Application of pressure-projection algorithms to a sharp projection immersed boundary method for the incompressible Navier-Stokes equations

In this thesis, we describe a globally second-order accurate sharp immersed boundary projection method with an algebraic structure parallel to the classic fractional step method for the unsteady, incompressible Navier-Stokes equations. While second-order accuracy in time and space is generally achievable for the velocity components, the pressure is usually first-order. To fully understand the source of this problem and the interplay between the pressure term and the overall fractional step method, we need to first look into the Navier-Stokes equations themselves. Perot demonstrated the possibility of higher order projection methods by means of LU approximations. This method seems applicable to any grid system and was widely accepted due to its ease of use and straightforward derivation of the order of accuracy for the pressure. It rendered another class of projection methods which relied on global pressure-updating hopeless as Strikwerda and Perot both speculated that these methods could inherently be first-order in pressure and simply cannot be improved to higher orders. Shortly after, Brown, Cortez and Minion provided insights to such pressure-updating schemes and proposed an entire class of global second-order accurate approximate projection methods. Despite Brown et al.'s success, it remains difficult to transfer these higher order pressure-updating methods to staggered grids. Discrete operators in staggered grids simply do not commute in the same way that colocated grids do (as in the case of approximate projection methods). We then continue to develop suitably accurate projection approaches for the immersed boundary method (IBM). The original IB method introduced by Peskin involves solving on an Eulerian grid (usually a uniform Cartesian mesh) which does not necessarily conform to the body's geometry. Some underlying mechanism is then needed to exchange information between the flow field described by the Eulerian discretization to the set of Lagrangian points which lie on the surface of the immersed boundary. The differences between the various IB methods lie in the way they implement this mechanism. Early adaptations of the IB method required arbitrary tuning parameters either for describing the forcing effect on the immersed boundary or characterizing the boundary velocity. Taira and Colonius were the first to formulate an IB method by means of a projection approach, also known as immersed boundary projection method (IBPM), which does not require any such parameters. Their formulation relied on Perot's LU factorization, where boundary forces and pressure terms act as Lagrange multipliers which enforce the no-slip and divergence-free constraints. This method quickly gained popularity due to its ease of use and is currently widely applied in modeling complex turbulent flows, multi-physics simulations and fluid-structure interaction. Its ability to model flow over complex geometries without the need for complex grids significantly reduced the time and effort needed for such simulations. This thesis addresses Perot's concern of commutativity of discrete operators and proposes a way to apply Brown et al.'s higher-order pressure updating schemes to staggered grid arrangements. Consequently, we present a fully second-order accurate immersed boundary projection method which employs similar updating schemes that Brown uses. We also improve upon Taira and Colonius's immersed boundary projection method by using multi-linear interpolation in place of the conventional use of the discrete Dirac delta function and create a new "sharp" immersed boundary projection method