An efficient finite differences method for the computation of compressible, subsonic, unsteady flows past airfoils and panels

Numerical investigation of the effects of compressibility on the flutter of a cantilevered plate in an inviscid, subsonic, open flow

“…where φ up (t, x) denotes the potential at y = 0 + . Substituting the latter relation into (8) and integrating by parts, the generalized aerodynamic forces read as [35] Using now (2) and writing φ(t, x, y) in the complex form φ(t, x, y) = Re φ(x, y) exp ((n + jω)t) , as well as q i (t) and Q aero i (t) (relations ( 9)-( 10)), the equations ( 13)-( 15),( 17) read as…”

“…Some other aerodynamic models are based on incompressible vortex-lattice methods, as happens in the works of Yamaguchi et al [23], D. Tang et al [3], Argentina and Mahadevan [24], L. Tang and Païdoussis [25,26], L. Tang et al [5,27], Gibbs et al [28], Zhao et al [29], Howell and Lucey [30,31], Alben [32], and Michelin and Llewellyn Smith [33]. Although these methods have been widely used in unsteady, incompressible aerodynamics -see, for example, [34]-, they present some inconveniences when extended to compressible flows [35,36] such as: (i) they are based on a fundamental solution -the so-called unsteady compressible vortex -whose velocity field is not well-known and lacks a clear physical meaning [37], (ii) when formulated in the time domain, they have to keep track of the history of the vortices' intensities in order to compute the pressure jumps at the airfoil at a given instant and (iii) their extension to three-dimensional flow is not clear, although some efforts have been done on the matter [38]. These drawbacks, which may also appear when using other kind of fundamental solutions, such as dipoles [39], or other boundary elements methods in general [40], considerably complicate the extension of the analysis of the incompressible flutter problem -usually performed using eigenvalue theory [30,31] or time-domain simulations [3,35,36]-to the compressible case.…”

“…Therefore, in order to deal with compressibility effects at higher Mach numbers, some investigators have tried an approach based on the direct discretization in space and in time of the linearized, compressible, potential flow equations. This is done, for example, in the works of Huang and Zhang [44] -who formulated a Chebyshev pseudospectral method-and of Colera and Pérez-Saborid [35,36] -who employed finite differences-. However, only the latter authors effectively applied their method to compressible flows -since Huang and Zhang only considered the case of zero Mach number-.…”

“…Compressibility effects on the flutter of a cantilevered, two-dimensional plate in a compressible, subsonic flow have been previously studied by Tanida [45] -who developed a compressible doublet-lattice method in the frequency domain for a plate in channel flow, but used only one Galerkin mode to describe its motionand by Colera and Pérez-Saborid -who used first a compressible vortex-lattice method [37,46,47] and later a finite differences method [35,36] to study a single case of a plate of given mass and stiffness-. The aim of this work is to generalize these investigations by performing a parametric study of the flutter speed and the flutter frequency of a two-dimensional, cantilevered plate in a compressible, subsonic flow as a function of its mechanical properties and the upstream Mach number.…”

“…The aim of this work is to generalize these investigations by performing a parametric study of the flutter speed and the flutter frequency of a two-dimensional, cantilevered plate in a compressible, subsonic flow as a function of its mechanical properties and the upstream Mach number. For that purpose, we have developed a numerical method based on expressing the plate equations of motion in terms of shape functions and generalized coordinates and on solving the equations for the linearized potential flow with a novel frequency-domain numerical scheme adapted from that presented in [35,36].…”