A method of immersed layers on Cartesian grids, with application to incompressible flows

The immersed boundary method (IBM) of Peskin (J. Comput. Phys., 1977), and derived forms such as the projection Taira Colonius 2007), have been useful for simulating flow physics in problems with moving interfaces on stationary grids. However, their interface treatment, these methods do not distinguish one side from other, but rather, apply motion constraint to both sides, associated force is an inseparable mix contributions each side. In this work, we define a discrete Heaviside function, natural companion familiar Dirac delta function (DDF), masked version field grid which, within error DDF, takes intended value respective sides interface. From foundation develop operators identities that are uniformly applicable any surface geometry. We use extended prototypical partial differential equations, including Poisson, convection-diffusion, incompressible Navier-Stokes, govern fields. These equations contain forcing term IBM, also additional terms regularize jumps quantities onto enable us individually specify constraints behavior Drawing connection between layer potentials elliptic problems, refer them generically layers. demonstrate application several representative two-dimensional flows inside rotating cylinder external square.