Marshall C. Galbraith

Introduction to COFFE: The Next-Generation HPCMP CREATE-AV CFD Solver

HPCMP CREATE-AV Conservative Field Finite Element (COFFE) is a modular, extensible, robust numerical solver for the Navier-Stokes equations that invokes modularity and extensibility from its first principles. COFFE implores a flexible, class-based hierarchy that provides a modular approach consisting of discretization, physics, parallelization, and linear algebra components. These components are developed with modern software engineering principles to ensure ease of uptake from a users or developers perspective. The Streamwise Upwind/Petrov-Galerkin (SU/PG) method is utilized to discretize the compressible Reynolds-Averaged Navier-Stokes (RANS) equations tightly coupled with a variety of turbulence models. The mathematics and the philosophy of the methodology that makes up COFFE are presented.

A Verification Driven Process for Rapid Development of CFD Software

Previous work by the authors has demonstrated a high-order fully-automated output-error based mesh adaptation method suitable for solving the Reynolds-Averaged Navier-Stokes equations. The high-order of accuracy is achieved with a discontinuous Galerkin discretization. While the adaptation method has proven to provide significant reduction in computational cost relative to second-order methods, the authors are currently exploring alternate high-order finite element discretizations to further reduce the computational cost. However, the previously developed software framework is not suitable for all discretizations of interest. Hence, a new software framework is being developed with enhanced maintainability and flexibility relative to the previous framework. This paper focuses on strategies employed to accelerate the development of the new software framework. A software development environment that promotes a verification driven process for software development is presented. The development environment encourages developers to incorporate the verification principles of Verification and Validation as part of the software development process to promote maintainability and collaboration. The software development is further accelerated through the use of automatic differentiation, which is used here to automatically compute the linearization of a mathematical model. This paper outlines an implementation of automatic differentiation with minimal computational overhead relative to manually written linearizations.

Implicit Solutions of Incompressible Navier-Stokes Equations Using the Pressure Gradient Method

*† A modified pressure gradient method is developed for solving the incompressible twoand three-dimensional Navier-Stokes equations in primitive variables. Velocity and pressure gradient vectors, rather than the velocity and pressure, are considered as dependent variables. This choice of the dependent variables is consistent with the flow physics as the velocity field of incompressible flows is not dependent on the pressure, but rather the pressure gradient. This introduces one and two additional dependent variables for two- and three-dimensions respectively. To close the system of equations, additional equations are obtained from the identity that the curl of the pressure gradient is zero. This choice of dependent variables also leads to Dirichlet boundary conditions for the pressure gradient, which improves numerical stability and convergence rates. The modified pressure gradient method includes a pressure gradient averaging term in the momentum equations to stabilize the solution. This term is extended here to general curvilinear coordinates. The system of equations is discritized using a conservative finite volume formulation. Time integration is achieved with a fully implicit scheme, including implicit boundary conditions. The implicit scheme guarantees that continuity is satisfied to machine zero. Numerical results compare well with solutions available in the literature for the cavity flow problem and flow about a circular cylinder at Reynolds number 40.

A space-time adaptive method for reservoir flows: formulation and one-dimensional application

This paper presents a space-time adaptive framework for solving porous media flow problems, with specific application to reservoir simulation. A fully unstructured mesh discretization of space and time is used instead of a conventional time-marching approach. A space-time discontinuous Galerkin finite element method is employed to achieve a high-order discretization on the anisotropic, unstructured meshes. Anisotropic mesh adaptation is performed to reduce the error of a specified output of interest, by using a posteriori error estimates from the dual-weighted residual method to drive a metric-based mesh optimization algorithm. The space-time adaptive method is tested on a one-dimensional two-phase flow problem, and is found to be more efficient in terms of computational cost (degrees-of-freedom and total runtime) required to achieve a specified output error level, when compared to a conventional first-order time-marching finite volume method and the space-time discontinuous Galerkin method on structured meshes.

Application of Higher-Order Adaptive Method to Reynolds-Averaged Navier–Stokes Test Cases

An output-based adaptive, higher-order discontinuous Galerkin finite element method is applied to the solution of standard turbulence modeling test cases drawn from the NASA Turbulence Modeling Resource website, specifically subsonic flow over a flat plate and a NACA 0012 airfoil. Results are presented for the current solver using linear, quadratic, and cubic solution approximations on both structured grids and adapted grids. Comparisons are also made with NASA Reynolds-averaged Navier–Stokes solvers FUN3D and CFL3D. Analysis is presented for the adaptive grids, particularly concerning grid refinement in the vicinity of flowfield singularities for both test cases.

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.

A Non-parametric Discontinuous Galerkin Formulation of the Integral Boundary Layer Equations with Strong Viscous/Inviscid Coupling

A non-parametric discontinuous Galerkin (DG) finite-element formulation is developed for the integral boundary layer (IBL) equations with strong viscous-inviscid coupling. This DG formulation eliminates the need of explicit curvilinear coordinates in traditional boundary layer solvers, and thus enables application to complex geometries even involving non-smooth features. The usual curvilinear coordinates are replaced by a local Cartesian basis, which is conveniently constructed in the DG finite-element formulation. This formulation is also applicable to the general convection-source type of partial differential equations defined on curved manifolds. Other benefits of DG methods are maintained, including support for high-order solutions and applicability to general unstructured meshes. For robust solution of the coupled IBL equations, a strong viscous-inviscid coupling scheme is also proposed, utilizing a global Newton method. This method provides for flexible and convenient coupling of viscous and inviscid solutions, and is readily extensible to coupling with more disciplines, such as structural analysis. As a precursor to the three-dimensional strongly-coupled IBL method, a two-dimensional IBL solver coupled with a panel method is implemented. Numerical examples are presented to demonstrate the viability and utility of the proposed methodology. Thesis Supervisor: Steven R. Allmaras Title: Research Engineer of Aeronautics and Astronautics