Modeling of Airplane Wings with Winglets

We describe methods and algorithms for automated generation of multiparted airplane wings with winglets. The wings are given by one or more cross-sections (e.g. by point clouds) and their top views. The relative thickness of the wing (thickness/chord) can be varied from section to section. A rounded tip with design parameters and GC1-continuity at the crossing to the wing is automatically computed. Additionally several types of winglets can be added at the tip. A simpli ed half of a fuselage is computed, too. The geometries can be modi ed by changing only a few signi cant parameters. Methods for the generation of surface grids for the di erent geometries are also presented. Introduction In the Collaborative Research Center SFB 401, "Modulation of Flow and Fluid-Structure Interaction at Airplane Wings", the aerodynamics of high lift and cruise con gurations and the interaction of structural dynamics and aerodynamics are presently being investigated. In the subproject "High Reynolds Number Aero-Structural Dynamics" stationary and unsteady wind tunnel readings with an elastic model have been carried out. The experiments were done in the European Transonic Wind-Tunnel (ETW) in December 2006. The wing corresponds to a cruise con guration of scale 1:28, whose supercritical cross-section is numerically de ned in two AGARD reports (BAC 3-11 aerofoil, [1], [2]) by its ordinates. It is modeled as a three parted back-swept wing with a rounded tip. To achieve realistic results a half-body is placed between the wing and the wind tunnel wall. In an upcoming project the outer part of the wing will be replaced by a part with a winglet. The winglet will have a tip according to the rules used for the original wing. We have developed algorithms for automatic generation of winglets with di erent bending radii, angles and top views. The surface grids are automatically deformed with the geometry. The developed tools are based on B-Spline representations and Catmull-Clark-subdivision. The approximation and fairing methods have to ful ll several constraints. These depend upon the manufacturing and the needs of the applied ow solver for the Navier-Stokes equations. The basic data exchange between the modeling, grid generation and manufacturing software is carried out by IGES les. The methods for approximation, fairing, modeling, grid generation and further details of the project are presented. The rest of the paper is organized as follows: We rst give same basic notations of B-Splines and show some results especially needed for our fairing methods. Then we explain our fairing method. The next section deals with the modi cations and bending of the wing for the winglet construction. Afterwards we describe a method to construct surface grids by subdivision methods. Basic B-Spline notations A B-Spline function of order k is piecewise a polynomial of degree k−1. For a given knot vector T = (t0, t1, ..., tk−1, tk, ..., tl, tl+1, ..., tl+k) the B-Spline functions can recursively be de ned/computed by k > 1 : N i (t) = t−ti ti+k−1−ti N k−1 i (t) + ti+k−t ti+k−ti+1 N k−1 i+1 (t) k = 1 : N i (t) = N 1 i (t) = { 1 ti ≤ t < ti+1 0 elsewhere (1) for i = 0, 1, ..., m− k. Sometimes they are called normalized in this form. For practical use we have to add Nm−1,1(tm) := 1, make sure that ti < ti+k for i = 0, 1, ..,m − k and, if for any i ∈ {0, 1, ..., m − 1} ti = ti+1 we have to add N j i (t) =    t−ti ti+j−1−ti N j−1 i (t)+ ti+j−t ti+j−ti+1 N j−1 i+1 (t) if ti < ti+j−1 and ti+1 < ti+j t−ti ti+j−1−ti N j−1 i (t) if ti+1 = ti+2 = ... = ti+j ti+j−t ti+j−ti+1 N j−1 i+1 (t) if ti = ti+1 = ... = ti+j−1 (2) From the above recursion formula all the properties of B-Spline functions can be derived. Only those theorems and formulas that are of interest for the further paper are stated. We start with the following thorem: Theorem 1 The normalized B-Spline functions of order k are a partition of unity on [tk, tm−k), that is ∑