An Adaptive Cartesian Grid Embedded Boundary Method for the Incompressible Navier Stokes Equations in Complex Geometry

We present a second-order accurate projection method to solve the incompressible Navier-Stokes equations on irregular domains in two and three dimensions. We use a finite-volume discretization obtained from intersecting the irregular domain boundary with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing a conservative discretization of the advective terms with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference to nearby cells. Our projection is based upon a finite-volume discretization of Poisson’s equation. We use a second-order, L∞-stable algorithm to advance in time. Block structured local refinement is applied in space. The resulting method is second-order accurate in L1 for smooth problems. We demonstrate the method on benchmark problems for flow past a cylinder in 2D and a sphere in 3D as well as flows in 3D geometries obtained from image data.