Effect of Edge-Based Discretization Schemes in Computations of the DLR F6 Wing-Body Configuration

Abstract

Two different edge-based spatial discretization schemes are investigated. In the first approach the discretization is applied to the edge control surface formed by the sum of the element sub-surfaces. In the second approach, the discretization is carried out on each sub- surface. It is shown theoretically that the discretization approach on the sub-surfaces may lead to enhanced accuracy at the expense of higher computational cost. The two approaches are applied in two dimensions over an airfoil and to the DLR F6 configuration. The accuracy is evaluated by conducting grid convergence studies. The discretization scheme on the sub- surfaces gives approximately first order accuracy in two dimensions, the other scheme is even less accurate. For the DLR F6 configuration in three dimensions, however, both approaches exhibit similar accuracy close to the expected second spatial order. 1 . Polar calculations and grid convergence studies were undertaken for the DLR-F6 wing-body (WB) and the WB configuration with a fairing (WB-FX2B) added to the junction between the wing trailing edge and fuselage. A summary of the results from the DPW3 contributions was given by Vassberg et al. 2,3 with a statistical analysis presented by Morrison and Hemsch 4 . The conclusion of that investigation was that, after removing outliers, both results with flow solvers using structured and unstructured grids give satisfactory grid convergence and hence accuracy. The most accurate results were obtained with finite volume solvers that rely on structured grids. The computed results on unstructured grids were slightly less accurate and, in particular, more sensitive to the selected family of grids. In this paper we investigate the grid convergence and solution accuracy of two different edge-based spatial discretization techniques in an unstructured, node-centered finite volume solver. The two approaches are compared primarily in terms of their grid convergence properties and hence solution accuracy but also steady state convergence and computational costs are discussed. They are applied to the WB configuration on three different groups of successively refined grids with different characteristics. The WB configuration was chosen since it has an area of flow separation at the wing-body trailing edge junction and the variation in the computed results of different participants was larger compared to the WB-FX2B where the flow is attached. In addition to the DLR F6 WB configuration, a similar grid convergence investigation is carried out over a two-dimensional airfoil. In the following section the flow solver with its solution algorithms is described together with the two discretization algorithms. The numerical accuracy of a convection and diffusion operator is investigated for a simple model grid in two dimensions. This is followed by the numerical results for the two test cases with concluding remarks at the end. II. Solution Algorithm The CFD solver employed in the calculations is the Edge code (http://www.edge.foi.se/), which is an edge- and node-based Navier-Stokes flow solver applicable for both structured and unstructured grids 5-7 . Edge is based on a finite-volume formulation where a median dual grid forms the control volumes with the unknowns allocated in the dual grid centres. The governing equations are integrated explicitly with a multi-stage Runge-Kutta scheme to