Accurate Reaction-Diffusion Operator Splitting on Tetrahedral Meshes for Parallel Stochastic Molecular Simulations

Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, meaning that the serial limit has already been reached in sub-cellular models. This calls for parallel simulations that can take advantage of the power of modern supercomputers; however exact methods are known to be inherently serial. We introduce an operator splitting implementation for irregular grids with a novel method to improve accuracy, and demonstrate potential for scalable parallel simulations in an initial MPI version. We foresee that this groundwork will enable larger scale, whole-cell stochastic simulations in the near future.Comment: 33 pages, 10 figure

Development of A Direct-Forcing Immersed-Boundary Method on Unstructured Meshes for Multi-Body Interactions in Air-Water Two-Phase Flows

A direct-forcing immersed boundary method (IBM) is developed in the framework of a finite-volume incompressible solver for high-Reynolds-number flows. The method solves governing equations on a background mesh whose grid lines do not conform to the concerned surface geometry, whereby the difficulty of generating high-quality body-fitted meshes is significantly reduced. The boundary conditions on the surface of the geometry are enforced through interpolation. A unique aspect of the proposed IBM is that the method is compatible with unstructured meshes, and as such can be combined with body-fitted meshes, so that some geometries can be represented by body-fitted meshes, and other geometries are represented by the IBM. The method provides an accurate solution for the cases of moving objects in both single-phase and air-water two-phase flows. The method can also be applied to both steady and unsteady, laminar and turbulent flows. In the current work, the method is implemented for solving the Reynolds-Averaged Navier-Stokes equations, and for turbulent flows, the Spalart-Allmaras turbulence model is used. A noticeable challenge of using IBMs is the difficulty in resolving boundary layers at high Reynolds numbers. In this thesis a universal wall function is implemented, which provides a smooth velocity profile from the outer edge of the logarithmic region down to the wall. The wall function improves accuracy when the mesh is not sufficiently fine to resolve the viscous sublayer. As a result, the stringent requirement of near-wall cell spacing for high-Reynolds-number flows is significantly alleviated. The Volume-of-Fluid (VoF) method is used for air-water two-phase flows. A field extension method is used to enforce the boundary condition of the volume fraction on the immersed surface. Detailed verification and validation studies are performed to demonstrate that the current method is second-order accurate. A careful comparison is presented between the results of the IBM, the experimental data, and other numerical results. The comparison fully demonstrates the accuracy and feasibility of the method by examining the flow field and the force on the immersed surface. The validation case of a ship advancing with a rotating rudder is also performed. The results demonstrate the accuracy, flexibility and efficiency when the IBM is used combined with unstructured body-fitted meshes.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162867/1/hxye_1.pd