Flow around simply and multiply connected bodies - A new iterative scheme for conformal mapping

Abstract

After discussing the techniques that can be adopted for numerical confprmal mapping of single airfoils, cascades, and doubly connected domains, either for the purpose of calculating the irrotational flowfield around these configurations or of constructing a calculation grid for the numerical computation of more general types of flow, an iterative algorithm is presented that displays quadratic convergence. This algorithm, made possible by the analytical solution of the integral equation that arises from linearizing the problem, is several times faster than the classical Theodorsen-Garrick procedure under most conditions and can handle more general shapes for which the older method does not converge at all.

A simplified Fornberg-like method for the conformal mapping of multiply connected regions—Comparisons and crowding

“…These linearizations were first proposed in [18,7] and were rigorously justified in [28]. The conditions in Theorem 3.2 are applied to require that (11) are the boundary values of a function analytic in D. These conditions give linear equations for U k , z k , k .…”

“…The other approach, first proposed by Fornberg [9] for the simply connected case and extended by DeLillo et al, derives the inner linear systems from conditions on the Fourier (Laurent) coefficients which guarantee analytic extension of functions on the boundaries into the computational domain; see [2] for an overview. Both approaches have been developed for the following canonical domains: the unit disk [9,24,8], the annulus [8,18,26], the ellipse [5,6,27,15] and multiply connected circular domains [1,7,15,28,29].…”

Finite element method for parallel slit mappings

“…Further the potential flow analysis and the grid generation of multielement airfoils have been studied by Ives [13], Halsey [8,9], Luchini and Manzo [14], Stewart [22], and Nielsen and Anderson [20].…”

On the use of conformal mapping for the computation of hydrodynamic forces acting on bodies of arbitrary shape in viscous flow. Part 2: multi-body configuration

“…Venkatesan underlined that conformality of the transformation must be checked. For more than two bodies Wegmann [14] and Luchini and Manzo [15] proposed some improvements of the transformations. As much work has already been done in the past, the main outlines of the technique is described in the sequel.…”