Asymptotical form of Possio integral equation in theoretical aeroelasticity

Abstract

The paper is the second in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. In the first paper "Reduction of the boundary-value problem to Possio integral equation in theoretical aeroelasticity", we reformulated the initial boundary-value problem, involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions, in the form of a singular integral equation, the Possio equation. In our derivation, we have used the ideas and techniques totally different from the ones used by C. Possio in his original work. In the present paper, we show that the integral equation can be split up into two parts in such a way that the first part is not small with respect to a complex parameter entering the equation, while the second part is asymptotically small and tends to zero as the above parameter tends to infinity. In this paper, we justify such a splitting and in the next paper, we will prove solvability of the nonvanishing part of the Possio equation and then prove that asymptotically small terms cannot destroy the aforementioned unique solvability. Keywords: compressible flow, velocity potential, Mikhlin multipliers, Hilbert integral transform There followed . . . two short outstanding contributions by Camillo Possio in Italy. In 1938, he applied the acceleration potential to the two-dimensional nonstationary problem and arrived at the integral equation . . . , the solution 0921-7134/09/$17.00 © 2009 -IOS Press and the authors. All rights reserved 214 M.A. Shubov / Asymptotical form of Possio equationto which gives the loading over a flat plate airfoil in an air stream for a known motion of the plate, i.e., for a given downwash. Possio indicated a procedure for its numerical solution, although several authors later contributed more convenient methods. Possio also gave an outline of the parallel problem for a supersonic mainstream. Possio's brief brilliant career ended with his death during the war years.

Evolution semigroups in supersonic flow-plate interactions

“…Flow-structure models have attracted considerable attention in the past mathematical literature, see, e.g., [2,3,8,9,10,14,16,17,22,31,32,36,37,39,41,43] and the references therein. However, the vast majority of the work done have been devoted to numerical and experimental studies, see, e..g., [1,6,7,20,21,22,24] and also the survey [34] and the literature cited there.…”

“…Much of the studies has been based on linear one-dimensional-special geometries plate models where the goal was to determine the speed at which flutter occurs, see [1,6,7,21,24,34] for instance. More recently the study of linear models with a one dimensional structure (beam) and Kutta-Jukovsky boundary conditions found renewed interest and have been extensively pursued in [2,39,40,41]. This line of work has focused on spectral properties of the system, with particular emphasis on identifying aeroelastic eigenmodes corresponding to the associated Possio integral equation.…”

Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures

“…However, the majority of the work that has been done on flow-structure interactions has been devoted to numerical and experimental studies, see, for instance, [2,9,41,34,44] and also the survey [55] and the literature cited there. Many mathematical studies have been based on linear, two dimensional, plate models with specific geometries, where the primary goal was to determine the speed at which flutter occurs [2,9,41,44,55], see also [3,61,5,6,62] for the recent studies of linear models with a one dimensional flag-type structure.…”

Mathematical Aeroelasticity: A Survey

“…Many mathematical studies have been based on linear, two dimensional, plate models with specific geometries, where the primary goal was to determine the flutter point (i.e., the flow speed at which flutter occurs) [2,11,41,45,61]. See also [3,5,6,69] for the recent studies of linear models with a one dimensional flag-type structure (beams). This line of work has focused primarily on spectral properties of the system, with particular emphasis on identifying aeroelastic eigenmodes corresponding to the associated Possio integral equation (addressed classically by [74]).…”