LBM Boundary Conditions

Typical lattice Boltzmann implementations of boundary conditions introduce an additional error that is dependent on the grid spacing and proportional to the square of the viscosity [1,2]. This error is of particular concern to flows where the Reynolds number is low and spatial resolution is limited. Building on the methodology of Sam Bennett and the analytical work of He, Zou, Luo, and Dembo, my excellent team of students and I have extensively benchmarked moment-based boundary conditions for no-slip flows in 2D rectangular geometries (with Seemaa Mohammed), used it to study natural convection (with Rebecca Allen),  developed it for 3D flows (with Ivars Krastins) and extended the method to wetting boundary conditions with multiphase models of Lee-Fischer type (with Andreas Hantsch).

Oblique dipole-wall collision at different Reynolds number using an LBM with moment-based boundary conditions. The two rows are snapshots of the flow at different times.

Seemaa completed her PhD in 2019 and her most detailed work with no-slip moment-based boundaries for LBM numerically examines the accuracy of the method and shows that, for flat boundaries aligned with grid points, it is extremely reliable when used with a two-relaxation-time collision operator. No restrictions are placed on the operator to impose the conditions.   Her work also sheds light on the physics of the wall-dipole collision problem at high Reynolds numbers. Another PhD student of Dave Graham and I – Zainab Bu Sinnah – has applied the method to pulsatile flow and extended it to 3D flows with slip.

Ivars Krastins, who completed his PhD in 2019 developed the moment-based boundary methodology to 3D flows and Zainab looked at slip flow in three dimensional geometries. It must be noted that, at present, the moment-based methodology is limited to flat boundaries aligned with grid points. In such cases it is very accurate for a wide variety of conditions, but of course lacks some geometric flexibility.

The slides below summarise the methodology of the moment-based approach to imposing boundary conditions. They are from a lecture I gave at the OpenLB Lattice Boltzmann Spring School.

Furthermore, I’ve investigated moment-based boundary conditions and the lattice Boltzmann stress field. I have shown that the stress field obtained from the lattice Boltzmann equation contains a contribution of second order in Knudsen number (i.e. a Burnett term) and shown how a stress condition for the LBM that is consistent with the underlying PDE can be imposed precisely and locally at grid point.  Moreover, I have solved the LBM in planar channel flow analytically for its stress field. This has consequences not just for the implementation of boundary conditions but also for the choice of “optimal” relaxation times. but this work was first shown at the 2014 ICMMES Conference, 2014.

Analytical solution to the LBM deviatoric stress in planar channel flow
Tangential LBM (T) and Newtonian stress along the centrelines in lid-driven cavity flow.  The LBM includes contributions to the deviatoric stress that are nor present in the Navier-Stokes equations (Reis 2019).

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[1] I. Ginzbourg and P. M. Adler, J. Phys. II France, 4:191 (1994)

[2]  X. He, Q. Zou, L.-S. Luo, and M. Dembo. J. Stat. Phys 87:115-136 (1997)