionless parameter allows prediction of whether dispersed

particles in a turbulent flow enhance or attenuate the turbulence.

### A Viewpoint on:

**Classification of Turbulence Modification by Dispersed Spheres Using a Novel Dimensionless Number**

Tomohiko Tanaka and John K. Eaton

Phys. Rev. Lett. 101, 114502 (2008) – Published September 08, 2008

Download PDF (free)

When

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.

Understanding

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~Re^{3/4}.

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~10^{6} , and in the atmosphere, flows with Re~10^{9} can easily occur.

The

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 Re–St

plane. Data sets showing either turbulent kinetic-energy attenuation or

augmentation were seemingly randomly scattered over the Re–St plane, suggesting that the Stokes number is not the correct control parameter for turbulence modification.

This

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=Re^{2}St(η/L)^{3}, which one would expect should scale as ~Re^{–1/4}St, given the aforementioned expression for L/η. Now, in the Re–Pa plane the 30 analyzed data sets do fall into different groups: For Pa<10^{3} the turbulence is augmented, for 10^{3}<Pa<10^{5} it is attenuated, and for Pa>10^{5}

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 Re^{–1/4} would not all of a sudden lead to a grouping of the data sets.

Without

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.

## Acknowledgments

I

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.