John W. Peterson

John W. Peterson, Alexander D. Lindsay, Fande Kong

The Multiphysics Object Oriented Simulation Environment (MOOSE) framework is a high-performance, open source, C++ finite element toolkit developed at Idaho National Laboratory. MOOSE was created with the aim of assisting domain scientists and engineers in creating customizable, high-quality tools for multiphysics simulations. While the core MOOSE framework itself does not contain code for simulating any particular physical application, it is distributed with a number of physics "modules" which are tailored to solving e.g. heat conduction, phase field, and solid/fluid mechanics problems. In this report, we describe the basic equations, finite element formulations, software implementation, and regression/verification tests currently available in MOOSE's navier_stokes module for solving the Incompressible Navier-Stokes (INS) equations.

John W. Peterson, Roy H. Stogner

The third-order Jeffery-Hamel ODE governing the flow of an incompressible fluid in a two-dimensional wedge is briefly derived, and a C^1 finite element formulation of the equation is developed. This formulation has several advantages, including a natural framework for enforcing the boundary conditions, a numerically efficient solution procedure, and suitability for implementation within well-established, open, scientific computing tools. The finite element formulation is shown to be non-coercive, and therefore not ideal for proving existence, uniqueness, or a priori error estimates, but the numerical solutions computed with quartic Hermite elements are nevertheless found to converge to reference solutions at nearly optimal rates (O(h^4) in both L^2 and H^1 norms). Further work is required to better understand the cause of the suboptimal convergence rates, and a linear model problem which exhibits analogous characteristics is also discussed as a possible starting point for future theoretical analyses.