Consider solid particles in a turbulent flow. When is it appropriate to use a velocity field for the particles?

To elaborate: In fluid mechanics, one uses the velocity field of the fluid as it is found from irreversible thermodynamics. According to the hypothesis of local equilibrium, one can divide the fluid into small control volumes such that each of these is large enough to contain many particles (in the sense that thermodynamic quantities like density, pressure etc. can be defined) and yet the control volumes are small enough such that later pressure, density etc. are continuous functions of space and time. Now if we consider particles in a turbulent flow and the particle number density is low, the volume elements we would have to choose for them to contain many particles would have to be rather large.

My question is: Is this even the right way to approach this for making a detailed model of particle transport? And how does one obtain objective criteria on the length scales on which the model is valid?

To add to that: If the stokes number is low, the particles follow the fluid. If it is not, we can presumably always find small volume elements where particles go in different directions. Especially near a reflecting boundary for the particles (i.e. near the ground in subaqueous bedload or aeolian saltation, or imagine hail bouncing off the side of a building), this seems to present a problem for defining a particle velocity field. How does one define a particle velocity field in this case? (If at all?)

Ultimately I'm thinking about descriptions of structures of particles agglomerating due to influences by the turbulent flow, such that these structures are visible to the human eye, yet because of the above-mentioned difficulties in defining a velocity field, number density etc. for the particles on different length scales might be elusive in terms of their theoretical description. Think of raindrops falling in gusty wind or aeolian streamers.

(Edit for clarity: I am thinking of "particles in a turbulent flow" as the layman would, i.e. a large number of small rigid bodies contained in a flowing medium that typically would be described as a fluid, like sand or silt transport in water at the bottom of a stream, volcanic dust blown by the wind, or seeds or grains blown through a tube by pressurized air.)

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    $\begingroup$ I am struggling to understand your question. Are you asking, what is the range of validity of Navier-Stokes equation? $\endgroup$
    – Pavlo. B.
    Commented Apr 29, 2021 at 20:40
  • $\begingroup$ Not at all, no. The validity of the Navier-Stokes is no problem, for that you need a continuum description of the fluid (cf. continuum hypothesis or local equilibrium hypothesis in irreversible thermodynamics), usually possible over a wide range of lengths (down to something like $10^{-6}\,\text{m}$). My question is whether and on which length scale there is an analog continuum description for particles added to the flow, especially when the particle number density is low. $\endgroup$
    – kricheli
    Commented Apr 29, 2021 at 21:37
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    $\begingroup$ As I remember from my papers and books about turbulent transport of aerosol particles in the atmosphere we used multiphase model to describe aerosol flow with different size of particles. Also in a case of rigid walls we added some new phase describing the reflecting particles flow. This is typical paper link.springer.com/article/10.1007/BF00915321 $\endgroup$ Commented May 2, 2021 at 17:30
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    $\begingroup$ I can recommend to use the kinetic equation in a case $Kn>>1$, the paper by my co-authors ui.adsabs.harvard.edu/abs/1986PMTF........93K/abstract $\endgroup$ Commented May 3, 2021 at 18:56
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    $\begingroup$ Thank you, Alex. Your remark has led me to have a look at doi.org/10.1007/978-3-319-66149-0 and so far this seems to be the next best thing to an answer. Good book with (among other things) a discussion of coupled fluid-particle systems. But early on in the chapter on those systems the assumption $d \ll V^{1/3} \ll L$ for the particle diameter $d$, an averaging volume $V$ and the scale $L$ of macroscopic variations is introduced. So the book is clearly helpful, but does not seem to provide an answer to my question. $\endgroup$
    – kricheli
    Commented May 5, 2021 at 10:42

1 Answer 1


Interesting question. This is essentially a research question with no canonical answer, so let me just give a few ideas for the lines of attack.

One of the essential assumptions of continuum mechanics is that the averaging lengths $L_{\rm avg}$ cannot be longer than the essential length-scales of the flow. In particular, if there are driving forces with a lengthscale $L_{\rm f}$, these will be imprinted in the flow variability. In your case the driving force is the turbulent length-scale, and if the particle density is such that only a small number of particles is in the turbulent volume, you simply cannot take a direct ensemble average to get a meaningful velocity field. Strictly speaking this seems to force you into a direct particle-by-particle simulation. However, whether you really need to do so ultimately depends on what you are trying to achieve.

For instance, you may be dealing with "test particles" that do not interact with one another and do not back-react on the flow. Then one can use a continuum Boltzmann-like transport equation, which also naturally treats the case of multiple velocities and reflected streams of particles. The point of being able to use the continuum equation is that the particle can be entirely virtual, you are just evolving the probability distribution of the test particle occurence into the future.

However, I assume that you are in fact not interested in all the details of the motion but really more in well-modelled statistics of the transport. In that case it may be admissible to use some form of Reynolds-averaged equations, which are equations that you obtain by averaging over the spatio-temporal scales of the turbulent problem to obtain a smooth(er) flow. (These are sometimes known as a large-eddy description of the flow.) The transported particles can then also have effective Reynolds-averaged velocity fields that are driven by the Reynolds-averaged quantities of the ambient flow. You would probably have to also keep track of a full stress tensor (tensor of $v^i v^j$ statistical moments, where $v^i$ are the particle velocities) and reflections off of surfaces would still probably require a multiphase treatment.

Getting a closed system of Reynolds-averaged equations is usually not a rigorous procedure, though. Specifically the turbulent driving of the particles on scales below the Reynolds-averaging scales is something that may not be possible to express only in terms of the Reynolds-averaged quantities. This is not an isolated issue in fluid mechanics and many scientists have been trying to work on it very hard, since the inexact closure of the multiphase Reynolds-averaged ocean-atmosphere system is currently one of the largest sources of uncertainties of climate modeling.

  • $\begingroup$ Thanks for your answer. I know this is a research question, I'm just trying to get ideas and references to literature. And I knew much of what you write already, in part it reads like part of the problem description. With your third paragraph I see a problem: What are you Reynolds averaging? If you have a substance with a reasonable understanding of its concentration and velocity field, all is well, but with disperse particles at low densities you may not have a concentration to begin with (on the relevant length scales). This is kind of the main point of my question. Ok, cf. your paragraph 1. $\endgroup$
    – kricheli
    Commented May 5, 2021 at 12:53
  • $\begingroup$ @kricheli You can translate the set of particles to a density and momentum field by using delta functions for the positions of the particles. The Reynolds average would then be done by convolving with an appropriate spatio-temporal mollifier. For example, the mollifier can be a spatial Gaussian with a half-width that has to be larger than typical interparticle distance. To gain a velocity field, you can simply divide the smoothened momentum field by the smooth density field. Of course, there will be cases when the inter-particle distance is too large even for a Reynolds averaging procedure. $\endgroup$
    – Void
    Commented May 5, 2021 at 19:02

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