Clarence O. E. Burg

Higher Order Variable Extrapolation For Unstructured Finite Volume RANS Flow Solvers

A new variable extrapolation formulation for unstructured finite volume codes is developed which closely resembles the MUSCL-scheme used within structured flow solvers. This new formulation is based on information currently available to the unstructured flow solvers, namely the variable information and the gradient information, and as such, it is trivial to implement within most finite volume flow solvers. This new variable extrapolation formulation represents a one-parameter family of equations and under certain circumstances, it is fully equivalent to the MUSCL-scheme, which is also a one-parameter family. A wide variety of results are presented, including theoretical analysis of the truncation error and numerical analysis of the truncation error for a one-dimensional problem, using the method of manufactured solutions, as well as for a two-dimensional problem. Inviscid three-dimensional results are presented which demonstrate that the numerical viscosity can be greatly decreased via this formulation, and the viscous results for a variety of cases indicate that the drag is better predicted and that the vortical structures within the flow field are captured more accurately and further downstream of their origination. Finally, the observed improved convergence levels and stability of the new formulation are discussed, as well as the issues involved with extending this approach to achieve third-order spatial accuracy for 2D and 3D grids.

Computationally efficient, numerically exact design space derivatives via the complex Taylor's series expansion method

Abstract Within numerical design optimization, discrete sensitivity analysis is often used to estimate the derivative of an objective function with respect to the design variables. Discrete sensitivity analysis estimates these derivatives by taking advantage of additional derivative information available in an implicit computational fluid dynamics (CFD) solver of the discretized governing partial differential equations. The key benefits of steady-state discrete sensitivity analysis are its computational efficiency and numerical accuracy. More recently, the complex Taylors series expansion (CTSE) method has been used to generate these design space derivatives to machine accuracy, by analyzing a complex perturbation of the objective function. For fortran codes, this method is quite easy to implement, for both implicit and explicit codes; unfortunately, the CTSE method can be quite time consuming, because it requires a complex solution of the governing partial differential equations. In this paper, the authors demonstrate that the direct formulation of discrete sensitivity analysis and the CTSE method solve the same iterative sensitivity equation , which sheds light on the most efficient use of the CTSE method. Finally, these methods are demonstrated via application to numerical simulations of one-dimensional and two-dimensional open-channel flows.

A Robust Unstructured Grid Movement Strategy using Three-Dimensional Torsional Springs

The ability to move a grid during a simulation is of critical importance to many types of simulations, including aero-elasticity simulations for wings, gradient-base d design optimization, dynamic movement of control surfaces for maneuvering, and surface tracking methods for free surface flows. For structured grids, transfinite interpolation is a robust method for propagating defo rmations on the surfaces into the volume grid, and its two-dimensional version is readily extendible to three-dimensions. For unstructured grids, two methods that have been used within two-dimensional grids are the use of linear springs and torsional springs to propagate the changes on the surface into the volume mesh. The use of linear springs provides an efficient method to move unstructured grids, but it is not very robust, in that moderate to large deformations will result in invalid meshes, where some of the elements have been inverted. Farhat developed the torsional spring method for two-dimensional grids, and has demonstrated that this method is much more robust, even for large deformations. Farhat and Murayama have attempted to extend the two-dimensional torsional springs method to threedimensions, and their results are impressive. However, both of their methods reduces the three-dimensional tetrahedra into simpler components without analyzing the dynamics of the three-dimensional object, and as such, their methods do not fully represent a three-dimensional extension of the two-dimensional torsional springs method. In this paper, a three-dimensional extension of the torsional springs method is derived by analyzing the equations of volume and area for a tetrahedron and its faces and edges.

Efficient Code Verification Using the Residual Formulation o f the Method of Manufactured Solutions

The Method of Manufactured Solutions is a code verification m ethod that modifies the governing equations solved within a code by adding a source term to drive the solution towards a predetermined analytic function. By solving the modified equations on a sequence of grids and co mparing the differences between the converged solution and manufactured solution, the order of accuracy of the implementation can be determined. The method of manufactured solutions combines the benefits of co mparing with an exact solution without the need to derive an exact solution to the governing equations. However, in its current form, highly converged solutions on a sequence of grids are required which can be quite costly and difficult to obtain. In this paper, the method of manufactured solutions is used in a different fashion that removes the need for converged solutions by considering only the residual of the discretized governing equations rather than the solution, thus avoiding the computational cost and difficulties inherent in obtaini ng highly converged solutions. Furthermore, this new approach is quite similar to the method for analyzing a discretization method to determine the order of accuracy of that method via Taylor’s series expansions. T his new approach is demonstrated to yield the same order of accuracy as the original method of manufactured solutions using three different cases - onedimensional porous media equation, one-dimensional St. Venant equations and two-dimensional unstructured Euler simulations.

VERIFICATION AND VALIDATION OF FORCES GENERATED BY AN UNSTRUCTURED FLOW SOLVER

A primary goal of computational fluid dynamics is the accurate prediction of the forces and moments on ships, aircraft, turbines and similar complicated geometries, especially when viscous effects are important. By using unstructured grids, much of the detail of these complicated geometries can be captured with the grid and hopefully with the solution generated by the flow solver. As unstructured flow solvers mature, the forces predicted by them should become more accurate. For unsteady maneuvering cases, the accuracy of the forces and moments is critical for accurate predictions of the location and orientation of the body in motion, because errors tend to accumulate and grow as the maneuver proceeds. Accurate simulations of maneuvers have been obtained with the structured flow solver UNCLE. However, its unstructured equivalent, U 2 NCLE, has produced anomalies that are not fully understood. In an effort to isolate and identify potential inaccuracies in the unstructured flow solver, the flow solver has undergone a thorough re-evaluation of each component and the interactions between the components. Particular attention has been paid to the effects of discretization error. The unstructured flow solver uses mixed element types to resolve the boundary, so the discretization error for the non-simplical element types was investigated. Several methods to discretize the viscous terms have been investigated, to determine whether they are linearity preserving. A new inviscid variable extrapolation method (Unstructured MUSCL) has been developed, which has a smaller discretization error than the previous method. Finally, effects of asymmetries in the grid and in the solution algorithm have been investigated.

CFD on the Sony PS3

The Sony PlayStation 3 uses the powerful Cell Broadband Engine processor which is used within the Department of Energy’s Roadrunner supercomputer, the world’s fastest. This revolutionary processor performs math intensive operations at roughly 10-50 times faster than current single-core PC processors, due to the vector-like operations of the processors and the use of specialized math co-processors. However, to take advantage of this processing power, a computationally-intensive code must be rewritten into a vector format and reorganized to access these co-processors. This paper describes the programming alterations required to port an existing 2D unstructured finite-volume flow solver to the Cell processor. The reduction in computation time for simulations running on the Sony PS3 is roughly a factor of 6 over the original non-vectorized code running on the PS3. Once the code has been ported to the Sony PS3 platform, it is expected that only minor alterations, in particular, compile-time alterations, would be necessary in order to run on other computational platforms based on the Cell processor.

Approximation of Surfaces and Solution of Partial Differential Equations Using B-Splines

B-Spline curves and surfaces are widely used by CAD systems to represent physical models and from which grids can be built for computational fluid dynamics simulations. These B-Spline entities can also be used in conjunction with flow solvers either to represent features in the solution or to discretize the governing equations. In this paper, the primary use of B-Splines is to represent the free surface about viscous hulls in conjunction with the generation quality viscous grids. After developing the tools for such a representation, B-Splines are used to develop a finite element framework using the basis functions for the B-Spline surfaces as the weight and interpolating functions. The evaluation of the resulting integral is simplified because the spatial derivatives of the B-Spline surfaces can be calculated exactly. These codes are tested on a set of algebraic cases where exact agreement is possible and then for actual free surfaces generated from a submerged threedimensional hydrofoil, for the Wigley parabolic hullform and for the prototypical naval destroyer DTMB Model 5415 hullform.