Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined the wind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFD has had its most favorable impact on the aerodynamic design of the high-speed cruise configuration of a transport. This success has raised expectations among aerodynamicists that the applicability of CFD can be extended to the full flight envelope. However, the complex nature of the flows and geometries involved places substantially increased demands on the solution methodology and resources required. Currently most simulations involve Reynolds-Averaged Navier-Stokes (RANS) codes although Large Eddy Simulation (LES) and Detached Eddy Suimulation (DES) codes are occasionally used for component analysis or theoretical studies. Despite simplified underlying assumptions, current RANS turbulence models have been spectacularly successful for analyzing attached, transonic flows. Whether or not these same models are applicable to complex flows with smooth surface separation is an open question. A prerequisite for answering this question is absolute confidence that the CFD codes employed reliably solve the continuous equations involved. Too often, failure to agree with experiment is mistakenly ascribed to the turbulence model rather than inadequate numerics. Grid convergence in three dimensions is rarely achieved. Even residual convergence on a given grid is often inadequate. This paper discusses issues involved in residual and especially grid convergence.

The need for accurate simulations of engine installations on modern commercial transport aircraft has led to consideration of several formulations capable of modeling engine exhausts. Since such exhausts often interact with wings, struts, and nacelles, a complex geometry computational fluid dynamics (CFD) capability is desirable. Engine exhausts often contain nonlinear effects such as weak shock waves. There are very few CFD codes that can model these effects for complex geometries in a timely way. However, a full potential formulation has been implemented in the general geometry code TRANAIR. This model incorporates certain assumptions, the main one being that the flowfield can be divided into a finite number of regions in each of which the total pressure and total temperature are constant. The purposes of this article are to state the theoretical assumptions made by the full potential and Euler models and to validate the methods on an axisymmetric test case. In the situations considered (typical of modern turbofan engines) the full potential and Euler results agree very well.

The adaptive grid method implemented in the 3D general geometry CFD code TRANAIR is described. Adaptive gridding in TRANAIR has been used to help solve many industrial aerospace analysis and design problems. Grids are adapted to numerical solutions by refining and coarsening local rectangular finite elements according to values of error indicators computed for the elements. The underlying finite element method is summarized and details of the adaptive gridding approach are given. Emphasis is placed on several principles used in developing and applying the approach. In particular, many computational comparisons are presented to explain the reasoning behind the local error indicator and gridding strategy used.

A new finite element method for solving important linear and nonlinear boundary value problems arising in computational physics is described in this paper. The method is designed to handle general three-dimensional regions, boundary conditions, and material properties. The boundaries are described by piecewise planar surfaces on which boundary conditions are imposed. The method uses box finite elements defined by a Cartesian grid that is independent of the boundary definition. Local refinements are performed by dividing a box element into eight similar box elements. The discretization uses trilinear approximations on the box elements with special element stiffness matrices for boxes cut by any boundary surface. This discretization process is automated and does not require the generation of a boundary conforming grid. The resulting (possibly nonlinear) discrete system is solved using a preconditioned GMRES algorithm. The primary preconditioner is a sparse matrix solver using a dynamic drop tolerance in the decomposition phase. Results are presented for aerodynamics problems with up to 400,000 elements, demonstrating the accuracy and efficiency of the method.

In this paper the use of a new out-of-core sparse matrix package for the numerical solution of partial differential equations involving complex geometries arising from aerospace applications is discussed. The sparse matrix solver accepts contributions to the matrix elements in random order and assembles the matrix using fast sort/merge routines. Fill-in is reduced through the use of a physically based nested dissection ordering. For very large problems a drop tolerance is used during the matrix decomposition phase. The resulting incomplete factorization is an effective preconditioner for Krylov subspace methods, such as GMRES. Problems involving 200,000 unknowns routinely are solved on the Cray X-MP using 64MW of solid-state storage device (SSD).

Attention is given to a new approach to solving full potential equations about arbitrary configurations. Numerical algorithms from such fields as finite elements, preconditioned Krylov subspace methods, discrete Fourier analysis, and integral equations are combined to take advantage of the size and speed of current and emerging supercomputers. On the basis of this appraoch, a robust, efficient and easy to use computer code referred to as TRANAIR has been developed for transonic analysis of complex geometries.

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson’s equation on irregular exterior regions. The method requires $O(N\log N)$operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

A method is presented that has the potential for solving transonic flow problems about the same complex aircraft configurations currently being analyzed by subsonic panel methods. This method does not require the generation of surface fitted grids. Instead it uses rectangular grids and subgrids together with embedded surface panels on which boundary conditions are imposed. Both the Euler and full potential equations are considered. The method of least squares is used to reduce the solution of these equations to the solution of a sequence of Poisson problems. The Poisson problems are solved using fast Fourier transforms and panel influence coefficient techniques. The overall method is still in its infancy but some two dimensional results are shown illustrating various key features.