W. Kyle Anderson

Navier-Stokes computations of vortical flows over low-aspect-ratio wings

An upwind-biased finite-volume algorithm is applied to the low-speed laminar flow over a low-aspect-ratio delta wing from 0 to 40 deg angle of attack. The differencing is second-order accurate spatially and a multigrid algorithm is used to promote convergence to the steady state. The results compare well with the detailed experiments of Hummel and others for an Ret of 0.95 x 106. The predicted maximum lift coefficient of 1.10 at 35 deg angle of attack agrees closely with the measured maximum lift of 1.06 at 33 deg. At 40 deg angle of attack, a bubble type of vortex breakdown is evident in the computations, extending from 0.6 of the root chord to just downstream of the trailing edge.

Recent improvements in aerodynamic design optimization on unstructured meshes

Recent improvements in an unstructured-grid method for large-scale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were dife cult to realize, particularly for viscous e ows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a signie cantly improved convergence rate is obtained. A new mesh movement algorithm is implemented, and several advantages over an existing technique are presented. Several design cases are shown for turbulent e ows in two and three dimensions.

Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables

The use of complex variables for determining sensitivity derivatives for turbulent flows is examined. Although a step size parameter is required, the numerical derivatives are not subject to subtractive cancellation errors and, therefore, exhibit true second-order accuracy as the step size is reduced. As a result, this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude. This behavior is in contrast to the use of finite differences, which suffer from inaccuracies due to subtractive cancellation errors. In addition, the complex-variable procedure is easily implemented into existing codes

Extension and applications of flux-vector splitting to unsteady calculations on dynamic meshes

The Van Leer method of flux-vector splitting for the Euler equations is extended for use on moving meshes and all the properties of the original splittings are maintained. The solution is advanced in time with an implicit, approximately factored algorithm. The use of multiple grids to reduce the computer time is investigated. A substantial reduction in computer time to resolve a pitching cycle is easily obtained with virtually no loss in accuracy. A subiterative procedure to eliminate factorization and linearization errors so that larger time steps can be used is also investigated. Subsequent computations show good agreement with experimental data for transonic and supersonic airfoils and wings undergoing forced pitching oscillation.

Petrov-Galerkin and discontinuous-Galerkin methods for time-domain and frequency-domain electromagnetic simulations

Finite-element discretizations for Maxwells first-order curl equations in both the time domain and frequency domain are developed. Petrov-Galerkin and discontinuous-Galerkin formulations are compared using higher-order basis functions. Verification cases are run to examine the accuracy of the algorithms on problems with exact solutions. Comparisons with other, well accepted, methodologies are also considered for problems for which exact solutions do not exist. Effects of several parameters, including spatial and temporal refinement, are also examined and the relative efficiency of each scheme is discussed. By considering test cases previously considered by other researchers, it is also demonstrated that the algorithms do not exhibit spurious solutions. Finally, three-dimensional results are compared with test results for a rectangular waveguide for which experimental data has been obtained with the explicit purpose of code-validation. The ability to predict changes in scattering parameters caused by variations in geometric and material properties are examined and it is demonstrated that the algorithms predict these changes with good accuracy.

Step-Size Independent Approach for Multidisciplinary Sensitivity Analysis

A multidisciplinary sensitivity analysis technique based on a complex Taylor series expansion method is implemented. This technique is shown to be independent of step-size selection as it pertains to cancellation errors. The primary focus of this research is to validate a new aero ‐structural analysis and sensitivity analysis procedure. This validation consists of comparing computed and experimental data obtained for an aeroelastic research wing. Because the aero ‐structural analysis procedure has the complex variable modie cations already included into the software, sensitivity derivatives are automatically computed. Other than for design purposes, sensitivity derivatives can be used for predicting the solution at nearby conditions. The use of sensitivity derivatives for predicting the aero ‐structural characteristics of this cone guration is demonstrated and compared with experimental measurements.

Three-Dimensional Stabilized Finite Elements for Compressible Navier–Stokes

In this paper, a stabilized finite-element approach is used in the development of a high-order flow solver for compressible flows. The streamline/upwind Petrov–Galerkin discretization is used for the Navier–Stokes equations, and a fully implicit methodology is used for advancing the solution at each time step. The order of accuracy is assessed for both inviscid and viscous flows using the method of manufactured solutions. For two-dimensional flow, a mesh-curving strategy is discussed that allows high-aspect-ratio curved elements in viscous flow regions. In addition, the effects of curved elements are evaluated in two dimensions using the method of manufacture solutions. Finally, test cases are presented in two and three dimensions and compared with well-established results and/or experimental data.

Mesh Generation Using Unstructured Computational Meshes and Elliptic Partial Differential Equation Smoothing

A novel appro ach for generating unstructured meshes using elliptic smoothing is presented. Like structured mesh generation methods, the approach begins with the construction of a computational mesh. The computational mesh is used to solve elliptic partial differential equations that control grid point distributions and improve mesh quality. Two types of elliptic partial differential equations are employed; modified linear -elastic theory and Winslow equation s, with or without forcing functions . Results are included to il lustrate the use of these methods for unstructured mesh generation, mesh boundary shape modification, mesh untangling and mesh movement.

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.

Extension and application of flux-vector splitting to calculations on dynamic meshes

THE Van Leer method of flux-vector splitting for the Euler equations is extended for use on moving meshes and all properties of the original splittings ate maintained. The solution is advanced in time with an implicit, approximately factored algorithm. A subiterative procedure to minimize factorization and linearization errors so that larger time steps can be used is investigated. Subsequent computations are shown for a transonic wing undergoing forced pitching oscillation.