Boundary Conditions

This page describes the treatment of boundary conditions for the vortex-in-cell method, based on Stock (2006), §3.3.2–3.3.5.

Overview

Flow simulations fall under two categories based on their boundaries:

The Poisson solver (HW3CRT/FFT) supports Dirichlet, Neumann, or periodic boundary conditions, enabling uniform inlet, wall, periodic, or open boundaries.

Periodic Boundaries

Periodic boundary conditions are commonly used to mimic free-space or to study spatially-developing instabilities.

Element Treatment

Because we use triangular elements with connectivity, extra care is required when elements "straddle" a periodic boundary:

1. Straddling Detection: For each element, check if any two nodes are separated by more than $2.5\Delta x$ in a periodic direction
2. Correction: Temporarily move nodes near the upper boundary down by one domain length
3. Operation: Perform calculations (area, normal, center, spreading)
4. Restoration: Return nodes to their "natural" positions

Vorticity Field Creation

When spreading vorticity near periodic boundaries:

• Grid nodes are indexed $[0..N]$, with index $N$ representing index $0$ of the next domain
• Vorticity at node $i_{\text{node}} < 0$ is added to $i_{\text{node}} + N$
• Vorticity at node $i_{\text{node}} > N-1$ is added to $i_{\text{node}} - N$
• After spreading, copy vorticity at index 0 to index $N$ for consistency

# Periodic wrap for vorticity spreading
function periodic_wrap_index(inode, N)
    if inode < 0
        return inode + N
    elseif inode > N - 1
        return inode - N
    else
        return inode
    end
end

Usage

using VortexMethod

# Create domain with periodic x and y
domain = DomainSpec(1.0, 1.0, 4.0)  # Lx, Ly, Lz
gr = GridSpec(64, 64, 256)

# Wrap nodes after any modification
wrap_nodes!(nodeX, nodeY, nodeZ, domain)

Open (Free-Space) Boundaries

Open boundaries allow vorticity to influence the flow beyond the computational domain. Implementation is more complex than periodic boundaries.

Dirichlet Boundary Values

When no vorticity is near open boundaries, use Dirichlet conditions with boundary velocities computed via direct Biot-Savart summation:

\[\mathbf{u}(\mathbf{x}_{\text{boundary}}) = \sum_{\text{interior cells}} \frac{\boldsymbol{\omega} \times (\mathbf{x}_{\text{boundary}} - \mathbf{x}_{\text{cell}})}{4\pi|\mathbf{x}_{\text{boundary}} - \mathbf{x}_{\text{cell}}|^3}\]

Optimization: Integrate over every 4th boundary node and interpolate the rest. Only sum over cells with non-zero vorticity.

Kinetic Energy Calculation

For open boundaries, kinetic energy requires special treatment because velocity is only computed inside the domain. Using the divergence theorem, assuming potential flow outside:

\[E = \frac{1}{2}\left[\int_V |\mathbf{u}|^2\,d\mathbf{x} + \int_S \phi\,(\mathbf{u}\cdot\mathbf{n})\,dA\right]\]

where $\phi$ is a scalar velocity potential computed on the boundary.

Element Entry/Exit

Elements crossing open boundaries require special handling—either removing them or using absorbing layers. Current simulations are designed to avoid this situation.

Wall Boundaries (Slip)

For high-Reynolds-number flows, wall-generated vorticity stays within a thin boundary layer, allowing slip-wall treatment.

Vorticity Near Walls

When elements are within the regularization distance of a wall:

On the wall ($i_{\text{node}} = 0$ or $N$):

• Double the wall-normal vorticity component
• Zero the wall-tangential components

Beyond the wall ($i_{\text{node}} < 0$ or $> N$):

• Reflect to $i_{\text{node}} \to -i_{\text{node}}$ or $2N - i_{\text{node}}$
• Zero the normal component
• Negate the tangential components (image vorticity)

This ensures near-wall vortex lines are smoothed as if they and their reflections existed.

Boundary Conditions for Poisson Solve

• Normal velocity: $u_n = 0$ (no penetration)
• Tangential velocity derivative: $\partial u_t / \partial n = 0$ (free slip)

Element-Wall Interaction

Two types of vortex sheet–wall interactions:

1. Edge attached to wall: The edge must stay on the wall; nodes within $\epsilon = 10^{-5}$ of the wall are constrained to remain there
2. Sheet approaching wall: Handled by the image vorticity treatment described above

Special Boundary Conditions

Inlet/Outlet (Jets, Mixing Layers)

For circular jets or mixing layers:

• Modify normal velocity component at inlet
• Insert shedding edges manually (see [Stock (2006), §3.6])

Traveling Window

For problems like propagating vortex rings, the computational domain can "follow" the structure:

# Domain moves with sheet centroid
domain_z_min = sheet_centroid_z - 2.0
domain_z_max = sheet_centroid_z + 2.0

Boundary Condition Selection Guide

References

• Stock, M. J. (2006). A regularized inviscid vortex sheet method for three dimensional flows with density interfaces. Ph.D. Thesis, California Institute of Technology.
• Swarztrauber, P., & Sweet, R. (1975). Efficient FORTRAN subprograms for the solution of elliptic partial differential equations. NCAR Technical Note.