Using Biot-Savart boundary conditions for unbounded external flow on Eulerian meshes

Abstract

We introduce a novel boundary condition for incompressible Eulerian simulations formulated using a Biot-Savart vorticity integral that maintains high-accuracy results even when the domain boundary is within a body-length of immersed solid boundaries. The key prerequisite to accurately couple the Biot-Savart condition to the Eulerian velocity and pressure fields is including the influence of the vorticity generated at the immersed boundaries during the incompressible-flow projection step. While the resulting linear operator for the pressure is non-local, it can be efficiently solved by partitioning it into the standard local Poisson operator and the Biot-Savart update. We use oct-tree clustering for the Fast Multi-level Method (FM$\ell$M) to reduce the computational cost of the evaluation of the Biot-Savart integral on the boundaries of a 3D simulation with $N$ points from $O(N^{5/3})$ to $O(N)$ and show that this has bounded errors. We show that the new method captures the analytical added-mass force of accelerating 2D and 3D plates exactly and matches experimentally measured wake development even when the entire domain only extends 1/2 diameter from the plate. The new method also predicts accurate time-varying forces for a 2D circle and 3D sphere regardless of domain size, while classical boundary conditions require a domain more than 100 times larger in 2D to converge on the new method's result. Finally, we study the highly sensitive 2D deflected wakes produced by high frequency flapping foils to the new boundary conditions and show that truncating the deflected wake within four cord-lengths of the body changes the body force amplitudes by 10-40%. Doubling the wake size recovers the asymptotic results to within 5%.