An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow

• The 2-D immersed interface solver computes the streamfunction in unbounded domains.
• The solver imposes a no through-flow condition along solid boundaries at high order.
• The body circulation can be imposed or found from the Kutta–Joukowsky condition.
• It is a crucial step for handling solid boundaries in vortex particle-mesh methods.
• The 2nd order method is iterative and thus well-suited for unsteady computations.

This paper presents a novel algorithm to solve the 2-D potential flow past complex geometries with circulation in unbounded domain and in the presence of a given vorticity field. It is based on a Poisson solver that combines two components: the immersed interface method to enforce the boundary condition on each inner boundary and the James–Lackner algorithm to compute the outer boundary condition consistent with the unbounded domain solution. The algorithm is here based on second order finite differences and it requires solely 1-D stencil corrections; this makes the immersed interface part of the present method easily extendable to higher dimensional problems. The treatment of the outer boundaries requires an iterative boundary potential method. The algorithm is validated, by means of grid convergence studies, on the flow past multiple bodies. The results confirm the second order accuracy everywhere. The algorithm is also self consistent as “all is done on the grid” (thus without using a vortex panel boundary element method in addition to the grid). For cusped airfoils, a consistent way to enforce the Kutta–Joukowsky condition is also presented. The present algorithm constitutes a crucial building block towards an immersed interface-enabled vortex particle-mesh method for the computation of unsteady viscous flows, with boundary layers, detached shear layers and wakes. A possible extension to 3-D problems is also briefly discussed.