Simple diffusion hopping model with convection

We present results from a new variant of a diffusion hopping model, the convective diffusive lattice model, to describe the behavior of a particulate flux around bluff obstacles. Particle interactions are constrained to an underlying square lattice where particles are subject to excluded volume conditions. In an extension to previous models, we impose a real continuous velocity field upon the lattice such that particles have an associated velocity vector. We use this velocity field to mediate the position update of the particles through the use of a convective update after which particles also undergo diffusion. We demonstrate the emergence of an expected wake behind a square obstacle which increases in size with increasing object size. For larger objects we observe the presence of recirculation zones marked by the presence of symmetric vortices in qualitative agreement with experiment and previous simulations.

Figure 1

Example of convective motion for a particle at cell ${\mathbf{r}}_i$ with velocity ${\mathbf{v}}_i$ on a two-dimensional grid. The dashed square represents the displaced position of the particle subject to ${\mathbf{v}}_i$. The shaded regions highlight the degree of overlap between the neighboring cells given as ${\mathbf{r}}_i+{\mathbf{e}}_{\alpha }$ where $\parallel {\mathbf{e}}_{\alpha }\parallel >0$ and the dashed square. Here ${\mathbf{e}}_{\alpha }$ is the displacement vector. The shaded areas are used to define displacement probabilities to the neighboring cells.

Figure 2

Schematic of a two-dimensional system studied with the CDLM. Particle flow takes place between two no-slip rigid boundaries, each of width $R$ and length $L_1$. The distance between the rigid boundaries $d_2=L_2-2R$. A bluff obstacle such as a square obstacle may be included in the flow channel. For the obstacle depicted, each side is $l$ lattice cells in length.

Figure 3

Variation of the mean square displacement of particles in a fully periodic system in the absence of convective flow. Results for three different values of $D_0$ are shown accompanied by a straight line fit (blue dashed line) given by Eq. (4) where $D$ is a fitting parameter.

Figure 4

Spatiotemporally averaged velocity profile (black) calculated in steady state for a system with $L_1=500\mathrm{\Delta }$ and $L_2=500\mathrm{\Delta }$. The dashed line (red) is a fit given by Eq. (7) to the simulation data.

Figure 5

Variation of $\nu$ with $L$. The open circles are the simulation data, while the solid black line is a fit given by $\nu \left(L\right)={\nu }_{\infty }[1-exp(-\alpha L)]$ where ${\nu }_{\infty }=0.0462\pm 0.0004$ and $\alpha =0.0176\pm 0.0001$. (Inset) Variation of $A_{g,x}$ with $L$, the external field required to achieve a constant flow velocity $U_{\text{max}}$. The open squares are the simulation data, while the dashed black line is a fit given by $A_{g,x}\left(L\right)=C_{Ag}{L}^{\beta }$ where $C_{Ag}=0.034\pm 0.001$ and $\beta =-1.98\pm 0.01$. For both figures $L=L_1=L_2$.

Figure 6

Variation of $\nu$ with $D$ for a flow channel with $L_1=L_2=300\mathrm{\Delta }$. The solid black line is a power law fit to the data such that $\nu \left(D\right)=E_{\nu }{D}^{\epsilon }$. Here $E_{\nu }=0.269\pm 0.001$ and $\epsilon =0.582\pm 0.002$.

Figure 7

Color maps for average particle occupancy during steady-state flow in a flow channel containing a square obstacle of side $l=200\mathrm{\Delta }$ with ${\tau }_w=5\times {10}^4\tau$. (a) Biased diffusion where $L_1=1500\mathrm{\Delta }$ and $L_2=500\mathrm{\Delta }$. (b) Convective motion where $L_1=1000\mathrm{\Delta }$ and $L_2=500\mathrm{\Delta }$. (c) Convective motion where $L_1=1000\mathrm{\Delta }$ and $L_2=500\mathrm{\Delta }$ (alternative scaling for color map). For this particle flow the Reynolds number $\text{Re}\approx 650$ as calculated using Eq. (10).

Figure 8

Spatiotemporally averaged velocity field in a flow channel with a square obstacle of side $l=300\mathrm{\Delta }$ with $U=0.01\mathrm{\Delta }/\tau$ so that $\text{Re}\approx 65$. To emphasize the recirculation zones behind the object, the arrows representing the velocity field where $|\langle {\mathbf{v}}_j\rangle |\le 0.003\mathrm{\Delta }/\tau$ have been multiplied by a factor of two. The full dimensions of the flow channel are $L_1=2000\mathrm{\Delta }\mathrm{and}{L}_2=1000\mathrm{\Delta }$. This velocity field is averaged over $1.25\times {10}^5\tau$. For the spatial coarse graining of the lattice the cells are of size $n\mathrm{\Delta }\times n\mathrm{\Delta }$ where $n=10$.

Figure 9

Color map for average particle occupancy for a particle flow in a flow channel with a square obstacle of side $l=300\mathrm{\Delta }$ with $U=0.01\mathrm{\Delta }/\tau$ such that $\text{Re}\approx 65$. The full dimensions of the flow channel are $L_1=2000\mathrm{\Delta }$ and $L_2=1000\mathrm{\Delta }$. The color map is averaged over $1.25\times {10}^5\tau$.

Figure 10

Spatiotemporally averaged velocity field in a flow channel with a square obstacle of side $l=300\mathrm{\Delta }$ with $U=0.1\mathrm{\Delta }/\tau$ such that $\text{Re}\approx 650$. Starting time for the velocity field is $2\times {10}^5\tau$, and the velocity field is averaged over a time scale of $2.5\times {10}^4\tau$. To emphasize the recirculation zones behind the object in each image, the arrows representing the velocity field where $|\langle {\mathbf{v}}_j\rangle |\le 0.02\mathrm{\Delta }/\tau$ have been multiplied by a factor of two. The full dimensions of the flow channel are $L_1=2000\mathrm{\Delta }$ and $L_2=1000\mathrm{\Delta }$.

Figure 11

Spatiotemporally averaged velocity field for the same flow channel presented in Fig. 10. Starting time for this velocity field is $2.25\times {10}^5\tau$, which is just after the velocity field presented in Fig. 10. To emphasize the recirculation zones behind the object in each image, the arrows representing the velocity field where $|\langle {\mathbf{v}}_j\rangle |\le 0.02\mathrm{\Delta }/\tau$ have been multiplied by a factor of two. The full dimensions of the flow channel are $L_1=2000\mathrm{\Delta }$ and $L_2=1000\mathrm{\Delta }$.

Figure 12

Color maps for average particle occupancy in a flow channel with a square obstacle of side $l=300\mathrm{\Delta }$ with $U=0.1\mathrm{\Delta }/\tau$ such that $\text{Re}\approx 650$. Each color map is averaged over a time scale of $2.5\times {10}^4\tau$. The starting time for each of the images is (a) $2\times {10}^5\tau$, (b) $2.25\times {10}^5\tau$, (c) $2.5\times {10}^5\tau$, (d) $2.75\times {10}^5\tau$. The full dimensions of the flow channel are $L_1=2000\mathrm{\Delta }$ and $L_2=1000\mathrm{\Delta }$.

Figure 13

Axial velocity profile $u\left(x\right)$ along the center line of the obstacle for different size obstacles and $U=0.01\mathrm{\Delta }/\tau$. The measure along the $x$ direction represents the distance from the right side of the obstacle.

Figure 14

Axial velocity profile $u\left(x\right)$ along the center line of the obstacle for different size obstacles and varying $U=0.1\mathrm{\Delta }/\tau$.