Boundary Conditions

Boundary conditions define the flow behavior at the edges of the computational domain and at solid surfaces. In the Lattice Boltzmann Method, boundary conditions operate at the mesoscopic level - they prescribe the unknown particle populations at boundary nodes rather than directly setting macroscopic quantities like velocity or pressure.

AeroSim provides boundary conditions for solid walls, inlets, outlets, and free surfaces. Each type is suited to a specific physical situation, and the choice affects both the accuracy and stability of the simulation.

See also

For turbulent inflow generation using the Synthetic Eddy Method, see Synthetic Eddy Method (SEM).

Solid walls

See also

Solid Wall - Theory - mathematical formulation of bounce-back and regularized bounce-back conditions.

Halfway bounce-back

The halfway bounce-back (HWBB) is the standard no-slip wall condition. Unknown populations at a wall node are obtained by reflecting the opposite post-collision populations back toward the fluid. The physical wall is located halfway between the boundary node and the first fluid node.

HWBB is fully local (it uses only data from the boundary node itself) and has second-order accuracy.

Note

The wall position does not coincide with the lattice node. This half-node offset must be considered when measuring distances from the wall or positioning probes near surfaces.

Regularized bounce-back

The regularized bounce-back reconstructs all populations at the wall node from macroscopic quantities instead of reflecting individual populations. The velocity is set to zero (no-slip), the density is extrapolated from the first interior node, and the rate-of-strain \(S_{\alpha\beta}\) is estimated using a second-order finite-difference scheme from the near-wall velocity field.

This approach provides increased stability compared to HWBB, particularly in turbulent simulations, but requires information from neighboring nodes (it is not fully local).

Wall models

See also

Wall Model - Theory - overview of wall model formulations and their coupling with LBM and IBM.

Wall Functions - log-law and thin boundary layer equation derivations.

In LES simulations, the near-wall grid resolution is often insufficient to resolve the turbulent boundary layer. Wall models bridge this gap by computing the wall stress from a modeled velocity profile between the wall and a reference point in the resolved flow region.

AeroSim provides two wall model formulations:

Equilibrium log-law

The log-law wall model computes the friction velocity \(u^*\) from the tangential velocity at a reference point using:

\[u^* = \frac{\kappa \, u_{t,P}}{\ln(z_P / z_0)}\]

where \(\kappa = 0.41\) is the von Karman constant and \(z_0\) is the aerodynamic roughness length. This model is well suited for atmospheric boundary layer simulations where the surface roughness is characterized by a terrain category.

Note

The roughness length \(z_0\) is the primary parameter that controls the velocity profile shape near the ground. Different terrain categories correspond to different \(z_0\) values, from open sea (\(z_0 = 0.003\) m) to dense urban areas (\(z_0 = 1.0\) m).

Thin boundary layer (TBL)

The TBL wall model solves a simplified thin boundary layer equation between the wall and the reference point. It uses the Johnson-King mixing-length model with Van Driest damping to compute the eddy viscosity profile, then solves the resulting system with a tridiagonal matrix algorithm (TDMA).

The TBL model can operate in two modes:

  • Equilibrium: the pressure gradient is set to zero. Suitable for regions where the streamwise pressure gradient is small.

  • Non-equilibrium: the pressure gradient is included in the boundary layer equation. Recommended for regions with significant pressure gradients, such as flow separation and reattachment zones.

Wall models with IBM

See also

Wall Model - IBM - Theory - algorithm for coupling wall models with the Immersed Boundary Method.

When wall models are used with the Immersed Boundary Method, the model operates through a velocity correction at an auxiliary point near the body surface. A reference point \(P\) is placed at a distance of at least \(3\Delta x\) from the wall - outside the diffusive layer - and the modeled velocity at a correction point \(Q\) (close to the wall) is imposed through an IBM interpolation-spread procedure.

Important

The simulation should first reach a stable flow state with the standard IBM before activating the wall model. Once activated, the wall model replaces the standard IBM forcing.

Inlet

See also

Inlet - Theory - mathematical formulation of uniform velocity and SEM inlet conditions.

Uniform velocity

The uniform velocity condition sets a constant velocity and density at the inlet by computing the equilibrium populations from the prescribed values:

\[f_{\bar{i}}(t + \Delta t) = f_i^{eq}(\rho, \mathbf{u})\]

This condition does not add mass to the domain (unlike velocity bounce-back), which makes it stable for long simulations. It is used when a constant, laminar inlet is needed, or as the base condition before adding turbulent fluctuations.

Synthetic Eddy Method (SEM)

The SEM generates a turbulent inlet by superimposing synthetic eddies onto a mean velocity profile. It produces transient velocity fluctuations that reproduce a prescribed Reynolds stress tensor, making it the standard inlet condition for atmospheric boundary layer and wind engineering simulations.

See also

The SEM has its own dedicated reference page: Synthetic Eddy Method (SEM).

Outlet

See also

Outlet - Theory - Neumann and regularized Neumann outlet formulations.

Neumann

The Neumann outlet enforces a zero normal gradient of all macroscopic quantities at the boundary. The boundary nodes replicate the macroscopic values from the first interior node. This provides a flexible open boundary that allows flow structures to exit the domain without strong reflections.

Regularized Neumann (outlet)

The regularized Neumann outlet extends the Neumann condition by fixing the density at a prescribed value \(\rho_{out} = \rho_0\) after applying the zero-gradient condition. This stabilizes the average density in the domain over long simulation times.

Important

The regularized Neumann outlet must be placed far enough from regions of high pressure gradient. If the outlet is too close to flow features that produce significant pressure variations (e.g., wakes of bluff bodies), the fixed-density constraint can cause reflections.

Free surface (slip)

See also

Free Surface - Theory - zero-gradient slip condition formulation.

The free surface condition applies a zero normal gradient of macroscopic properties at the boundary:

\[\frac{\partial A}{\partial x_n} = 0\]

where \(A\) represents any macroscopic variable (\(\rho\), \(u_\alpha\), \(S_{\alpha\beta}\)). This is implemented through a regularized Neumann formulation and is typically used at the top boundary of atmospheric flow domains where the flow should pass freely without friction.

Periodic

See also

Periodic - Theory - periodic boundary implementation in LBM.

Periodic boundaries connect opposite faces of the domain so that populations streaming out of one side re-enter from the other. This is the simplest boundary condition to implement in LBM since it only requires redirecting the streaming step across the domain boundary.

Periodic conditions are useful for simulations of infinite or repeating geometries, and are commonly applied in the lateral (crosswind) directions.

See also