Numerical analysis of multiple element high lift devices by Navier Stokes equation using implicit TVD finite volume method

Abstract

This paper deals with the analysis of multiple element high lift devices by solving the Navier Stokes equations using the TVD(Tota1 Variation Diminishing) finite difference method. In order to generate a computational grid around the multiple element airfoils automatically, the grid generator using the elliptic method, in which Poisson equations are by the finite difference method, combined with 2-D panel method is developed. As to the flow solver, some improvements are added to the TVD scheme to calculate low Mach number flows efficiently. Numerical calculations are carried out for the single slotted flap configuration. Multiple element H.L.D.(High Lift Devices) are commonly used in many transport aircrafts to obtain high lift forces necessary in landing and taking off. In designing H.L.D., however, the accurate estimation of their performances by wind tunnel testings is a difficult problem since high Reynolds number flows are hard to obtain in a wind tunnel. The analysis by CFD(Computationa1 Fluid Dynamics) is thus expected as a powerful tool in designing H.L.D. Recent advancements of super computers and numerical algorithms have made it possible to use the finite difference calculation of Navier-Stokes equations as a practical design tool, but some problems are still remained to use it for predicting the performance of H.L.D. Among these problems, the most urgent problem is to establish an automated numerical grid generation method around the multiple element airfoils. In the present work, we develop an automated numerical grid generator for multiple airfoils utilizing the elliptic method combined with 2 dimensional panel method and an implicit TVD(Tota1 Variation Diminishing) scheme suited for analyzing the flow field of multiple element H.L.D. The finite difference calculation of Navier Stokes equations for H.L.D. using these methods are carried out. Calculated results compared with experimental data are shown in the latter sections. Research engineer. 2.GRID GENERATION In order to calculate the flow around multiple element high lift devices by a finite difference method, it is necessary to generate a computational grid around multiple body configuration. As is usual to move each element in parametric design process of high lift devices , the numerical grid for each configuration is also needed. Thus, in order to use the Navier-Stokes analysis as a practical design tool, it is highly desirable that the grid generator can treat each configuration automatically. We utilize the solution of potential flow around airfoils as a building block of the automated grid generator. Since the dividing stream lines contain body surface and the potential lines are orthogonal to those stream lines, we can make orthogonal body fitted coordinate around airfoils. Using a panel method, a solution of the potential flow around arbitrary multiple bodies can be obtained so that we can generate a grid system aut~rnaticall~.[ ']~[~] The stream function Ij, and the potential qhof the p e tential flow are governed by following differential equations and boundary conditions: nldx + nz& = 0 on the body surfaces, (2.3) 4 = const on the body surfaces, (2.4) where (n l , n z ) is a unit vector normal to the body surfaces. The solution of these equations is easily obtained by the panel method. Now consider the strip region enclosed by two dividing stream lines and two potential lines. The stream function and the potential in this region are governed by equations(2.1) to (2.4) and the additinal boundary conditions: n l& + n2du = 0 on the strean lines, (2.5) $J = const on the stream lines, (2.6) nl$, + n2& = 0 on the potential lines, (2.7) $ = const on the potential lines, (2.8) where (nl, nz) is a unit vector normal to a stream line or a potential line. Although it is possible to generate a grid by tracing stream lines and potential lines in principle, the algorithm becomes complicated and the accuracy of panel method is not sufficient near the bodv surface to generate fine grids appropriate for Navier Stokes calculation. Therefore the following indirect method is adopted instead. After Thompson, Thames and Mastin 13],equations (2.1) and (2.3) can be transformed as follows: