  We introduce a novel boundary condition formulated using a Biot-Savart
vorticity integral inside an Eulerian fluid domain that maintains high-accuracy
results even when the domain boundary is within a body-length of immersed solid
boundaries. The Biot-Savart condition is incorporated into a general solver
with Eulerian velocity-pressure fields by including the pressure-induced body
vorticity during the incompressible-flow projection-step. We use a multilevel
approach to reduce the computational cost of the evaluation of the Biot-Savart
integral from O(N) to O(\log N) and formally show that the error associated
with this approach is bounded and scales with the inverse power of the
problem's dimension. We use two-dimensional and three-dimensional analytic
vortex problems to demonstrate this scaling is correct and select appropriate
multilevel coarsening parameters accordingly. We show that the new method
exactly captures the analytical added-mass force of accelerating plates and
matches experimentally measured wake development even when the entire domain
only extends 1/2 diameter from the plate. The new method also predicts
essentially identical time-varying drag forces for a two-dimensional circle
regardless of domain size, while classical boundary conditions require a domain
nearly 48 times larger to converge on the new method's result. Finally, we
study the sensitivity of 2D deflected wakes to the new domain boundary
conditions and show that only in minimal domains, 25 times smaller than the
reference domain, the omission of the far-field wake influences the flow field
and the forces on the body. Doubling the domain size recovers the correct
results.
