On the Use of High Order Methods for Complex Aerodynamic Problems

Abstract

Finite-difference, finite-volume and finite-element methods implemented in CFD codes require high grid resolution and smooth grids in order to obtain highresolution computations of complex aerodynamic flows. For certain applications, the grid resolution requirements with second-order accurate methods make these computations prohibitively expensive. For example, numerical solutions of flows over wings at high incidence [1] and helicopter rotors [2] require large number of grid points in order to preserve tip vortices, vortical wakes, and detached shear layers which have a significant influence on loading. The ever-increasing demand to capture more complex flow phenomena makes necessary the use of high-resolution numerical solutions. The recently developed high-order accurate methods for the compressible flow equations in complex domains are expected to play a key role in efforts to achieve further progress towards this direction. In this review, a brief description of available high-order accurate methods is given. The first class of methods uses high-order accurate, finite-difference discretizations. The second class of methods is the high-order finite-volume schemes including essentially nonoscillatory (ENO) and WENO reconstructions. The third class is the discontinuous Galerking and the spectral volume discretizations. Accuracy and efficiency characteristics and the potential impact of each method to aerodynamic applications is assessed. Examples of high-order accurate numerical solutions of compressible flow applications appeared in the literature are presented. For these aerodynamic problems, an attractive alternative to high grid resolution is the use of highorder accurate in space methods. High-order methods typically provide at least third-order spatial accuracy. Several investigations [3], [4], [5], [6] showed that use of numerical methods with order of accuracy higher than second can significantly benefit this class of problems. These methods, however, are often perceived as less robust, costly to run, and complicated to understand and code. As a result, there are very few working CFD codes that use higher order accurate schemes. This is especially true for codes designed to compute steady flows. In this paper, we attempt to dispel this negative impression about high-order methods and show the potential benefit from their use in aerodynamic computations.