Boundary Conditions (BC)

Boundary nodes are those in the immediate vicinity of a numerical boundary. A boundary node carries many populations but only a few macroscopic quantities have physical meaning at the boundary, so defining a boundary condition in LBM is a question of which macroscopics to impose.

In Nassu’s macroscopic-storing RR-BGK solver this is the foundation of every boundary condition: a BC is defined by the hydrodynamic moments imposed at the boundary node, and the populations are reconstructed from those moments by the collision operator on the next step. The node positions coincide with the boundaries. This regularized, moment-based formulation is developed in the Moment-Based Boundary Conditions page, which the per-type pages below specialise.

Population bounce-back - reflecting individual populations across the link - is the local, viscosity-dependent legacy form. It places the wall half a node off the boundary, so it is documented alongside the moment-based families for completeness rather than as a co-equal method.

Warning

A subtle trap: edges and corners belong to more than one boundary. A node shared by two faces would otherwise receive conflicting populations from each BC. Nassu resolves this by imposing a fixed overwriting order, applied first to last:

SLIP (z) \(\Rightarrow\) SLIP (y) and WALL (y) \(\Rightarrow\) OUTLET and INLET

Changing this order silently changes which BC wins at every edge and corner. To know more about this order and its impact, check BC order validation test.

These pages cover the boundary conditions that act directly on the fluid lattice. The boundary conditions specific to other modules live with their physics: turbulent inlets in the Turbulent inflow chapter, near-wall stress closures in the Wall model chapter, and scalar/thermal BCs in their respective chapters.

The implemented fluid boundary conditions, as moment closures (impose / derive / reconstruct), are listed below:

  • Solid Wall

    • Regularized (moment-based): impose \(\mathbf{u}=\mathbf{0}\), fix density, derive the rate-of-strain from the one-sided finite difference, reconstruct populations. Node-coincident wall, increased stability; uses non-local interior values.

    • Halfway Bounce-Back (legacy, local): second-order accurate but the wall sits half a node off the boundary with a relaxation-time dependence.

  • Moving Wall

    • Regularized (moment-based): impose \(\mathbf{u}=\mathbf{u}_\mathrm{w}\), fix density, derive the stress, reconstruct. Same treatment as the no-slip wall with a non-null wall velocity.

    • Velocity Bounce-Back (legacy, local): second-order but steadily raises the domain-average density, which can destabilise long runs.

  • Free Surface

    • Regularized Neumann (slip): impose zero normal velocity, fix density, zero-gradient the tangential velocity and stress. High stability and local.

  • Inlet

    • Uniform Velocity: impose density and velocity, equilibrium state (\(S_{\alpha\beta}=0\)). Does not raise the average density; operates at constant density.

  • Outlet

    • Regularized Neumann: fix the outlet density (pressure); take velocity and stress zero-gradient from the interior. Holds a stable domain-average density; place it far from regions of high pressure gradient.

    • Neumann: zero-gradient on all macroscopics including density, which keeps it flexible but does not guarantee a stable domain-average density.

    • Anti Bounce-Back (legacy, local): fully local and keeps a stable pressure, but with stability issues that can cause divergence.