Isotropic stochastic rotation dynamics

Stochastic rotation dynamics (SRD) is a widely used method for the mesoscopic modeling of complex fluids, such as colloidal suspensions or multiphase flows. In this method, however, the underlying Cartesian grid defining the coarse-grained interaction volumes induces anisotropy. We propose an isotropic, lattice-free variant of stochastic rotation dynamics, termed iSRD. Instead of Cartesian grid cells, we employ randomly distributed spherical interaction volumes. This eliminates the requirement of a grid shift, which is essential in standard SRD to maintain Galilean invariance. We derive analytical expressions for the viscosity and the diffusion coefficient in relation to the model parameters, which show excellent agreement with the results obtained in iSRD simulations. The proposed algorithm is particularly suitable to model systems bound by walls of complex shape, where the domain cannot be meshed uniformly. The presented approach is not limited to SRD but is applicable to any other mesoscopic method, where particles interact within certain coarse-grained volumes.

Figure 1

The upper curves correspond to plane Poiseuille flow simulated using SRD without grid shift. The discretization of the channel, which is 20 cells wide, is shown by the vertical thin lines. The particle velocities are averaged within slices oriented parallel to the channel walls. The symbols ($\circ$) show the average particle velocity within slices having width $a$. The upper dashed line is a parabola fitted to this averaged velocity profile in the interval $[-5,5]$, i.e., distant from the walls. The red solid line represents the detailed flow profile, obtained by averaging within slices of width $a/20$. The two lower curves show the results for SRD with grid shift, using the same parameters as before. To reduce velocity slip at the walls, for the solid blue line, ghost particles (see Sec. 5e) were used in addition to the grid shift.

Figure 2

Plane Poiseuille flow simulated using SRD with grid shift and two different grid orientations. The channel walls are modeled following the approach described in Ref. [18].

Figure 3

Mean squared displacement measurement of a force-free system obtained by iSRD. The dashed line shows a linear fit to the simulation data, disregarding the initial transient. The auxiliary grid consists of $32\times 32\times 32$ cubic cells with side length 1.

Figure 4

Couette flow simulated using iSRD with and without ghost particles. The dashed line shows the analytical solution for the given boundary velocities. The size of a coarse-graining sphere is shown at the top left.

Figure 5

Plane Poiseuille flow simulated using iSRD with and without ghost particles. The dashed lines are parabolas fitted in the interval $[-5,5]$. For the viscosity, we obtain $\nu =0.568$ with ghost particles and $\nu =0.569$ without. In this example, the Reynolds number is approximately $15.5$. In dimensional quantities, this would, for example, correspond to water at room temperature (${20}^{\circ }\text{C}$) flowing through a 2-mm-wide channel at velocity $7.8\text{mm}/\text{s}$.

Figure 6

Plane Poiseuille flow simulated using iSRD, including ghost particles, for the three different auxiliary grids depicted in panel (a). The mesh on the left consists of cubic cells with constant width, as indicated by the black scale in panel (b). The red scale refers to the mesh in the center, comprising nonuniform hexahedra with square basis and varying extent in the direction of the velocity gradient, i.e., normal to the walls. The mesh on the right is composed of nonuniform tetrahedral cells.

Figure 7

Redistribution of momentum within one coarse-graining sphere in stationary Couette flow. The coarse-graining sphere is divided into two subvolumes by a horizontal plane at $y={\tilde{y}}_0$.

Figure 8

The characteristic area, $A$, is the average area that the momentum transport within one single coarse-graining sphere refers to. The size of area $A$ can be obtained considering the average number of coarse-graining spheres, $N^{★}$, in a reference volume $V^{★}\equiv A^{★}d$, which is continued periodically in $x$ and $z$ directions.

Figure 9

Diffusion coefficient as a function of the rotation angle, $\alpha$, for two different time steps and $M=10$ particles per coarse-graining sphere on average. The left and right vertical axes refer to the data for $\mathrm{\Delta }t=0.1$ and $\mathrm{\Delta }t=1$, respectively. The dashed line shows the theoretical result, Eq. (23).

Figure 10

Diffusion coefficient as a function of the average number of particles per coarse-graining sphere, $M$, for two different time steps and $\alpha =\pi /2$. The left and right vertical axes refer to the data for $\mathrm{\Delta }t=0.1$ and $\mathrm{\Delta }t=1$, respectively. The dashed line shows the theoretical result, Eq. (23).

Figure 11

Relative deviation between the diffusion coefficient measured in simulations, $D_{\text{sim}}$, and the analytical result from Eq. (23), denoted as $D_{\text{theo}}$, over the mean free path, $\lambda$, which is equal to the time step for the chosen parameters. To yield a pronounced correlation contribution, $\alpha$ and $M$ are chosen as ${130}^{\circ }$ and 5, respectively [35].

Figure 12

Kinematic viscosity as a function of the rotation angle, $\alpha$, for two different time steps and $M=10$ particles per coarse-graining sphere on average. The left and right vertical axes refer to the data for $\mathrm{\Delta }t=0.1$ and $\mathrm{\Delta }t=1$, respectively. The dashed line shows the theoretical result, Eqs. (22) and (36).

Figure 13

Kinematic viscosity as a function of the average number of particles per coarse-graining sphere, $M$, for two different time steps and $\alpha =\pi /2$. The left and right vertical axes refer to the data for $\mathrm{\Delta }t=0.1$ and $\mathrm{\Delta }t=1$, respectively. The line shows the theoretical result, Eqs. (22) and (36).