Hybrid computational fluid dynamic algorithms based on analytic and finite volume methods

Abstract

A new approach for generating computational fluid dynamics (CFD) algorithms is introduced. This approach combines traditional analytical techniques with modern numerical methods to generate accurate solutions on very coarse grids. The flow field is subdivided into very coarse rectilinear partitions. Asymptotic solutions to the Euler equations are developed within each rectilinear partition and coupled to finite volume solutions in neighboring partitions. In this way complex flowfield phenomena such as shock waves can be modeled with a traditional CFD method in one partition while modeling the flowfield in the remaining partitions with efficient analytic methods. This new approach has particularly strong potential for aerodynamic design optimization since the analytic solutions can be differentiated to provide direct estimates of aerodynamic design sensitivities. A hybrid analytic/CFD algorithm is developed and solutions for several subsonic and transonic test problems are developed. Order-ofmagnitude or greater increases in efficiency and a significant simplification of grid requirements are demonstrated over standard CFD methods. Introduction After more than two decades of intense research and development, Computational Fluid Dynamics (CFD) has reached a state of near maturity. What has emerged is an approach based on discretization of the fluid dynamic (i.e., Navier-Stokes, Euler, etc.) and turbulence modeling equations. Equation discretization necessitates fine computational grids to achieve the required accuracy and flowfield resolution. Generation of these computational grids about * Senior Project Engineer; Senior Member AIAA ** MDC Research Fellow; Assoc. Fellow AIAA Copyright ©1997 by McDonnell Douglas Corporation Published by the American Institute of Aeronautics and Astronautics, Inc. with permission complex geometries is time consuming and can lead to extremely large grids containing several million grid points. Solution times on these grids can become quite large particularly for CFD-based design optimization and timeaccurate analyses. A single design analysis can require hundreds of CFD solutions, and timeaccurate CFD solutions typically require two or three orders of magnitude more CPU time than a steady-state solution. Several new approaches based on traditional analytic methods have been investigated in an attempt to reduce the high cost of CFD analysis. Dramatic efficiency improvements (i.e., orders of magnitude) have been demonstrated with these analytic based CFD algorithms for a number of applications. In References [1] and [2] a method for flowfield analysis and airfoil design and optimization was developed; Time-accurate analysis was demonstrated in References [3] and [4]. Of equal importance, significant simplification of grid requirements was also demonstrated. The typical CFD approach relies on discretization of the governing flowfield equations and approximating their behavior over small field cells (i.e., the grid) using low-order polynomials. The partial differential equation system is thereby reduced to a very large system of algebraic equations which is subsequently solved by iteration. If time accuracy is not required, convergence acceleration techniques, such as local time stepping, can be used to reduce significantly the CPU time for a steadystate analysis. In contrast, analytic based CFD methods subdivide the flowfield into a small number of coarse partitions and develop higher-accuracy flowfield representations within each partition using efficient classical analytic techniques appropriate to each partition. These techniques include Fourier analysis, integral transforms, and orthogonal function decomposition. Full continuous coupling of the local analytic solutions along cell interface boundaries is enforced providing high resolutions over the entire flowfield. There are several advantages to the analytic-based computational approach. Because of their intrinsic analytic nature, these solutions can be differentiated with respect to geometric design variables to efficiently yield aerodynamic sensitivity derivatives for use in design optimization. Surface boundary conditions (e.g., slope, curvature, etc.) can be analytically included in the solutions as opposed to the pointwise enforcement used in the conventional CFD approach. Engineering data need be computed only at those points of interest in the field whereas the CFD approach requires calculation of the solution at every grid point. Far field boundary conditions can be imposed rigorously at infinity by using semi-infinite cells in the outer portions of the field. These improved boundary conditions provide greater solution accuracy. Coarse partitioning of the flowfield greatly simplifies the grid generation process and may lend itself to fully-automated grid generation. One disadvantage of the analytic-based approach is that it is difficult to apply to flowfields containing complex phenomena. For instance to compute transonic solutions, complex shock fitting methods must be developed which are applicable to a limited set of problems [Ref 8]. Traditional CFD methods have been developed to the point were they routinely handle these types of problems. In this paper, a new approach is presented that combines the analytic and traditional CFD approaches to develop an efficient algorithm that is applicable to complex flowfield problems. The solution domain is divided into very coarse rectilinear partitions. The best computational algorithm is then selected for use within each partition. For instance a finite-volume CFD method may be used to capture a solution discontinuity in one partition while utilizing an analytic-based computational algorithm in the remaining partitions. Solutions in analytic and CFD partitions are coupled together by enforcing continuity of the flowfield variables at the partition boundaries. The hybrid computational algorithm is developed in two parts. First, an analytic-based algorithm applicable for multiple partitions is developed and applied to some sample test problems. Next coupling relationships are developed that provide continuity between CFD and analytic partitions. Hybrid CFD/analytic solutions are presented and solution accuracy and efficiency is compared to traditional CFD methods. Analytic-Based Computational Approach The analytic-based solution consists of a collection of individual analytic solutions valid over each flowfield subdivision. All of these solutions are linked together through their boundary conditions to provide a continuous solution throughout the entire domain. The analytic-based computational approach will be presented by first developing an analytic solution valid over a single partition and then developing the coupling relationships to provide a smooth transition between this solution and its neighboring partitions.