^{1,a)}, S. B. Pope

^{2}and P. K. Yeung

^{3}

### Abstract

We have extended the “well-mixed” two-particle stochastic models for 3D Gaussian turbulence to n particles, and have performed calculations for clusters of n ⩽ 6 particles. The particle joint motions are Gaussian and are constrained by pair-wise spatial correlations. This neglects non-Gaussian properties of the two-point velocity distribution and also neglects multi-point correlations. It also takes no account of intermittency. Although the models do not predict the growth of the separation of particles in the cluster satisfactorily, we find that they do give a good representation of the shape statistics for the cluster in comparison with direct numerical simulation results. We conclude that the pair-wise spatial structure of the turbulence accounts for most of the observed characteristics of the shape of multi-particle clusters in turbulence, and that non-Gaussian and multi-point features of the turbulence are of secondary importance.

Direct numerical simulation data used for model comparisons in this paper were generated using advanced computing resources provided to the third author (P. K. Yeung) by the National Institute of Computational Sciences (at the University of Tennessee, USA) and the National Center for Computational Sciences (at Oak Ridge National Laboratory, USA). This work is also supported in part by NSF Grant No. CBET-1235906. This work was initiated while the first author (B. L. Sawford) was supported as a Visiting Professor by Cornell University.

I. INTRODUCTION

II. THEORY

A. “Well-mixed” models in primitive coordinates

B. “Well-mixed” models in reduced coordinates

C. Inertial sub-range scaling

1. Diffusion limit

D. Shape statistics

III. NUMERICAL RESULTS

A. Dispersion statistics

1. Mean-square dispersion

2. Relative dispersion PDFs

B. Shape statistics

1. Triangles

2. Tetrads

3. Trends with number of particles

IV. CONCLUSIONS

### Key Topics

- Diffusion
- 34.0
- Lagrangian mechanics
- 22.0
- Turbulence simulations
- 18.0
- Turbulent flows
- 15.0
- Dispersion relations
- 13.0

## Figures

Cubed local slope plot of difference relative dispersion. Dashed line is Batchelor range independent motion limit 2C 0 = 14 and solid red line is a fit to the Richardson range (here with g = 1.08).

Cubed local slope plot of difference relative dispersion. Dashed line is Batchelor range independent motion limit 2C 0 = 14 and solid red line is a fit to the Richardson range (here with g = 1.08).

Schematic of initial positions for n ⩽ 14 particles.

Schematic of initial positions for n ⩽ 14 particles.

Model estimates of Richardson's constant as a function of C 0. Symbols are (bottom to top) for Borgas two-particle model (red cross), Borgas three-particle model (red circle), Thomson two-particle model (blue star), Thomson three-particle model (blue circle), diffusion two-particle model (orange triangle), and diffusion three-particle model (orange circle). The solid line is the analytical result (Eq. (39) ) for the two-particle diffusion model.

Model estimates of Richardson's constant as a function of C 0. Symbols are (bottom to top) for Borgas two-particle model (red cross), Borgas three-particle model (red circle), Thomson two-particle model (blue star), Thomson three-particle model (blue circle), diffusion two-particle model (orange triangle), and diffusion three-particle model (orange circle). The solid line is the analytical result (Eq. (39) ) for the two-particle diffusion model.

Richardson's constant g estimated for Thomson's model (filled symbols; solid lines) and the diffusion equation (open symbols; dashed lines) as a function of the number of particles n. Upper two (black) lines are for C 0 = 7, and the lower two (red) lines are for C 0 = 14.

Richardson's constant g estimated for Thomson's model (filled symbols; solid lines) and the diffusion equation (open symbols; dashed lines) as a function of the number of particles n. Upper two (black) lines are for C 0 = 7, and the lower two (red) lines are for C 0 = 14.

Richardson's distance-neighbour function estimated for Thomson's model for n = 4 for a range of times. Left panel shows times t ⩽ 0.27t 0 and right panel shows times t ⩾ 7.29t 0 (in both cases the precise order is not important). The initial separation has been subtracted, r ′ = r i − r i0, and the results are averaged over all separations in the cluster. Richardson's result from Eq. (40) is plotted as a solid black line, which is almost completely obscured by the model results in the right panel. The dashed line is a Gaussian function and is almost completely obscured by the model results in the left panel.

Richardson's distance-neighbour function estimated for Thomson's model for n = 4 for a range of times. Left panel shows times t ⩽ 0.27t 0 and right panel shows times t ⩾ 7.29t 0 (in both cases the precise order is not important). The initial separation has been subtracted, r ′ = r i − r i0, and the results are averaged over all separations in the cluster. Richardson's result from Eq. (40) is plotted as a solid black line, which is almost completely obscured by the model results in the right panel. The dashed line is a Gaussian function and is almost completely obscured by the model results in the left panel.

Richardson's distance-neighbour function (left panel) estimated for “un-mixed” version of Borgas' model for n = 2 and a range of times t ⩾ 7.29t 0 (the precise order is not important), representative of the Richardson scaling range. The initial separation has been subtracted, r ′ = (r i − r i0). Richardson's result from Eq. (40) is plotted as a solid black line. Right panel shows the corresponding PDF in linear coordinates.

Richardson's distance-neighbour function (left panel) estimated for “un-mixed” version of Borgas' model for n = 2 and a range of times t ⩾ 7.29t 0 (the precise order is not important), representative of the Richardson scaling range. The initial separation has been subtracted, r ′ = (r i − r i0). Richardson's result from Eq. (40) is plotted as a solid black line. Right panel shows the corresponding PDF in linear coordinates.

Model estimates of mean shape statistics for triangles as a function of non-dimensional time for an initial separation r 0/L = 10−9 and C 0 = 7, corresponding to . Left panel: the mean shape factors ⟨I 1⟩ and ⟨I 2⟩. Open red circles and (overlapping) solid red curve (lower) are ⟨I 2⟩ for Borgas and Thomson models, respectively, for initially right isosceles triangles and open blue circles and (overlapping) solid blue line (upper) are corresponding results for ⟨I 1⟩. The red and blue dashed curves are for Thomson's model for initially equilateral triangles. The solid horizontal lines are DNS estimates in the inertial sub-range (⟨I 1⟩ = 0.88 and ⟨I 2⟩ = 0.12) and the dotted horizontal lines are the analytical independent motion estimates (⟨I 1⟩ = 5/6 and ⟨I 2⟩ = 1/6). Right panel: the mean symmetry parameter ⟨χ⟩. Blue circles are for Borgas' model for initially right isosceles triangles and the dashed red curve is for Thomson's model for initially equilateral triangles. The solid horizontal line is the DNS estimate in the inertial sub-range (6⟨χ⟩/π = 0.45) and the dotted horizontal line is the Monte Carlo independent motion estimate (6⟨χ⟩/π = 1/2).

Model estimates of mean shape statistics for triangles as a function of non-dimensional time for an initial separation r 0/L = 10−9 and C 0 = 7, corresponding to . Left panel: the mean shape factors ⟨I 1⟩ and ⟨I 2⟩. Open red circles and (overlapping) solid red curve (lower) are ⟨I 2⟩ for Borgas and Thomson models, respectively, for initially right isosceles triangles and open blue circles and (overlapping) solid blue line (upper) are corresponding results for ⟨I 1⟩. The red and blue dashed curves are for Thomson's model for initially equilateral triangles. The solid horizontal lines are DNS estimates in the inertial sub-range (⟨I 1⟩ = 0.88 and ⟨I 2⟩ = 0.12) and the dotted horizontal lines are the analytical independent motion estimates (⟨I 1⟩ = 5/6 and ⟨I 2⟩ = 1/6). Right panel: the mean symmetry parameter ⟨χ⟩. Blue circles are for Borgas' model for initially right isosceles triangles and the dashed red curve is for Thomson's model for initially equilateral triangles. The solid horizontal line is the DNS estimate in the inertial sub-range (6⟨χ⟩/π = 0.45) and the dotted horizontal line is the Monte Carlo independent motion estimate (6⟨χ⟩/π = 1/2).

Model estimates for the PDFs of shape statistics for initially right isosceles triangles for an initial separation r 0/L = 10−9 and C 0 = 7. Left panel: the shape factor I 1, for which the initial condition is a delta function at I 1 = 3/4, for a range of non-dimensional times t/t 0 = 0.1 (black curve with peak at I 1 ≈ 0.75), 0.3 (blue curve with peak at I 1 ≈ 0.85), 0.9 (cyan curve just above the dashed curve), 2.7, …, 5.9 × 103 (remaining curves collapsing to a self-similar form). The black dashed line is the average over Richardson scaling times from DNS and the blue dotted line is the independent motion diffusive limit. Right panel: the triangle symmetry parameter χ for which the initial condition is a delta function at χ = π/6, for a range of non-dimensional times t/t 0 = 0.1 (black curve with P(χ) ranging from 0.4 to 4.5), 0.3 (blue curve with P(χ) ranging from 1.7 to 2.3), 0.9 (cyan curve close to the blue dotted line), 2.7, …, 5.9 × 103 (remaining curves collapsing to a self-similar form). The dashed and dotted lines are as in the left panel.

Model estimates for the PDFs of shape statistics for initially right isosceles triangles for an initial separation r 0/L = 10−9 and C 0 = 7. Left panel: the shape factor I 1, for which the initial condition is a delta function at I 1 = 3/4, for a range of non-dimensional times t/t 0 = 0.1 (black curve with peak at I 1 ≈ 0.75), 0.3 (blue curve with peak at I 1 ≈ 0.85), 0.9 (cyan curve just above the dashed curve), 2.7, …, 5.9 × 103 (remaining curves collapsing to a self-similar form). The black dashed line is the average over Richardson scaling times from DNS and the blue dotted line is the independent motion diffusive limit. Right panel: the triangle symmetry parameter χ for which the initial condition is a delta function at χ = π/6, for a range of non-dimensional times t/t 0 = 0.1 (black curve with P(χ) ranging from 0.4 to 4.5), 0.3 (blue curve with P(χ) ranging from 1.7 to 2.3), 0.9 (cyan curve close to the blue dotted line), 2.7, …, 5.9 × 103 (remaining curves collapsing to a self-similar form). The dashed and dotted lines are as in the left panel.

Model estimates of the mean shape factors ⟨I 1⟩, ⟨I 2⟩, and ⟨I 3⟩ for initially right tetrahedra as a function of non-dimensional time for an initial separation r 0/L = 10−9 and C 0 = 7, corresponding to . Red (top), blue (middle), and cyan (bottom) lines are for ⟨I 1⟩, ⟨I 2⟩, and ⟨I 3⟩, respectively. Solid curves are for Thomson's model and dashed curves are for the diffusion model. The solid horizontal lines are our DNS estimates in the inertial sub-range (⟨I 1⟩ = 0.825 and ⟨I 2⟩ = 0.16 and ⟨I 3⟩ = 0.015) and the dotted horizontal lines are the Monte Carlo independent motion estimates (⟨I 1⟩ = 0.75, ⟨I 2⟩ = 0.22, and ⟨I 1⟩ = 0.03).

Model estimates of the mean shape factors ⟨I 1⟩, ⟨I 2⟩, and ⟨I 3⟩ for initially right tetrahedra as a function of non-dimensional time for an initial separation r 0/L = 10−9 and C 0 = 7, corresponding to . Red (top), blue (middle), and cyan (bottom) lines are for ⟨I 1⟩, ⟨I 2⟩, and ⟨I 3⟩, respectively. Solid curves are for Thomson's model and dashed curves are for the diffusion model. The solid horizontal lines are our DNS estimates in the inertial sub-range (⟨I 1⟩ = 0.825 and ⟨I 2⟩ = 0.16 and ⟨I 3⟩ = 0.015) and the dotted horizontal lines are the Monte Carlo independent motion estimates (⟨I 1⟩ = 0.75, ⟨I 2⟩ = 0.22, and ⟨I 1⟩ = 0.03).

Estimates of the PDFs for the shape factors I 1, I 2, and I 3 from Thomson's model for initially right tetrads for t/t 0 and r 0/L as in Fig. 8 . The initial conditions are delta functions at I 1 = I 2 = 4/9 and I 3 = 1/9 as indicated by the vertical dashed lines. In each panel, the curves for t/t 0 = 0.1, 0.3, …, 8.1 increase in width as the peak value moves further from the initial condition, and the remaining curves for t/t 0 = 24.3, …, 5.9 × 103 collapse to a self-similar form. The black dashed curve is the average over Richardson scaling times from DNS and the blue dotted curve is the independent motion diffusive limit.

Estimates of the PDFs for the shape factors I 1, I 2, and I 3 from Thomson's model for initially right tetrads for t/t 0 and r 0/L as in Fig. 8 . The initial conditions are delta functions at I 1 = I 2 = 4/9 and I 3 = 1/9 as indicated by the vertical dashed lines. In each panel, the curves for t/t 0 = 0.1, 0.3, …, 8.1 increase in width as the peak value moves further from the initial condition, and the remaining curves for t/t 0 = 24.3, …, 5.9 × 103 collapse to a self-similar form. The black dashed curve is the average over Richardson scaling times from DNS and the blue dotted curve is the independent motion diffusive limit.

Mean shape factors as a function of the cluster size n. DNS results for triangles and tetrads are shown as the + symbol, filled circles are for Thomson's model and open circles are for the diffusion model.

Mean shape factors as a function of the cluster size n. DNS results for triangles and tetrads are shown as the + symbol, filled circles are for Thomson's model and open circles are for the diffusion model.

Shape factor PDFs as a function of the cluster size n for Thomson's model (solid lines) and DNS (dashed lines). In the tails of the distributions, n increases from 3 to 6 from the lower to upper curves (black, red, blue, and cyan).

Shape factor PDFs as a function of the cluster size n for Thomson's model (solid lines) and DNS (dashed lines). In the tails of the distributions, n increases from 3 to 6 from the lower to upper curves (black, red, blue, and cyan).

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