Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion

Abstract

This work demonstrates that in advection-diffusion Lattice Boltzmann schemes, the local mass-conserving boundary rules, such as bounce-back and local specular reflection, may modify the transport coefficients predicted by the Chapman-Enskog expansion when they enforce to zero not only the normal, but also the tangential boundary flux. In order to accommodate it to the bulk solution, the system develops a Knudsen-layer correction to the non-equilibrium part of the population solution. Two principal secondary effects-(i) decrease in the diffusion coefficient, and (ii) retardation of the average advection velocity, obtained in a closed analytical form, are proportional, respectively, to freely assigned diagonal weights for equilibrium mass and velocity terms. In addition, due to their transverse velocity gradients, the boundary layers affect the longitudinal diffusion coefficient similarly to Taylor dispersion, as they grow as the square of the Péclet number. These numerical artifacts can be eliminated or reduced by a proper space distribution of the free-tunable collision eigenvalue in two-relaxation-time schemes.

Truncation effect on Taylor–Aris dispersion in lattice Boltzmann schemes: Accuracy towards stability

“…However, any non-zero diagonal weights, as the hydrodynamic weights, create the artificial Knudsen-type boundary layers [46], as the system response on the boundary restriction of the tangential advection-diffusion flux by the bounce-back rule [45]. Furthermore, the transverse velocity gradients in those induced boundarylayers produce the longitudinal numerical dispersion, in agreement with the Taylor mechanism and the three types of dispersions, namely, physical, truncation and boundary, are superposed.…”

“…Further boundary analysis [45] has shown that the bounce-back drawback originates in its restriction of the tangential boundary flux, handled in hydrodynamic velocity sets by their diagonal links. In the accompanying work [46], we exactly quantify the difference in mean advection velocity and diffusion/dispersion for the specular reflection and bounce-back, in plug and parabolic flows. It has been also noted [45] that the specular reflection delivered identical solutions with the periodic condition on the straight solid wall in plug flow.…”

“…All in all, the results from this section have demonstrated the combined role of the equilibrium velocityweight and Λ on the accuracy and stability in d2Q9-SNL scheme, in the presence of the high mechanical dispersion and in the absence of the boundary effect. The truncation and spurious boundary effects of the bounce-back rule in the presence of the diagonal links are examined in parallel study [46] and summarized in Section 5.…”

“…Furthermore, the transverse velocity gradients in those induced boundarylayers produce the longitudinal numerical dispersion, in agreement with the Taylor mechanism and the three types of dispersions, namely, physical, truncation and boundary, are superposed. A simple remedy [46] consists in using of the specific (typically very small) Λ values at the boundary nodes only. The rational behind is that the boundary Λ−values control the amplitude of the Knudsen layers, in agreement with the prediction [50].…”

Mass-balance and locality versus accuracy with the new boundary and interface-conjugate approaches in advection-diffusion lattice Boltzmann method

“…16,29,52,54,97 However, it was recognized 14,107 that although the mirror reflection is suitable, the BB and "local" specular reflection (which returns the mass to departure node), enforce to zero not only the normal but also the tangential mass-flux components through the wall-inclined discrete-velocities. 26,33 The implicit interface tracking shares a similar tangential deficiency, because its advective-diffusive flux and scalarfield continuity conditions intrinsically couple 29 the BB and the (anti-BB) ABB Dirichlet rule. 26,56 The distribution moments can be regarded as the best indicators of the nonphysical solution behavior.…”