Prediction of the moments in advection-diffusion lattice Boltzmann method. II. Attenuation of the boundary layers via double-$\mathrm{\Lambda }$ bounce-back flux scheme

Impact of the unphysical tangential advective-diffusion constraint of the bounce-back (BB) reflection on the impermeable solid surface is examined for the first four moments of concentration. Despite the number of recent improvements for the Neumann condition in the lattice Boltzmann method–advection-diffusion equation, the BB rule remains the only known local mass-conserving no-flux condition suitable for staircase porous geometry. We examine the closure relation of the BB rule in straight channel and cylindrical capillary analytically, and show that it excites the Knudsen-type boundary layers in the nonequilibrium solution for full-weight equilibrium stencil. Although the d2Q5 and d3Q7 coordinate schemes are sufficient for the modeling of isotropic diffusion, the full-weight stencils are appealing for their advanced stability, isotropy, anisotropy and anti-numerical-diffusion ability. The boundary layers are not covered by the Chapman-Enskog expansion around the expected equilibrium, but they accommodate the Chapman-Enskog expansion in the bulk with the closure relation of the bounce-back rule. We show that the induced boundary layers introduce first-order errors in two primary transport properties, namely, mean velocity (first moment) and molecular diffusion coefficient (second moment). As a side effect, the Taylor-dispersion coefficient (second moment), skewness (third moment), and kurtosis (fourth moment) deviate from their physical values and predictions of the fourth-order Chapman-Enskog analysis, even though the kurtosis error in pure diffusion does not depend on grid resolution. In two- and three-dimensional grid-aligned channels and open-tubular conduits, the errors of velocity and diffusion are proportional to the diagonal weight values of the corresponding equilibrium terms. The d2Q5 and d3Q7 schemes do not suffer from this deficiency in grid-aligned geometries but they cannot avoid it if the boundaries are not parallel to the coordinate lines. In order to vanish or attenuate the disparity of the modeled transport coefficients with the equilibrium weights without any modification of the BB rule, we propose to use the two-relaxation-times collision operator with free-tunable product of two eigenfunctions $\mathrm{\Lambda }$. Two different values ${\mathrm{\Lambda }}_v$ and ${\mathrm{\Lambda }}_b$ are assigned for bulk and boundary nodes, respectively. The rationale behind this is that ${\mathrm{\Lambda }}_v$ is adjustable for stability, accuracy, or other purposes, while the corresponding ${\mathrm{\Lambda }}_b\left({\mathrm{\Lambda }}_v\right)$ controls the primary accommodation effects. Two distinguished but similar functional relations ${\mathrm{\Lambda }}_b\left({\mathrm{\Lambda }}_v\right)$ are constructed analytically: they preserve advection velocity in parabolic profile, exactly in the two-dimensional channel and very accurately in a three-dimensional cylindrical capillary. For any velocity-weight stencil, the (local) double-$\mathrm{\Lambda }$ BB scheme produces quasi-identical solutions with the (nonlocal) specular-forward reflection for first four moments in a channel. In a capillary, this strategy allows for the accurate modeling of the Taylor-dispersion and non-Gaussian effects. As illustrative example, it is shown that in the flow around a circular obstacle, the double-$\mathrm{\Lambda }$ scheme may also vanish the dependency of mean velocity on the velocity weight; the required value for ${\mathrm{\Lambda }}_b\left({\mathrm{\Lambda }}_v\right)$ can be identified in a few bisection iterations in given geometry. A positive solution for ${\mathrm{\Lambda }}_b\left({\mathrm{\Lambda }}_v\right)$ may not exist in pure diffusion, but a sufficiently small value of ${\mathrm{\Lambda }}_b$ significantly reduces the disparity in diffusion coefficient with the mass weight in ducts and in the presence of rectangular obstacles. Although ${\mathrm{\Lambda }}_b$ also controls the effective position of straight or curved boundaries, the double-$\mathrm{\Lambda }$ scheme deals with the lower-order effects. Its idea and construction may help understanding and amelioration of the anomalous, zero- and first-order behavior of the macroscopic solution in the presence of the bulk and boundary or interface discontinuities, commonly found in multiphase flow and heterogeneous transport.

Figure 1

This figure illustrates the coordinate and the diagonal equilibrium stencils of the $\text{d2Q9}$ scheme for, from the left to the right, the mass weight, the velocity weight, and the anti-diffusion weight in Eq. (2). The three families of weights are independent, but all of them satisfy the isotropic constraint ${\sum }_{q=1}^{Q_m}{t}_q^{(·)}{c}_{q\alpha }{c}_{q\beta }={\delta }_{\alpha \beta }$. The $\text{d3Q15}$ is composed from 6 coordinate discrete velocities $\left|\right|{\mathbf{c}}_q{\left|\right|}^2=1$ with the weight $t_c^{(·)}$, and 8 diagonal velocities $\left|\right|{\mathbf{c}}_q{\left|\right|}^2=3$ with the weight $t_d^{(·)}$. The $\text{d2Q5}$ and $\text{d3Q7}$ yield $t_c^{(·)}=\frac{1}{2}$, $t_d^{(·)}=0$. The hydrodynamic choice is $t_c^{\left(m\right)}=t_c^{\left(a\right)}=\frac{1}{3}$, $t_d^{\left(m\right)}=t_d^{\left(a\right)}=\frac{1}{12}$ ($\text{d2Q9}$), and $t_d^{\left(m\right)}=t_d^{\left(a\right)}=\frac{1}{24}$ ($\text{d3Q15}$).

Figure 2

This figure illustrates construction of the nonequilibrium boundary layers along the vertical axis in straight channel and along a coordinate axis in circular cross section. The grid nodes are those whose centers lie inside the bounded domain; the boundary nodes ${\mathbf{r}}_{\mathrm{b}}=\{{\mathbf{r}}_0,{\mathbf{r}}_N\}$ have two ($\text{d2Q9}$) and four ($\text{d3Q15}$) incoming diagonal populations with $c_{qx}=\pm 1$.

Figure 3

This figure plots the functions (a) $P\left(n\right)$ and (b) $M\left(n\right)$ defined in Eq. (24), then (c) $\mathcal{P}\left(n\right)$ and (d) $\mathcal{M}\left(n\right)$ from Eq. (25), when $\mathrm{\Lambda }=\frac{1}{12}$ (dotted line, “squares,” black), $\mathrm{\Lambda }=\frac{1}{6}$ (dotted-dashed line,“lozenges,” blue), $\mathrm{\Lambda }=\frac{1}{2}$ (dashed line, “circles,” magenta). The $\mathrm{\Lambda }=\frac{1}{4}$ (solid line, “triangles,” red) is constrained to the boundary points. The two functions (a) and (b) produce (for $n=0,1,...,7$) the boundary-layer solution in straight channel of $H=8$ displayed in Figs. 5 and 6.

Figure 4

This figure compares pure-diffusion numerical results (symbols) by $\text{d3Q15}\text{bounce-back}$ scheme for its relative error in diffusion coefficient to its theoretical prediction ${\text{err}}_D^{\left(bb\right)}={\delta }_m$ (straight lines) given by Eq. (27) with Eq. (35) for $k_T=0$. From the top to the bottom: $R=\{5,10,20\}$. The results are plotted for $\mathrm{\Lambda }=\{\frac{1}{12},\frac{1}{6},\frac{1}{4},\frac{1}{2}\}$ with dotted line (black), dotted-dashed line (blue), solid line (red), dashed line (magenta).

Figure 5

The pure diffusion in straight channel is modeled by $\text{d2Q9}\text{BB}$ with $\mathrm{\Lambda }=\frac{1}{12}$, $t_c^{\left(m\right)}=\frac{1}{4}$. The two first diagrams compare, respectively, the predicted projections (lines) for (i) $m=\frac{[g_q^{-}(x_c,y)·{\mathbf{c}}_x]}{\left|\right|{\mathbf{c}}_x{\left|\right|}^2}$, ${\mathbf{c}}_x=\left\{c_{qx}\right\}$, and (ii) $m=\frac{[g_q^{+}(x_c,y)·{\mathbf{v}}_{xy}]}{\left|\right|{\mathbf{c}}_x{\left|\right|}^2}$, ${\mathbf{v}}_{xy}=\left\{c_{qx}{c}_{qy}\right\}$, to their numerical values (symbols) at grid points $y=1,2,...,H=8$. The two last diagrams display results for $H=40$, with $m=\frac{[g_q^{-}(x_c,y)·{\mathbf{q}}_x]}{\left|\right|{\mathbf{q}}_x{\left|\right|}^2}$, ${\mathbf{q}}_x=c_{qx}(3c_{qy}^2-2)$, and $m=\frac{[g_q^{+}(x_c,y)·{\mathbf{v}}_{xy}]}{\left|\right|{\mathbf{v}}_{xy}{\left|\right|}^2}$. The nonequilibrium solution $g_q^{\pm }=G_q^{\pm }+\delta g_q^{\pm }$ is predicted by Eqs. (28) and (30); it is monitored at grid node $x_c=x_0-1$ at some time $t$ and normalized by ${max}_x{\partial }_t\overline{C}(x,t)$.

Figure 6

Similarly as in Fig. 5 but for $\mathrm{\Lambda }=\frac{1}{4}$.

Figure 7

Pure diffusion in straight channel. In two first diagrams, the (horizontal) lines plot $t_c^{\left(m\right)}$- and $H$-independent truncation prediction (38) for ${\text{Ku}}_{☆}^{\left(\mathrm{tr}\right)}$ when $\{c_e,{\mathrm{\Lambda }}^{-}\}=\{\frac{1}{3},\sqrt{\frac{1}{12}}\}$ and $\mathrm{\Lambda }=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [solid line (red), dotted-dashed line (blue), dotted line (black)]. In two last diagrams, the lines plot $t_c^{\left(m\right)}$- and $H$-dependent bounce-back prediction (46). The numerical results of the $\text{d2Q9}\text{BB}$ (“empty” symbols) are computed with $t_c^{\left(m\right)}=\frac{1}{4}$ in the first (third) diagrams and with $t_c^{\left(m\right)}=0$ in the second (fourth) diagrams.

Figure 8

Pure diffusion in straight channel is modeled with the $\text{d2Q9}\text{BB}$ scheme for $\mathrm{\Lambda }=\frac{1}{4}$ and $t_c^{\left(m\right)}=\frac{1}{4}$. The first diagram displays the dependency $C^{\prime }(x,y)$ in three points from center $x_0$, $x=\{x_0,x_0-50,x_0-100\}$ (red, magenta, blue), versus $y\in [-H/2,H/2]$, $H=40$. The second diagram plots together numerical solution $C^{\prime }(x,y=0)$ (blue) and its predicted parabolic approximation (magenta) $C^{\prime }(x,y=0)\approx \frac{{\delta }_m}{24}(1+2H^2){\partial }_x^2\overline{C}\left(x\right)$. The third diagram plots together ${\partial }_x^4\overline{C}\left(x\right)$ and the fourth-order correction ${\partial }_t\overline{C}-{\mathrm{\Lambda }}^{-}(1+{\delta }_m){\partial }_x^2\overline{C}$, divided by $A_3^{bb}$ from Eq. (39). The $C^{\prime }(x,y)$ is normalized with ${max}_x{\partial }_t\overline{C}(x,t)$ in all diagrams.

Figure 9

The first diagram compares two pure-diffusion solutions by $\text{d2Q5}$ with ${\text{err}}_D=0$ (dotted-dashed line, black) to Gaussian distribution (dotted line, magenta) in straight channel at $t^{\prime }=t/T=\{\frac{625\sqrt{3}}{96},\frac{625\sqrt{3}}{48}\}$, $T=H^2/c_e{\mathrm{\Lambda }}^{-}$, $c_e{\mathrm{\Lambda }}^{-}=\frac{1}{3}\sqrt{\frac{1}{12}\phantom{{}^{l^l}}}$, $H=8$. The second diagram compares profiles by $\text{d2Q5}$ (dotted-dashed line, black) and “rotated” $\text{d2Q9}$ scheme $\{t_c^{\left(m\right)}=0,t_d^{\left(m\right)}=\frac{1}{4}\}$ (dotted line, blue) where ${\text{err}}_D={\delta }_m=-1/H$ (or $-12.5\%$) according to Eq. (35a) for $\mathrm{\Lambda }=\frac{1}{4}$. The last diagram compares the centered profiles of these two schemes in parabolic flow $\text{Pe}\approx 9.5$. They correspond to the largest discrepancy in moments displayed in Fig. 18 (bottom row), when ${\text{err}}_D$, $\text{err}\left(\text{Sk}\right)$, and $\text{err}\left(\text{Ku}\right)$ differ in two schemes by about $-3.5\%$, $-130\%$, and $50\%$ for $H=20$.

Figure 10

The two first diagrams (a) $\mathrm{\Lambda }=\frac{1}{12}$ and (b) $\mathrm{\Lambda }=\frac{1}{4}$ illustrate the bounce-back retardation effect in cylindrical capillary of $R=5$ due to velocity weight, when $t_c^{\left(a\right)}=\{\frac{1}{2},\frac{1}{3},0\}$ [dotted line (black), dashed line (blue), dotted-dashed line (red)]. The third diagram (c) displays together for $\mathrm{\Lambda }=\frac{1}{12}$ and $\frac{1}{4}$, and for three weights, the profiles obtained with the double-$\mathrm{\Lambda }$ $\text{BB}$ scheme. The simulations are run at $\text{Pe}\approx 9.5$ starting from the uniform plume $C(x=x_0,r)=1$ and output after ${10}^4$ time steps; the mass weight $t_c^{\left(m\right)}=\frac{1}{3}$.

Figure 11

The two first diagrams (a) $R=5$ and (b) $R=10$ compare numerical results (symbols) for the relative mean-velocity error ${\text{err}}_U$ in plug flow prescribed in cylindrical capillary to its theoretical prediction ${\text{err}}_U^{\left(bb\right)}(t_c^{\left(a\right)},\mathrm{\Lambda },R)$ (straight lines) given by Eq. (55), with ${\text{err}}_U(t_c^{\left(a\right)}=\frac{1}{2})=0$. The last diagram (c) compares numerical results for ${\text{err}}_D/{\text{Pe}}^2$ (symbols) to theoretical prediction ${\text{err}}_D^{(bb,U)}(t_c^{\left(a\right)},\mathrm{\Lambda },R)/{\text{Pe}}^2$ (lines) for boundary-layer relative dispersion correction given by Eq. (60) when $R=10$ (data are in percents, $t_c=t_c^{\left(a\right)}$). The results are plotted for uniform distribution $\mathrm{\Lambda }=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dotted line (black), dotted-dashed line (blue), solid line (red)].

Figure 12

This figure quantifies bounce-back retardation of the Poiseuille profile in cylindrical capillary of $R=\{5,10,20\}$ (from the top to the bottom). The numerical values ${\text{err}}_U-{\text{err}}_U^{\left(\mathrm{sum}\right)}$ in Eq. (50) (symbols) are compared to its theoretical prediction ${\text{err}}_U^{\left(bb\right)}(t_c^{\left(a\right)},\mathrm{\Lambda },R)$ from Eq. (57) (straight lines) The results are plotted for uniform distribution $\mathrm{\Lambda }=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dotted line (black), dotted-dashed line (blue), solid line (red)].

Figure 13

This figure illustrates construction of the double-$\mathrm{\Lambda }$ $\text{BB}$ scheme in (a) channel and (b) circular shape, where ${\mathrm{\Lambda }}_v$ is set in all bulk nodes while ${\mathrm{\Lambda }}_b$ is prescribed in all boundary nodes. The bounce-back closure conditions are set at $n=n_b$, the interface conditions are set for $n=\{n_b,n_v\}$, and the axis-symmetry condition is set for $n=\{n_i^{+},n_i^{-}\}$.

Figure 14

Straight channel of $H=12$ is modeled with the double-$\mathrm{\Lambda }$ $\text{BB}$ scheme for (arbitrary) small value ${\mathrm{\Lambda }}_b=\frac{1}{4}\times {10}^{-2}$. The numerical results for relative velocity error ${\text{err}}_U^{\left(bb\right)}={\text{err}}_U-{\text{err}}_U^{\left(\mathrm{sum}\right)}$ are displayed versus velocity-weight $t_c=t_c^{\left(a\right)}$, plug flow in the left diagram and parabolic flow in the right diagram. Their predicted respective solutions (69) and (77) are plotted by lines for $\mathrm{\Lambda }=\{\frac{1}{6},\frac{1}{4},\frac{1}{2}\}$ [dashed-dotted line (blue, “lozenges”), solid line (red, “triangles”), dashed line (magenta, “squares”)].

Figure 15

The two diagrams show (a) exact solution ${\mathrm{\Lambda }}_b({\mathrm{\Lambda }}_v,H)$ given by Eq. (71) for plug flow in straight channel, $H=\{4,8,16\}$ (solid, dashed, dotted lines) and (b) approximate solution ${\mathrm{\Lambda }}_b({\mathrm{\Lambda }}_v,R)$ for plug flow in a capillary, $R=\{2,4,8\}$ (solid, dashed, dotted lines). The ${\mathrm{\Lambda }}_b({\mathrm{\Lambda }}_v,\mathcal{L})$ is negative meaning that the desired solution does not exist in plug flow.

Figure 16

The two first diagrams (a) and (b) plot solution (71) for ${\mathrm{\Lambda }}_b^{\left(\mathrm{sol}\right)}({\mathrm{\Lambda }}_v,H)$ in double-$\mathrm{\Lambda }$ scheme derived for parabolic profile in 2D channel. (a) ${\mathrm{\Lambda }}_b\left({\mathrm{\Lambda }}_v\right)$ for $H=\{4,8,16\}$ (solid, dashed, dotted lines) and (b) ${\mathrm{\Lambda }}_b\left(H\right)$ for ${\mathrm{\Lambda }}_v=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dotted line (black), dotted-dashed line (blue), solid line (red)]. The two last diagrams (c) and (d) plot solution ${\mathrm{\Lambda }}_b({\mathrm{\Lambda }}_v,R)$ for parabolic flow in a capillary. (c) ${\mathrm{\Lambda }}_b^{\left(\mathrm{sol}\right)}\left({\mathrm{\Lambda }}_v\right)$ for $R=\{5,10,20\}$ (solid, dashed, dotted lines) and (d) ${\mathrm{\Lambda }}_b^{\left(\mathrm{sol}\right)}\left(R\right)$ for ${\mathrm{\Lambda }}_v=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dotted line (black), dotted-dashed line (blue), solid line (red)].

Figure 17

This figure displays numerical results (lines with symbols) obtained with the $\text{d3Q15}$ double-$\mathrm{\Lambda }$ $\text{BB}$ scheme in cylindrical capillary with the help of solution ${\mathrm{\Lambda }}_b^{\left(\mathrm{sol}\right)}({\mathrm{\Lambda }}_v,R)$ from Table 3. The relative velocity error ${\text{err}}_U^{\left(bb\right)}={\text{err}}_U-{\text{err}}_U^{\left(\mathrm{sum}\right)}$ in Poiseuille flow is displayed when $R=\{5,10,20\}$, from the top to the bottom. To be compared with Fig. 12 for uniform $\mathrm{\Lambda }$. The numerical results are plotted for bulk values ${\mathrm{\Lambda }}_v=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [“squares” (black), “lozenges” (blue), “triangles” (red)].

Figure 18

The ${\text{err}}_D$, $\text{err}\left(\text{Sk}\right)$, and $\text{err}\left(\text{Ku}\right)$ (from the left to the right) at $\text{Pe}\approx 9.5$ in straight Poiseuille profile. The results are plotted together for $\text{d2Q9}\text{SNL}$ (“empty” symbols) and $\text{d2Q9}\text{BB}$ (“filled” symbols) and compared to bulk predictions (lines) when velocity weight $t_c^{\left(a\right)}=\frac{1}{2}$ [dashed line (blue), “circles”] and $t_c^{\left(a\right)}=0$ [dotted line (magenta), “triangles”]. Top row: mass weight $t_c^{\left(m\right)}=\frac{1}{2}$. Bottom row: $t_c^{\left(m\right)}=0$. In the first column, the $\text{d2Q9}\text{SNL}$ agrees with the bulk prediction ${\text{err}}_D^{\left(\mathrm{bulk}\right)}\left(t_c^{\left(a\right)}\right)$; the $\text{d2Q9}\text{BB}$ agrees with ${\text{err}}_D^{\left(\mathrm{bulk}\right)}\left(t_c^{\left(a\right)}\right)+{\text{err}}_D^{\left(bb\right)}\left(t_c^{\left(m\right)}\right)+{\text{err}}_D^{(bb,U)}\left(t_c^{\left(a\right)}\right)$, with account of the boundary-layer diffusion and dispersion corrections.

Figure 19

Similarly as in Fig. 18 but for $\text{Pe}\approx 95$.

Figure 20

Similarly as in Fig. 18, $\text{Pe}\approx 9.5$, but the $\text{d2Q9}\text{SNL}$ and $\text{d2Q9}\text{BB}$ are both applied with the double-$\mathrm{\Lambda }$ scheme (78). Top row: $t_c^{\left(m\right)}=\frac{1}{2}$. Bottom row: $t_c^{\left(m\right)}=0$. The $\text{d2Q9}\text{BB}$ results for ${\text{err}}_D$ are compared to ${\text{err}}_D^{\left(\mathrm{bulk}\right)}+{\text{err}}_D^{\left(bb\right)}\left(t_c^{\left(m\right)}\right)$.

Figure 21

Similarly as in Fig. 19, $\text{Pe}\approx 95$, but the $\text{d2Q9}\text{SNL}$ and $\text{d2Q9}\text{BB}$ are both applied with the double-$\mathrm{\Lambda }$ distribution (78).

Figure 22

The numerical estimate of the apparent bulk dispersion ${\text{err}}_D^{\left(\mathrm{num}\right)}-({\text{err}}_D^{\left(bb\right)}+{\text{err}}_D^{(bb,U)})$ is compared to its bulk prediction ${\text{err}}_D^{\left(\mathrm{bulk}\right)}$ from [30] (dashed lines of the same color as the numerical data), in Poiseuille flow in cylindrical capillary of $R=20$, with parameters of Expt. I–Expt. IV (Table 2) from the left to the right and from the top to the bottom. The numerical simulations are run with the uniform $\mathrm{\Lambda }=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$.

Figure 23

The four diagrams are similar to Fig. 22 but ${\text{err}}_D^{(bb,U)}$ is intentionally omitted and ${\text{err}}_D^{\left(\mathrm{num}\right)}-{\text{err}}_D^{\left(bb\right)}$ is compared with the bulk estimate ${\text{err}}_D^{\left(\mathrm{bulk}\right)}$ (the dashed lines of the same color as ${\text{err}}_D^{\left(\mathrm{bulk}\right)}$).

Figure 24

This figure illustrates the depreciation of the boundary-layer dispersion component ${\text{err}}_D^{(bb,U)}$ with the double-$\mathrm{\Lambda }$ $\text{BB}$ scheme from Table 3. The computations are run with ${\mathrm{\Lambda }}_v=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dotted line (black), dotted-dashed line (blue), solid line (red)]. Numerical results for ${\text{err}}_D^{\left(\mathrm{num}\right)}-{\text{err}}_D^{\left(bb\right)}$ are plotted together with the bulk prediction ${\text{err}}_D^{\left(\mathrm{bulk}\right)}$ (dashed lines of the same color). To be compared with Fig. 23 where the uniform-$\mathrm{\Lambda }$ $\text{d3Q15}\text{BB}$ scheme is employed.

Figure 25

Cylindrical pipe of $R=20$, Expt. II [dotted line (red), “squares”] and Expt. IV [dotted-dashed line (blue), “circles”] from Table 2. This figure displays the difference in the results obtained with $t_c^{\left(a\right)}=\frac{1}{3}$ and $t_c^{\left(a\right)}=\frac{1}{2}$ versus bulk value $\mathrm{\Lambda }={\mathrm{\Lambda }}_v$. The numerical data (symbols) and truncation estimate [30] (lines) are displayed together for (a) $\left|\right|{\text{err}}_D\left|\right|$, (b) $\left|\right|\text{err}\left(\text{Sk}\right)\left|\right|$ and (c) $\left|\right|\text{err}\left(\text{Ku}\right)\left|\right|$. The double-$\mathrm{\Lambda }$ scheme is applied; it attenuates the boundary-layer advection effects due to $t_c^{\left(a\right)}=\frac{1}{3}$ and enables the truncation estimate.

Figure 26

Three regular arrangements in unit cell $X\times Y={60}^2$ are examined: (a) centered square cylinders of porosity $\theta =0.36$ (obstacle length is $\frac{4}{5}$ of the cell width); (b) centered circular cylinders of porosity $\theta =0.8365$ (the diameter halves the cell width); (c) staggered square cylinders of porosity $\theta =0.56$ (the neighbor corners are distanced by $2\sqrt{2}$ l.u.)

Figure 27

This figure displays the relative error ${\text{err}}_D$ of the relative dispersivity $D_r$ to its reference value in three regular porous arrangements from Fig. 26. The results are presented versus the diagonal mass weight $t_d^{\left(m\right)}\in [0,\frac{1}{4}]$. The computations are performed with the uniform-$\mathrm{\Lambda }$ scheme (“empty symbols”) and the double-$\mathrm{\Lambda }$ scheme ${\mathrm{\Lambda }}_b={10}^{-4}$ (“filled symbols”), when ${\mathrm{\Lambda }}_v=\frac{1}{4}$ (black, “squares”) and ${\mathrm{\Lambda }}_v=\frac{1}{6}$ (blue, “triangles”).

Figure 28

The diagram in the right illustrates streamlines of the velocity field computed in streamwise-periodic porous cell $[X\times H=30\times 20]$ sketched left. The cell is bounded by the horizontal walls; the circular obstacle of $R=7$ is placed at $(x,y)=(14.5,11.5)$.

Figure 29

Similar as in Fig. 28 but the cell is periodic at all ends.

Figure 30

This figure displays numerical results of the uniform-$\mathrm{\Lambda }$ $\text{d2Q9}\text{BB}$ scheme in porous system depicted in Fig. 28, for (a) ${\text{err}}_U={\mathcal{U}}^{\left(\mathrm{num}\right)}/\mathcal{U}-1$ and (b) $\left|\right|{\text{err}}_U\left|\right|={\text{err}}_U\left(t_d^{\left(a\right)}\right)-{\text{err}}_U(t_d^{\left(a\right)}=0)$. The results are plotted versus the diagonal velocity-weight value $t_d^{\left(a\right)}\in [0,\frac{1}{4}]$ when $\mathrm{\Lambda }=\{\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dotted line (black) triangles, dotted-dashed line (blue) lozenges, solid line (red) squares].

Figure 31

This figure examines ${\text{err}}_U$ (left) and $\left|\right|{\text{err}}_U\left|\right|$ (right) in simulations with the double-$\mathrm{\Lambda }$ $\text{d2Q9}\text{BB}$ scheme in porous system bounded by the two horizontal walls. The bulk value ${\mathrm{\Lambda }}_v=\frac{1}{6}$ is fixed for all results; the uniform-$\mathrm{\Lambda }$ solution is recalled from Fig. 30 [dotted-dashed line (blue) lozenges]. The ${\mathrm{\Lambda }}_b$ takes three values: ${\mathrm{\Lambda }}_b^{\left(i\right)}\approx \{0.033726,{10}^{-4},0.011\left(6\right)\}$, $i=1,2,3$ [dashed line (magenta) squares, dotted line (black) triangles, solid line (red) circles]; ${\mathrm{\Lambda }}_b^{\left(3\right)}$ solves the linear approximate $\left|\right|{\text{err}}_U\left({\mathrm{\Lambda }}_b\right)\left|\right|=0$ when ${\mathrm{\Lambda }}_b\in [{\mathrm{\Lambda }}_b^{\left(2\right)},{\mathrm{\Lambda }}_b^{\left(1\right)}]$.

Figure 32

This figure displays numerical results ${\text{err}}_U\left(t_d^{\left(a\right)}\right)$ (left) and $\left|\right|{\text{err}}_U\left(t_d^{\left(a\right)}\right)\left|\right|$ (right) for simulations in fully periodic flow depicted in Fig. 29. The results are plotted for uniform distribution $\mathrm{\Lambda }=\{{10}^{-4},\frac{1}{12},\frac{1}{6},\frac{1}{4}\}$ [dashed line (magenta) circles, dotted line (black) triangles, dotted-dashed line (blue) lozenges, solid line (red) squares]. To be compared with the similar simulations in bounded porous channel in Fig. 30.

Figure 33

This figure examines ${\text{err}}_U$ (left) and $\left|\right|{\text{err}}_U\left|\right|$ (right) in simulations with the double-$\mathrm{\Lambda }$ scheme in fully periodic porous system. The bulk value ${\mathrm{\Lambda }}_v=\frac{1}{4}$ is fixed for all results; the uniform-$\mathrm{\Lambda }$ solution is recalled from Fig. 32 [solid line (red), squares]. The ${\mathrm{\Lambda }}_b$ takes two values: ${\mathrm{\Lambda }}_b^{\left(1\right)}\approx 0.04365$ from Eq. (78) [dashed-dotted line (blue), triangles] and ${\mathrm{\Lambda }}_b^{\left(2\right)}\approx 0.05$ [dotted line (black), lozenges]; ${\mathrm{\Lambda }}_b^{\left(2\right)}$ solves linear approximate $\left|\right|{\text{err}}_U\left({\mathrm{\Lambda }}_b\right)\left|\right|=0$ when ${\mathrm{\Lambda }}_b\in [{\mathrm{\Lambda }}_v,{\mathrm{\Lambda }}_b^{\left(1\right)}]$.