High-order BEM for potential transonic flows

Abstract

A high-order boundary element method (BEM) for the analysis of the steady two-dimensional full-potential transonic equation is presented. The use of a highorder (piecewise cubic on the boundary and piecewise bi-cubic in the ®eld) numerical formulation, the main novelty of the present work, is important in two respects: ®rst, the convergence of the solution as h vanishes is faster than the zeroth-order (the piecewise constant) one, yielding more accurate results with coarser grid resolutions. In addition, in supercritical¯ows, the derivation of the velocity ®eld from the high-order representation for potential gives, in the vicinity of the shock, a sharper discontinuity, and allows for an in-depth analysis of the shock properties. Both aspects are analyzed in the present paper through applications to steady, two-dimensional, full-potential¯ows. All the results obtained using the present method are validated through comparison to other computational¯uid dynamics (CFD) solutions of the fullpotential¯ows and, when applicable, Euler equations. A comparison to a zeroth-order BEM, based on the same integral formulation is also included. IntroductionA high-order (piecewise cubic on the boundary and piecewise bi-cubic in the ®eld) boundary element method (BEM) for the analysis of steady two-dimensional fullpotential transonic¯ows is presented. To the authors knowledge, this represents the ®rst high-order BEM formulation in transonic aerodynamics.The boundary integral formulation used here is related to that of Tseng and Morino (1982), where the equation for the velocity potential has the form of a non-homogeneous wave equation. The unsteady formulation has been applied to steady two-and three-dimensional¯ows, under the assumption of transonic small perturbation, in Tseng and Morino (1982), and Tseng (1983, 1984, and extended to the full-potential steady two-and three-dimensional analysis in Morino and Iemma (1993), Iemma (1994), and Iemma and Morino (1994.This work is limited to steady, two-dimensional¯ows. Accordingly, the linear differential operator, in the Prandtl Glauert space, is the Laplacian, as all the nonlinear terms are moved to the right hand side of the differential equation and treated as ®eld sources. These are written in conservative form (see Morino and Iemma 1993). Note that the linear part of the equation is formally identical to the equation governing potential incompressible aerodynamics. This yields an integral formulation for the potential which is particularly simple when compared to the full unsteady formulation presented in Iemma (1994) and Iemma and Morino (1997). In particular, the linear part of the resulting integro-differential equation is identical to the one presented in Gennaretti et al. (1997) for the analysis of three-dimensional steady incompressible potential¯ows, which represents the ®rst successful application of the high-order BEM presented here. Although the method of Gennaretti et al. (1997) is three-dimensional, in this preliminary phase we limit our attention to the twodimensi...

Numerical simulation and analysis of surface scattering – part 1

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