6.2 Clustering
6.2.2 Extended particles in a Kraichnan flow field
After having investigated the clustering of point particles in a Kraichnan flow field, we will focus on extended, hard-core particles in a Kraichnan flow field and inspect the differences between the two models to determine the influence of hard cores. Our calculations are shown in Fig.6.6in the same manner as it was done for point particles in Fig. 6.4. Again, we compare the results of the three measures for clustering:
Patch concentration enhancement factor|Q|, exponentαof number fluctuations, and variance of the cluster analysis N2. |Q| is again nearly vanishing in the isotropic phase. For a constant Péclet number, it reaches a maximum as a function of the Stokes number at about Sω ≈3. Instead for constant Sω, |Q| seems to increase with increasing P and from the given data it is impossible to state whether there is a maximum as well, since the data are rather noisy.
The exponent αof the number fluctuations is equal to the equilibrium valueαeq. = 0.5 in the isotropic phase. In the nematic phase, it increases as bothP andSω increase and does not possess any local maxima or minima. In the given range of Stokes and Péclet numbers it does reach a value similar to the theoretical prediction for nematic αnematic = 5/6 or even slightly above. This shows that the nematic phase is subject to giant number fluctuations with their strength being a function of Stokes and Péclet number. The variance of the cluster analysis, N2, shows a very similar behavior to α. It is nearly vanishing in the isotropic phase and grows gradually with growing P and Sω indicating the onset of clustering.
Again, we compare typical snapshots of the system in the steady state to the different values of the clustering measures to understand what kind of clustering occurs. The results in Fig. 6.7 show the curves of |Q|, α, and N2 as a function of Sω for a fixed Péclet number (P = 1.38×107). The snapshot at Sω = 0.02 is in the isotropic phase and shows a homogeneous distribution of the particles. Consequently, the corresponding values of all three clustering measures are small, indicating that there are no patches; the system behaves like an equilibrium system, and the cluster analysis has no increase in its variance.
6.2 Clustering
patchenhancement factor|Q|
P´ecletnumberP
vortical Stokes number Sω
10−2 10−1 100 101 102 103
(a) Patch concentration enhancement factor Sω,v
exponentα
P´ecletnumberP
vortical Stokes number Sω
10−1 100 101 102 103
(b) Exponent of analysis of number fluctuations
N2
P´ecletnumberP
vortical Stokes number Sω
10−2 10−1 100 101 102 103
(c) Second moment of cluster size distribution
Figure 6.6: Clustering of hard-core particles in the Kraichnan flow field (analogously to Fig.6.4) ((a) and (b) to be published in Breier et al., 2017, (b) in supplementary information).
N2
vortical Stokes number Sω
α|Q|
10−2 10−1 100 101 102 103 100
102 104 0.4 0.6 0.8 1 0 5 10
(a) Different measures of clustering as a function of vortical Stokes number for hard-core particles in Kraichnan flow field (P = 1.38×107). Red circles mark simulations for which snapshots are shown below.
normalized local density ρlocal/ρ
Sω = 181.30 Sω = 5.82
Sω = 1.04 Sω = 0.18
Sω = 0.02
0 5 10 15
(b) Typical snapshots with 40% of the particles plotted as dots. The color indicates the local density ρi as it is calculated from the Voronoi tessellation (ρi = 1/vi). A few orientations are given to illustrate the (nematic) order.
Figure 6.7: Typical snapshots of hard-core particles in Kraichnan flow field analo-gously to Fig.6.5.
6.2 Clustering
The snapshot at Sω = 0.18, however, shows global nematic alignment but hardly any changes in the local densities compared to the first snapshot. Hence, both |Q|
and N2 are at a similar low level. However, the exponent of the number fluctuations is larger than the equilibrium value (but smaller thanαnematic).
The values of α and N2 for Sω = 1.04,5.82,181.30 all exhibit a similarly high level with a slight increase upon growing Sω. Furthermore, the value of |Q| is much larger than for the homogeneous systems but with a maximum corresponding to the Sω = 5.82 snapshot. The snapshots for Sω = 1.04,5.82,181.30 are similar in that they all show global nematic alignment with variations in the local density. However, the largest values of local density are reached by the system at Sω = 5.82 and also the structure differs to some extent between the three snapshot: TheSω = 1.04 snapshot has only very small voids where no particles are positioned and dense streams are visible which percolate through the periodic boundary conditions. At Sω = 5.82, dense streams also form but they are accompanied by larger voids. Moreover, the local density within the stream is not homogeneous but reaches a maximum in the center of the stream. Finally, the snapshot at Sω = 181.30 reveals similar dense streams which are rather homogeneous in density along and across each stream.
Several differences appear in the comparison of the simulations of point-like and extended, hard-core particles in a Kraichnan flow field as a function of Sω (while the behavior as a function of P does not change drastically). We will discuss in the following these differences as a function of the vortical Stokes number (see Fig.6.5for the point particles and Fig.6.7for hard cores). The patch concentration enhancement factor|Q|shows similar behavior for both models with an increase from the isotropic to the nematic phase and a “sweet spot” at Sω∗ ∈ [3,4]. However, the values which
|Q|reaches in the clustered state are very different: The value for point particles can extend to the order of 104 while it is only 101 for hard-core particles. The number fluctuations, instead, reveal a completely different behavior if we compare point-like particles with extended ones. The hard cores lead to a monotonously increasing α with Sω while the number fluctuations increase for point particles but with a local minimum where small-scale patches form. The results of the variance of the cluster analysis is similar in both models since N2 increases with increasing Sω. The only difference is that in the case of point particles a local maximum is found where a dense stream forms in a rather homogeneous surrounding.
The systems exhibit the same sequence of transitions between collective steady-state organization; they first show a homogeneous and isotropic system, then a homo-geneous and nematic system and finally clusters within the nematic order. However, the clusters in the case of hard-core particles always percolate through the periodic boundary conditions while the point particles form much smaller patches. Of course, the hard cores do not allow for the formation of arbitrarily dense clusters (as opposed to point particles), so an investigation of systems of hard cores with lower filling fraction is necessary.
vortical Stokes number Sω
|Q|
10−1 100 101 102 103
0 20 40
(a) Patch concentration enhancement factor as a function of vortical Stokes number. The red circles denote the simulations which are shown as snapshots below.
normalized local density ρlocal/ρ
Sω = 128.76 Sω = 32.67
Sω = 4.12 Sω = 0.52
Sω = 0.13
0 5 10 15
(b) Typical snapshots. Here, only 40% of the particles are shown as dots with the local density (from Voronoi tesselation) as color. A few orientations are given as small arrows.
Figure 6.8: Extended particles in a Kraichnan flow field at low filling fraction (N = 8000, φ= 0.24%, P = 1.38×105,v0/urms= 8.5).
6.2 Clustering
A lower filling fraction is achieved by keeping the box size fixed and using fewer particles (N = 8000). The resulting clustering analysis and characteristic snapshots are shown in Fig.6.8. All results are within the nematic phase since this is where the clustering occurs. We see a similar behavior of|Q|(Sω) as before, namely a maximum around Sω ≈ 4 even though the data are rather noisy. The patch concentration enhancement factor is roughly a factor of two larger than for the hard-core particles at higher filling fraction. It is still roughly three orders of magnitude lower than for point particles, though. All snapshots show nematic order and different degrees of clustering. The particles in the Sω = 0.13 snapshot are homogeneously distributed.
In the Sω = 0.52 snapshot, a dense stream is formed within a rather homogeneous gas. The particles in the Sω = 4.12 snapshot form small distinct patches with polar alignment. The patches in theSω = 32.67 snapshot span a larger volume than in the third snapshot, and finally, in the Sω = 128.76 snapshot they percolate through the periodic boundary conditions. The snapshots of hard-core particles at low density are thus more similar to point-particles than to high density hard-core particles as they also show the formation of small-scale clusters.