An Arbitrary Curvilinear Coordinate Method for Particle-In-Cell Modeling

A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in- Cell (PIC) simulation method, is explored. In the two dimensional (2d) electrostatic version of our method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out in 2d on a uniform grid on the unit square logical domain, and mapped to a nonuniform boundary-fitted grid on the physical domain. As the resulting logical grid equations of motion are not separable, we have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. A generalized, curvilinear coordinate formulation of Poisson's equations to solve for the electrostatic fields on the uniform logical grid is also developed. By our formulation, we compute the plasma charge density on the logical grid based on the particles' positions on the logical domain. That is, the plasma particles are weighted to the uniform logical grid and the self-consistent mean electrostatic fields obtained from the solution of the logical grid Poisson equation are interpolated to the particle positions on the logical grid. This process eliminates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid and allows us to run the PIC method on arbitrary boundary-fitted meshes.Comment: Submitted to Computational Science & Discovery December 201

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

A comparison of finite difference methods for solving Laplace's equation on curvilinear coordinate systems

Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied

Rocket injector anomalies study. Volume 1: Description of the mathematical model and solution procedure

The capability of simulating three dimensional two phase reactive flows with combustion in the liquid fuelled rocket engines is demonstrated. This was accomplished by modifying an existing three dimensional computer program (REFLAN3D) with Eulerian Lagrangian approach to simulate two phase spray flow, evaporation and combustion. The modified code is referred as REFLAN3D-SPRAY. The mathematical formulation of the fluid flow, heat transfer, combustion and two phase flow interaction of the numerical solution procedure, boundary conditions and their treatment are described

An arbitrary curvilinear coordinate particle in cell method

A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in-Cell (PIC) simulation method, is explored. In this method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out on a uniform, unitary square logical domain and mapped to a nonuniform, boundary fitted physical domain. We utilize an elliptic grid generation technique known as Winslow\u27s method to generate boundary-fitted physical domains. We have derived the logical grid macroparticle equations of motion based on a canonical transformation of Hamilton\u27s equations from the physical domain to the logical. These equations of motion are not seperable, and therefore unable to be integrated using the standard Leapfrog method. We have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. We constructed a proof to show that the ML integrator is symplectic for systems of arbitrary dimension. We have constructed a generalized, curvilinear coordinate formulation of Poisson\u27s equations to solve for the electrostatic fields on the uniform logical grid. By our formulation, we supply the plasma charge density on the logical grid as a source term. By the formulations of the logical grid particle mover and the field equations, the plasma particles are weighted to the uniform logical grid and the self-consistent mean fields obtained from the solution of the Poisson equation are interpolated to the particle position on the logical grid. This process coordinates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid. In this work, we explore the feasibility of the ACC-PIC method as a first step towards building a production level, time-adaptive-grid, 3D electromagnetic ACC-PIC code. We begin by combining the individual components to construct a 1D, electrostatic ACC-PIC code on a stationary nonuniform grid. Several standard physics tests were used to validate the accuracy of our method in comparison with a standard uniform grid PIC code. We then extend the code to two spatial dimensions and repeat the physics tests on a rectangular domain with both orthogonal and nonorthogonal meshing in comparison with a standard 2D uniform grid PIC code. As a proof of principle, we then show the time evolution of an electrostatic plasma oscillation on an annular domain obtained using Winslow\u27s method

A study of three dimensional turbulent boundary layer separation and vortex flow control using the reduced Navier Stokes equations

A reduced Navier Stokes (RNS) initial value space marching solution technique was applied to vortex generator and separated flow problems and demonstrated good predictions of the engine face flow field. This RNS solution technique using FLARE approximations can adequately describe the topological and topographical structure flow separation associated with vortex liftoff, and this conclusion led to the concept of a subclass of separations which can be called vorticity separations: separations dominated by the transport of vorticity. Adequate near wall resolution of vorticity separations appears necessary for good predictions of these flows