Reformulation of Possio's kernel with application to unsteady wind tunnel interference

Abstract

An efficient. method for computing the Possio kernel has remained elusive up to the present time. In this paper we reformulate the Possio kernel so that it can be computed accurately using existing high precision numerical quadrature techniques. Convergence to the correct values is demonstrated and optimization of the integration procedures is discussed. Since more general kernels such as those associated with unsteady flows in ventilated wind tunnels are analytic perturbations of the Possio free air kernel, a more accurate evaluation of their collocation matrices results with an exponential improvement in convergence. An application to predicting frequency response of an airfoil-trailing edge control system in a wind tunnel compared with that in free air is given showing strong interference effects.

On the numerical resolution of the generalized airfoil equation with Possio kernel

“…The importance of the construction of rules which require a low number of function evaluations, such as those we present in Sect. 2, appears more evident if we notice that the kernels Ko K1 and K2 above are given by very complicated expressions which also involve the evaluation of special functions (see for example [2] and [7]). It is this final remark that in Sect.…”

“…As stated in [11, p. 142], "at present the only methods which appear capable of handling weak singularities in the kernels" Ko(x-t) "are either Galerkin or collocation methods". However, even in these cases the evaluation of the integrals required by these two methods in general does not seem to have been considered properly, except in [7] and in [20].…”

“…the Jacobi polynomials orthogonal in (-1, 1) with respect to the weight function 1Vq~-t/]//1 + t. To construct the elements of the final collocation matrix he integrates exactly the 1 kernel elements --and log Ix-t I, but approximates with a Gauss-Jacobi [7] have shown that Possio's kernel can be written in the form In this paper we consider the decomposition (1.3) and propose a rule, based on the zeros of P,( 89 which appears much more efficient than those of Fromme and Golberg. Also when one prefers to use a Galerkin method, our formula can be easily extended to compute the coefficients of the final Galerkin matrix and thus produce a less expensive approach than the one given by Moss in [20].…”