Using Biot-Savart to shrink Eulerian domains while maintaining or improving external flow accuracy

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 by the total vorticity contained in the lower levels and that it 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 a square plate accelerated from rest 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 10 times larger to converge on the new method's result. Finally, we study the sensitivity of deflected wakes to the new domain boundary conditions and show that only in minimal domains, 5 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.