Design of a two-element airfoil in a range of angles of attack

Abstract

A numerical-analytical solution of an inverse boundary-value problem of aerohydrodynamics is obtained for a two-element airfoil in the full formulation, based on the velocity distribution defined on the sought airfoil contours in a range of angles of attack. It is demonstrated that flow separation does not occur in the entire range considered for a specified non-separated velocity distribution on the upper surfaces at the maximum angle of attack and on the lower surface at the minimum angle of attack. An example of constructing a sectional airfoil is given; verification of the results obtained is performed with the use of the Fluent software package.Key words: inverse boundary-value problems, airfoil, range of angles of attack.Design of multi-element airfoils with optimal aerodynamic characteristics is an urgent problem [1,2]. The approach used in [1, 2] implies airfoil modification by the method of adjoint gradients. The calculations are performed on multiprocessor supercomputers by multiple solutions of the problem of a viscous flow around a multielement airfoil. Such a problem, however, can also be solved on a usual personal computer by applying the theory of inverse boundary-value problems of aerohydrodynamics [3].Abzalilov et al.[4] solved the problem of design of a two-element airfoil in an ideal incompressible fluid flow, based on the velocity or pressure distribution specified on the airfoil surface. From the practical viewpoint, it seems of interest to design airfoils with specified aerodynamic characteristics for a certain range of angles of attack rather than for one angle of attack. Let us consider the problem solved in [4], with the only difference that the initial data of the problem, namely, the velocity distributions, are specified for two different angles of attack.The problem is solved in the following formulation. In a physical plane z (Fig. 1a), the sought two-element airfoil A k B k (k = 1, 2) is exposed to a steady irrotational flow of an ideal incompressible fluid at two angles of attack (α and α * ); the difference δ = α * − α > 0 is assumed to be given. The airfoil contours L zk are assumed to be smooth, except for the trailing edges B k , where the internal (with respect to the flow domain) angles are 2π.The origin of the Cartesian coordinate system is chosen on the trailing edge B 1 of the contour L z1 , and the abscissa axis is parallel to the direction of the specified free-stream velocity vector v ∞ . The contour perimeters are known and equal to l k . The arc abscissas s k of the airfoil contours are counted from zero at the points of B k to l k at the same points, so that the flow domain remains on the left with increasing s k . Each contour is divided into two parts (upper and lower surfaces) by the point C k (s = s ck ). The velocity distributions on the airfoil contour L zk are specified for the angle of attack α on the lower surface and for the angle of attack α * on the upper surface:An example of such a parametric distribution of velocities v k (s k ) and v * k (s k ) is give...