Enhanced single-node lattice Boltzmann boundary condition for fluid flows

We propose a procedure to implement Dirichlet velocity boundary conditions for complex shapes that use data from a single node only, in the context of the lattice Boltzmann method. Two ideas are at the base of this approach. The first is to generalize the geometrical description of boundary conditions combining bounce-back rule with interpolations. The second is to enhance them by limiting the interpolation extension to the proximity of the boundary. Despite its local nature, the resulting method exhibits second-order convergence for the velocity field and shows similar or better accuracy than the well-established Bouzidi's scheme for curved walls [M. Bouzidi, M. Firdaouss, and P. Lallemand, Phys. Fluids 13, 3452 (2001)]. Among the infinite number of possibilities, we identify several meaningful variants of the method, discerned by their approximation of the second-order nonequilibrium terms and their interpolation coefficients. For each one, we provide two parametrized versions that produce viscosity independent accuracy at steady state. The method proves to be suitable to simulate moving rigid objects or surfaces moving following either the rigid body dynamics or a prescribed kinematic. Also, it applies uniformly and without modifications in the whole domain for any shape, including corners, narrow gaps, or any other singular geometry.

Figure 1

2D representation of the boundary nodes, normalized distance $q$, discrete lattice velocities ${\vec{c}}_i,{\vec{c}}_{\overline{ı}}$, links (dashed segments), and locations of boundaries at the intersections with links ($•$). $\mathrm{\Delta }x$ is the space step.

Figure 2

1D linkwise representation of bounce-back procedure in the BFL method: (left) $q<1/2$ and (right) $q\ge 1/2$. The coefficients $a_1,a_2,a_3$ and the populations are defined in Eqs. (11). The arrows represent the populations and their color is related to the color of the node from which they depart at time $t$ after the collision: light blue for node FF, magenta for node F, black for an off-lattice position. The dashed curved arrows represent the bounce-back rule. ($a$) the bounce-back rule is applied after the interpolation. ($b$) the bounce-back rule is applied before the interpolation.

Figure 3

Equivalences of the representation of population (arrows) through the standard space-time coordinates (LHS of the equivalences) and through the generalized coordinate (RHS of the equivalences). On the left: arrows represent populations at time $t$ in their post-collision state. On the right: arrows represent populations at time $t+1$ in their precollision state.

Figure 4

Representation of computational domain. In blue, the intercylinders Couette flow. The external cylinder is resting, and the inner cylinder is moving with angular velocity $\omega$.

Figure 5

$E_{\mathrm{rrs}}$ error of NELI–FL and NELI–FQ, and BFL methods against the resolution in lattice units $\left[\mathrm{lu}\right]$ with a D3Q27 lattice and TRT collision model ($\mathrm{\Lambda }=3/16$). The results are relative to steady-state condition for $\mathrm{Re}=1$.

Figure 6

$E_{\mathrm{rrs}}$ error at steady state of ELI variants, and ZY methods for intercylinder distance $r_2-r_1$ in lattice units $\left[\mathrm{lu}\right]$ (with $\beta =1/2$). Results obtained using a D2Q9 lattice, BGK collision model, and linear equilibrium.

Figure 7

$E_{\mathrm{rrs}}$ error at steady state of ELI variants, BFL, CLI, and FH methods for increasing values of the symmetric relaxation time ${\tau }^{+}$. The simulations for which instabilities have been detected are listed in the table in the central figure (NB detected instabilities are relative to nonparametrized methods only). Results obtained using a D2Q9 lattice, TRT collision model ($\mathrm{\Lambda }=1/8$) and linear equilibrium.

Figure 8

Average density fluctuation, at dimensionless time $t^{★}=5$ after the impulsive motion of the internal cylinder, as a function of the channel height in lattice units under diffusive scaling using a D2Q9 lattice, TRT collision model ($\mathrm{\Lambda }=1/8$, ${\tau }^{+}=2$) and linear equilibrium ($\mathrm{Re}\ll 1$).

Figure 9

Space convergence of $E_{\mathrm{rrs}}$ at time is $t^{★}=0.001$ of ELI variants, and for BFL, FH, CLI, ZY (in the central figures). The simulation was carried using a D2Q9 lattice, a TRT collision model ($\mathrm{\Lambda }=1/8$) and $\mathrm{Re}=1$.

Figure 10

Comparison of the error of the ELI method variants for different nonequilibrium component treatment (N,S,C), BFL, FH, and CLI, for two values of the magic parameter $\mathrm{\Lambda }$ as a function of the symmetric relaxation time ${\tau }^{+}$. Simulations characterized by channel height of $h=91.6\mathrm{lu}(q=0.8)$ and $\mathrm{Re}=1$.

Figure 11

Sensitivity of the $E_{\mathrm{rrs}}$ error to symmetric relaxation time ${\tau }^{+}$, in the case of impulsively started Couette flow. The experiment refers to a dimensionless time of $t^{★}=0.001$ for $q=0.2$ and $\mathrm{Re}=1$. The simulation was carried using a D2Q9 lattice with 60 nodes in the vertical direction and a TRT collision model ($\mathrm{\Lambda }=1/8$). Results for BFL, CLI, and FH are shown for comparison.

Figure 12

Comparison of accuracy at transient state for different parametrization of the boundary scheme ($K_1^{-}$ of $K_2^{-}$), in the case of impulsively started Couette flow. The experiment is referred to a dimensionless time of $t^{★}=0.001$ for $q=0.2$ and $\mathrm{Re}=1$. The simulation was carried using a D2Q9 lattice and a TRT collision model ($\mathrm{\Lambda }=1/8$).

Figure 13

Sensitivity of the $E_{\mathrm{rrs}}$ error to normalized distance from the wall $q$. The experiment refers to a dimensionless time of $t^{★}=0.001$. The results of BFL are shown for comparison. The simulation was carried using a D2Q9 lattice with 60 nodes in the vertical direction and a TRT collision model ($\mathrm{\Lambda }=1/2$, ${\tau }^{+}=2$).

Figure 14

Sensitivity of the $E_{\mathrm{rrs}}$ error to the magic parameter $\mathrm{\Lambda }$, in the case of impulsively started Couette flow, for CELI, BFL, and FH. The experiment is referred to a dimensionless time of $t^{★}=0.001$. Left: $q=0.2$; Right: $q=0.8$.

Figure 15

$E_{\mathrm{rrs}}$ error as a function of the symmetric relaxation time ${\tau }^{+}$ and of the TRT magic parameter $\mathrm{\Lambda }$. Each mesh vertex corresponds to a different simulation at $t^{★}=0.001$ and channel height $h=60.4\mathrm{lu}$.

Figure 16

Angular velocity of the Jeffery's orbit described by a 3D prolate ellipsoid characterized by a ratio of radii $\beta =r_a/r_b=2$. The solutions of HW, BFL, NELI-FQ, and NELI-ULT schemes (nonparametric) are compared with the analytical solution (black dashed line). Parameters of the simulations: $\tau =2$, BGK collision model, $\mathrm{Re}=4$, $H=10r_e$ (channel height), $r_e=\sqrt{r_a^2+2·r_b^2}=30$ (equivalent radius).

Figure 17

Jeffery's orbit described by a 3D prolate ellipsoid characterized by a ratio of radii $\lambda =r_a/r_b=2$. Plots of the squared fluctuation of the torque obtained with Eq. (E4). (left) Without LIR refilling, (right) with LIR refilling.

Figure 18

The modified LIR algorithm.