An adaptive discretization for incompressible and compressible flow using a multitude of moving Cartesian grids with gap flow treatment [electronic resource]
Keywords: Electrical Engineering.

This dissertation presents novel methods for simulating incompressible and compressible ﬂow on a multitude of Cartesian grids that can rotate and translate in order to decompose the domain into diﬀerent regions with varying spatial resolutions. While there are a wide variety of methods for adaptive discretization, many of these methods suﬀer from issues with costly remeshing and domain decomposition when they are scaled to solve large problems that require the use of large distributed systems. Block structured approaches such as Adaptive Mesh Reﬁnement (AMR) and Chimera grid methods have been successful in alleviating these issues by utilizing structured grids patched upon one another. However, typical AMR methods constrain grid patches to be axis-aligned greatly increasing the number of patches required. With so many small patches, typical AMR methods are often more akin to unstructured grids with respect to parallelization and scalability. Chimera grid methods allow the grid patches to rotate allowing one to resolve interesting features with far fewer degrees of freedom. Moreover, unlike typical AMR methods which require the coarse grid lines to match up with the ﬁne grid lines along patch boundaries, Chimera grid methods do not have this requirement allowing the grids to move in order to for example follow the motion of moving solids with no need of remeshing. The presented computational framework can be categorized as a Chimera grid method, and new ideas are proposed regarding conservation, linear systems for implicit solve, and alleviating time step restrictions. The incompressible Navier-Stokes equations are discretized on overlapping grids by ﬁrst performing advection on each grid with ﬁrst or second order accurate semi-Lagrangian schemes extended to Chimera grids in order to alleviate any time step restrictions associated with small cells which are introduced due to adaptivity. In order to solve for the stiﬀ terms such as the pressure or viscous forces implicitly on overlapping grids, local Voronoi meshes are constructed along intergrid boundaries to connect the various degrees of freedom across diﬀerent grids in a contiguous manner, resulting in a symmetric positive-deﬁnite system that can be solved via the preconditioned conjugate gradient method. In order to handle free surface ﬂow on overlapping grids, the particle level set method is modiﬁed, including devising particle treatment across grid boundaries with disparate cell sizes, and designing strategies to deal with locality in the implementation of the level set and fast marching algorithms. The resulting method is highly scalable on distributed parallel architectures with minimal communication costs. The Euler equations for compressible ﬂow are discretized using a semi-implicit formulation that splits the time integration into an explicit step for advection followed by an implicit solve for the pressure. A second order accurate ﬂux based scheme is devised to handle advection on each moving Cartesian grid using an eﬀective characteristic velocity that accounts for the grid motion. In order to avoid the stringent time step restriction imposed by small cells, strategies are proposed in order to allow for a ﬂuid velocity CFL number larger than 1. The stringent time step restriction related to the sound speed is alleviated by formulating an implicit linear system in order to ﬁnd a pressure consistent with the equation of state, again utilizing the Voronoi mesh obtaining a symmetric positive-deﬁnite system. Since a straightforward application of this technique contains an inherent central diﬀerencing which can result in spurious oscillations, a new high order diﬀusion term is introduced similar in spirit to ENO-LLF but solved for implicitly in order to avoid any associated time step restrictions. The method is conservative on each grid, as well as globally conservative on the background grid that contains all other grids. Moreover, a conservative interpolation operator is devised for conservatively remapping values in order to keep them consistent across diﬀerent overlapping grids. Additionally, the method is extended to handle two-way solid ﬂuid coupling in a monolithic fashion. In solid ﬂuid coupling problems, the ﬂuid in the thin gap between solids in close proximity is diﬃcult to resolve with ﬂuid grids. Although one might attempt to address this diﬃculty using an adaptive, body-ﬁtted, or ALE ﬂuid grid, the size of the ﬂuid cells can shrink to zero as the solids collide. The inability to apply pressure forces in a thin lubricated gap tends to make the solids stick together, since collision forces stop interpenetration but vanish when the solids are separating leaving the ﬂuid pressure forces on the surfaces of the solids unbalanced in regard to the gap region. This problem is addressed by adding ﬂuid pressure and velocity degrees of freedom onto solids' surfaces, and subsequently using the resulting pressure forces to provide solid ﬂuid coupling in the thin gap region. These ﬂuid pressure and velocity degrees of freedom readily resolve the tangential ﬂow along the solid surface inside the gap and are two-way coupled to the ﬂuid degrees of freedom on the grids allowing the ﬂuid to freely ﬂow into and out of the gap region, which again results in a symmetric positive-deﬁnite implicit linear system.