Analysis of the clustering of inertial particles in turbulent flows

An asymptotic solution is derived for the motion of inertial particles exposed to Stokes drag in an unsteady random flow. This solution provides an estimate for the sum of Lyapunov exponents as a function of the Stokes number and Lagrangian strain- and rotation-rate autocovariance functions. The sum of exponents in a Lagrangian framework is the rate of contraction of clouds of particles, and in an Eulerian framework, it is the concentration-weighted divergence of the particle velocity field. Previous literature offers an estimate of the divergence of the particle velocity field, which is applicable only in the limit of small Stokes numbers [Robinson, Comm. Pure Appl. Math. 9, 69 (1956) and Maxey, J. Fluid Mech. 174, 441 (1987)] (R-M). In addition to reproducing R-M at this limit, our analysis provides a first-order correction to R-M at larger Stokes numbers. Our analysis is validated by a directly computed rate of contraction of clouds of particles from simulations of particles in homogeneous isotropic turbulence over a broad range of Stokes numbers. Our analysis and R-M predictions agree well with the direct computations at the limit of small Stokes numbers. At large Stokes numbers, in contrast to R-M, our model predictions remain bounded. In spite of an improvement over R-M, our analysis fails to predict the expansion of high Stokes clouds observed in the direct computations. Consistent with the general trend of particle segregation versus Stokes number, our analysis shows a maximum rate of contraction at an intermediate Stokes number of $O\left(1\right)$ and minimal rates of contraction at small and large Stokes numbers.

Figure 1

Schematic of a cloud of particles, ${\mathrm{\Omega }}^t$, defined as an infinitesimal 3D manifold occupied by a collection of particles. This cloud, which undergoes deformation characterized by $\mathbf{\Xi }$, may expand or contract toward a central point $\mathbf{\xi }$.

Figure 2

The dynamic adjustment of the linear forcing coefficient $A^t$ based on the measured ${\tau }_{\eta }^t$ using Eq. (34).

Figure 3

Deviation of the instantaneous Kolmogorov time scale ${\tau }_{\eta }^t$ from the target ${\tau }_{\eta }$ in percent over the entire simulation history for controlled $A^t$ with $k=100$ (solid line) and uncontrolled $A^t$ with $k=0$ (dashed line).

Figure 4

The Lagrangian velocity gradients as a function of time for an arbitrary particle interpolated from an Eulerian velocity field. As a particle crosses a cell boundary, a jump is observed in the velocity gradients due to the finite support of the interpolation scheme (dashed lines). This issue, originally caused by performing interpolation and then differentiation, is resolved by first creating a spatially consistent $C^0$ continuous field for the velocity gradient and then performing the Lagrangian interpolation (solid lines).

Figure 5

The ensemble-averaged Lagrangian autocovariance functions (a) $\langle {\rho }^{\mathrm{I}}\rangle$, (b) $\langle {\rho }^{\mathrm{II}}\rangle$, and (c) $\langle {\rho }^{\mathrm{III}}\rangle$ defined in Eq. (30) at different Stokes numbers.

Figure 6

The schematic of four canonical flows and their effect on various autocovariance functions.

Figure 7

The ensemble-averaged velocity gradients $\overline{u_{1,1}{u}_{1,1}}$ (dashed triangles), $\overline{u_{1,2}{u}_{1,2}}$ (dashed squares), $\overline{u_{1,2}{u}_{2,1}}$ (dashed circles), rotation rate (solid dots), strain rate (solid stars), and $Q$ criterion (solid plus) as a function of St. Note $Q=-\overline{u_{i,j}{u}_{j,i}}={\parallel \mathbf{\Omega }\parallel }^2-{\parallel \mathit{S}\parallel }^2$ is two times the traditional definition of the $Q$ criterion [26], here defined for inertial particles in a Lagrangian framework. Each point in this plot corresponds to an autocorrelation function at zero-time separation, which is the intercept of individual curves in Figs. 5 and 8.

Figure 8

The ensemble-averaged Lagrangian (a) rotation-rate and (b) strains-rate autocovariance functions and (c) their difference versus time at different Stokes numbers.

Figure 9

The Fourier transformation of the autocovariance of the second invariant of the velocity gradient tensor, $\langle {\tilde{\rho }}^{\mathrm{Q}}\rangle$, computed along the trajectory of particles with different Stokes number. Note ${\tau }_{\eta }\langle {\tilde{\rho }}^{\mathrm{Q}}({\tau }_{\eta }\omega \ll 1)\rangle \propto 1/\mathrm{St}$ as St $\to \infty$. Error bars (not shown) are of $O\left({10}^{-4}\right)$.

Figure 10

Deformation of three arbitrary spherical clouds in a HIT flow (particles are not shown). These clouds are constructed using a set of inertial particles initially seeded on a spherical shell. From top to bottom: St $={16}^{-1}$, 1, and 16. While the cloud with St $={16}^{-1}$ deforms significantly, its volume remains within 30% of the initial volume. The St $=1$ cloud contracts with a twofold decrease in volume. The St $=16$ cloud remains almost spherical with a 30% change in volume at $t=8{\tau }_{\eta }$.

Figure 11

The ensemble-averaged finite-time contraction rate $\langle {\mathcal{C}}^t\rangle$ at different St. To ensure a fully stationary condition when computing $\mathcal{C}$ from ${\mathcal{C}}^t$, integration is continued to $2000{\tau }_{\eta }$ (only $t\le 100{\tau }_{\eta }$ is shown here). For symbols, see Fig. 9.

Figure 12

The PDF of the rate of contraction ${\tau }_{\eta }{\mathcal{C}}^t$ for different Stokes numbers computed by the direct integration of Eq. (6) (solid), using the present analysis, or Eq. (28) (solid dotted), and using R-M or Eq. (3) (dashed). Note that the spread of these PDF varies depending on the sampling period $t$, which is $200{\tau }_{\eta }$ in this case.

Figure 13

The theoretically predicted ensemble-averaged rate of contraction $\langle \mathcal{C}\rangle$ using the present analysis that is Eq. (28) (solid dot) and R-M that is Eq. (3) (dashed) at different St. The reference quantities (solid circle) are obtained from the direct computation of $\langle \mathcal{C}\rangle$. Inset: ${log}_2(-\mathcal{C})$ versus ${log}_2\left(\mathrm{St}\right)$ for St $\le 1$. Lines with a slope of 1 and 2 are shown for reference. To visualize clustering, particles are shown in a slab of size $300\eta \times 300\eta \times 10\eta$ for St $={16}^{-1}$ (a), $4^{-1}$ (b), 1 (c), 4 (d), 16 (e).

Figure 14

Standardized PDF of the finite-time rate of contraction ${\mathcal{C}}^t$ obtained from the direct integration of Eq. (6) at different St. The solid line is a Gaussian distribution, shown as a reference.

Figure 15

The standard deviation of the rate of contraction ${\mathcal{C}}_{\mathrm{rms}}^t$ using the present analysis (solid dot), R-M (dashed), and the direct computations (solid circle) as a function of St. To make values independent of $t,{\mathcal{C}}_{\mathrm{rms}}^t$ is multiplied by $\sqrt{t}$.