UC Merced

This dissertation develops a family of projection-based numerical methods for incompressible flow on adaptive, collocated grids. Here, projection methods act as the primary computational mechanism through which numerical stability is enforced, physical features are represented sharply, and the geometric structure of incompressible flow is respected. In a broader sense, these methods constitute a conceptual framework for modeling incompressible flows, enabling the development of more accurate, flexible, and physically consistent simulations of complex fluid phenomena. Chapter 2 develops a stable collocated adaptive mesh framework for the incompressible Navier-Stokes equations. While collocating variables simplifies algorithmic design and data structures, it disrupts the discrete representation of analytical operators that is typically automatic in staggered grid formulations. This challenge is addressed through the design of a stable nodal projection method tailored to node-based adaptive grids, demonstrating that collocated formulations are stable and can achieve high-order accuracy. Chapter 3 extends this collocated projection framework to immiscible two-phase incompressible flows. Interfacial jump conditions are incorporated directly into the projection operator through a hybrid finite difference–finite volume formulation, enabling a sharp and stable treatment of interfacial stresses. Numerical results demonstrate accurate interface dynamics and strong volume conservation for complex two- and three-dimensional flows. Chapter 4 introduces the characteristic bending method, a structure-preserving advection scheme that applies a volume-preserving projection in a Lagrangian frame to attenuate compressible modes. This projection can be viewed as bending characteristics toward the divergence-free space, using local information from the reference map to construct a global correction. The resulting method yields a consistent representation of incompressible advection and integrates naturally into the collocated framework of the previous chapters. Together, these developments establish a cohesive and extensible, projection-based framework for enforcing incompressibility on adaptive, collocated grids, unifying stability, sharp feature representation, and structure preservation across single-phase fluid flow, multiphase fluid flow, and general advection problems.