Thomas H. Pulliam

A Diagonal Form of an Implicit Approximate-Factorization Algorithm

Abstract A modification of an implicit approximate-factorization finite-difference algorithm applied to partial differential equations is presented. This algorithm is applied to the two- and three-dimensional Euler equations in general curvilinear coordinates. The modification transforms the coupled system of equations into an uncoupled diagonal form that requires less computational work. For steady-state applications, the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. The diagonal algorithm reduces the storage requirement of the implicit solution process and therefore has an important effect on the application of implicit finite-difference schemes to vector processors. Results are presented for realistic two-dimensional transonic flow fields about airfoils. Computation costs are reduced 24–34%.

Artificial dissipation models for the Euler equations

Various artificial dissipation models that are used with central difference algorithms for the Euler equations are analyzed for their effect on accuracy, stability, and convergence rates. In particular, linear and nonlinear models are investigated using an implicit approximate factorization code (ARC2D) for transonic airfoils. Fully implicit application of the dissipation models is shown to improve robustness and convergence rates. The treatment of dissipation models at boundaries will be examined. It will be shown that accurate, error free solutions with sharp shocks can be obtained using a central difference algorithm coupled with an appropriate nonlinear artificial dissipation model. I. Introduction T HE solution of the Euler equations using numerical techniques requires the use of either a differencing method with inherent dissipation or the addition of dissipation terms to a nondissipative scheme. This is because the Euler equations do not provide any natural dissipation mechanism (such as viscosity in the Navier-Stokes equations) that would eliminate high frequencies which are caused by nonlinearitie s and especially shocks. A variety of numerical algorithms and computer codes for the Euler equations have been developed. Methods such as MacCormacks1 explicit

Multipoint and Multi-Objective Aerodynamic Shape Optimization

A Newton‐Krylov algorithm is presented for the aerodynamic optimization of singleand multi-element airfoil configurations. The flow is governed by the compressible Navier‐Stokes equations in conjunction with a one-equation turbulence model. The preconditioned generalized minimum residual method is applied to solve the discreteadjoint equation, leading to a fast computation of accurate objective function gradients. Optimization constraints are enforced through a penalty formulation, and the resulting unconstrained problem is solved via a quasi-Newton method. Design examples include lift-enhancement and multi-point lift-constrained drag minimization problems. Furthermore, the new algorithm is used to compute a Pareto front for a multi-objective problem, and the results are validated using a genetic algorithm. Overall, the new algorithm provides an ecient and robust approach for addressing the issues of complex aerodynamic

Time accuracy and the use of implicit methods

Some of the approximations used to make implicit methods more efficient and practical for the solution of the Euler and Navier-Stokes equations are addressed. In particular, approximate factorizations, diagonalizations, and linearization approximations are reviewed and categorized. A subiteration correction scheme commonly in use at present is introduced, improved, demonstrated, and analyzed. This scheme is used to produce a second-order accurate, more robust implicit method for unsteady flow computations. The subiteration approach can be employed to recover time accuracy without increasing computational time (in most cases producing substantial savings).

Numerical Solution of the Azimuthal-Invariant Thin-Layer Navier-Stokes Equations

NUMERICAL solutions have been obtained for a twodimensional azimuthal- (or circumferentially) invariant form of the thin-layer Navier-Stokes equations. The governing equations which have been developed are generalized over the usual two-dimensional and axisymmetric formulation by allowing nonzero velocity components in the invariant direction. The equation formulation along with the solution method is described, and results for spinning and nonspinning bodies are presented. Contents The three-dimensional flow field equations are frequently simplified for flowfields which are invariant in one coordinate direction. In the usual axisymmetric approximation, the azimuthal velocity is assumed to be zero, and the momentum equation in that direction can be eliminated. Thus, only four equations are required to be solved for four unknowns. However, for a variety of interesting flowfields, the velocity component in the invariant direction (here taken as TJ) is not zero although the governing equations are still twodimensional. Examples include viscous flow about an infinitely swept wing, the viscous flow about a spinning axisymmetric body at 0-deg angle of attack, and axisymmetric swirl flows. Each of these flows can be solved as a twodimensional problem although all three momentum equations have to be retained, and source terms replace the derivative of the flux terms in the rj-direction. Azimuthal-invariant equations are obtained from the threedimensional equations1 by making use of two restrictions: 1) all body geometries are of axisymmetric types and 2) the state variables and the contravariant velocities do not vary in the azimuthal direction. Here, TJ is used for the azimuthal coordinate, and the terms azimuthal and rj-invariant will be used interchangeably. A sketch of a typical axisymmetric body is shown in Fig. la. In order to determine the circumferential variation of typical flow and geometric parameters, we first establish correspondence between the

Aerodynamic Shape Optimization Using A Real-Number-Encoded Genetic Algorithm

A new method for aerodynamic shape optimization using a genetic algorithm with real number encoding is presented. The algorithm is used to optimize three different problems, a simple hill climbing problem, a quasi-one-dimensional nozzle problem using an Euler equation solver and a three-dimensional transonic wing problem using a nonlinear potential solver. Results indicate that the genetic algorithm is easy to implement and extremely reliable, being relatively insensitive to design space noise.

CAD-Based Aerodynamic Design of Complex Configurations using a Cartesian Method

1 Abstract A modular framework for aerodynamic optimization of complex geometries is developed. By working directly with a parametric CAD system, complex-geometry models are modified and tessellated in an automatic fashion. The use of a component-based Cartesian method significantly reduces the demands on the CAD system, and also provides for robust and efficient flowfield analysis. The optimization is controlled using either a genetic or quasi‐Newton algorithm. Parallel efficiency of the framework is maintained even when subject to limited CAD resources by dynamically re-allocating the processors of the flow solver. Overall, the resulting framework can explore designs incorporating large shape modifications and changes in topology.

Comparison of Evolutionary (Genetic) Algorithm and Adjoint Methods for Multi-Objective Viscous Airfoil Optimizations

A comparison between an Evolutionary Algorithm (EA) and an Adjoint-Gradient (AG) Method applied to a two-dimensional Navier-Stokes code for airfoil design is presented. Both approaches use a common function evaluation code, the steady-state explicit part of the code,ARC2D. The parameterization of the design space is a common B-spline approach for an airfoil surface, which together with a common griding approach, restricts the AG and EA to the same design space. Results are presented for a class of viscous transonic airfoils in which the optimization tradeoff between drag minimization as one objective and lift maximization as another, produces the multi-objective design space. Comparisons are made for efficiency, accuracy and design consistency.

Fundamental Algorithms in Computational Fluid Dynamics

Intended as a textbook for courses in computational fluid dynamics at the senior undergraduate or graduate level, this book is a follow-up to the book Fundamentals of Computational Fluid Dynamics by the same authors, which was published in the series Scientific Computation in 2001. Whereas the earlier book concentrated on the analysis of numerical methods applied to model equations, this new book concentrates on algorithms for the numerical solution of the Euler and Navier-Stokes equations. It focuses on some classical algorithms as well as the underlying ideas based on the latest methods. A key feature of the book is the inclusion of programming exercises at the end of each chapter based on the numerical solution of the quasi-one-dimensional Euler equations and the shock-tube problem. These exercises can be included in the context of a typical course and sample solutions are provided in each chapter, so readers can confirm that they have coded the algorithms correctly.

Drag Prediction for the NASA CRM Wing-Body-Tail Using CFL3D and OVERFLOW on an Overset Mesh

In response to the fourth AIAA CFD Drag Prediction Workshop (DPW-IV), the NASA Common Research Model (CRM) wing-body and wing-body-tail configurations are analyzed using the Reynolds-averaged Navier-Stokes (RANS) flow solvers CFL3D and OVERFLOW. Two families of structured, overset grids are built for DPW-IV. Grid Family 1 (GF1) consists of a coarse (7.2 million), medium (16.9 million), fine (56.5 million), and extra-fine (189.4 million) mesh. Grid Family 2 (GF2) is an extension of the first and includes a superfine (714.2 million) and an ultra-fine (2.4 billion) mesh. The medium grid anchors both families with an established build process for accurate cruise drag prediction studies. This base mesh is coarsened and enhanced to form a set of parametrically equivalent grids that increase in size by a factor of roughly 3.4 from one level to the next denser level. Both CFL3D and OVERFLOW are run on GF1 using a consistent numerical approach. Additional OVERFLOW runs are made to study effects of differencing scheme and turbulence model on GF1 and to obtain results for GF2. All CFD results are post-processed using Richardson extrapolation, and approximate grid-converged values of drag are compared. The medium grid is also used to compute a trimmed drag polar for both codes.