The method of immersed layers, with application to internal and external flows

The immersed boundary method (IBM) has served for several decades as a versatile tool for simulating and studying flows with complex moving boundaries on stationary grids. In this work, we show that the IBM is a special case of a broader and more powerful approach in which the interior and exterior of any surface can be distinguished from one another. To show this, we make use of generalized functions on level sets to reformulate the incompressible Navier--Stokes equations and related partial differential equations so that jumps in surface quantities are incorporated exactly into the equations. In this manner, we can enforce any well-posed boundary or interface condition in a straightforward manner and readily compute the associated surface forces. Because of their resemblance to single and double layers in the Poisson equation, we denote these jump terms as `immersed layers'. We discretize the equations with second-order differencing operators on a staggered Cartesian grid. These operators' mimetic properties in clearly-defined inner product spaces---coupled with the use of the lattice Green's function for inverting the Laplace operator---ensure that various continuous identities are exactly preserved in the discrete sense and that other results are simple to derive. We highlight an open-source, extendable framework of tools that implement the operators and methods, and demonstrate the framework's use on various flows, including past multiple moving bodies and through internal ducts.