A modification of the generalized airfoil equation and the corresponding numerical methods

• Main Author: Ioakimidis, Nikolaos
• Other Authors: Ιωακειμίδης, Νικόλαος
• Format: Technical Report
• Language: English
• Published: 2018
• Subjects: Steady flow; Inviscid flow; Irrotational flow; Subsonic flow; Thin airfoils; Cracks; Wind tunnels; Airfoil equation; Generalized airfoil equation; Airfoil polynomials; Chebyshev polynomials; Cauchy type integrals; Cauchy type singular integral equations; Numerical methods; Numerical integration; Galerkin method; Gauss–Chebyshev method; Lobatto–Chebyshev method; Σταθερή ροή; Μόνιμη ροή; Ροή χωρίς ιξώδες; Αστρόβιλη ροή; Υποηχητική ροή; Λεπτές αεροτομές; Ρωγμές; Αεροδυναμικές σήραγγες; Εξίσωση της αεροτομής; Γενικευμένη εξίσωση της αεροτομής; Πολυώνυμα της αεροτομής; Πολυώνυμα Chebyshev; Ολοκληρώματα τύπου Cauchy; Ιδιόμορφες ολοκληρωτικές εξισώσεις τύπου Cauchy; Αριθμητικές μέθοδοι; Αριθμητική ολοκλήρωση; Μέθοδος του Galerkin; Μέθοδος των Gauss–Chebyshev; Μέθοδος των Lobatto–Chebyshev

The two-dimensional problem of steady, inviscid, irrotational, subsonic flow around a straight or curvilinear thin airfoil or an array of such airfoils inside a wind tunnel is generally reduced to a one-dimensional Cauchy type real or complex singular integral equation called generalized airfoil equation. Here a new form of this equation is suggested (with no change in the unknown function) with index equal to 1 instead of 0. The new equation is supplemented by an integral condition assuring the uniqueness of its solution. This modification of the generalized airfoil equation permits the application of the theoretical results and the algorithms for the numerical solution of Cauchy type singular integral equations with index equal to 1 (mainly appearing in crack problems in the theory of plane elasticity) to the generalized airfoil equation and it establishes the relationship between crack and airfoil problems. Moreover, it permits the utilization of the classical Chebyshev polynomials instead of the airfoil polynomials. Three applications are also made and numerical results are presented.