Unsteady Differentiation of Aerodynamic Coefficients: Methodology and Application

The evaluation of aerodynamic forces and of the relevant coefficients is a fundamental task in aircraft design. A number of numerical methods have been investigated and some of them, the most reliable and cost effective, are daily applied in the Aerospace field for design, development and research purposes. Conversely, not so many methods have been considered reliable enough for computing aerodynamic derivatives, which are fundamental to evaluate sensitivities to aerodynamic or shape parameters. Despite the efforts deployed by former researchers, the basic mathematical complexity to evaluate derivatives strongly limits the application of several methods. Numerical differentiation is surely a task widely studied and most of the schemes which have been developed are based on the finite differences method. Finite differences are flexible and easily applicable, but they are just an algebraic transposition of the Taylor series expansion, and as such they are an approximation. As a matter of fact, the attempt to apply the mathematical theory of differentiation to computer programs has to be handled differently. Furthermore, while steady derivatives allow quantifying different sensitivities when flying in steady condition, dynamic derivatives are supposed to provide useful information on what happens in unsteady condition, but this introduces the extra complexity of the time dependency. It is hence again evident that the need of a solid mathematical basis is crucial when dealing with these kinds of problems. A possible approach is the experimental use of physical models in wind tunnels; data acquired while simulating different steady flight conditions can be interpolated to compute steady derivatives. While unsteady derivatives can be quantified by both mathematically handling steady values or moving the model in the wind tunnel and then managing the data streams that have been acquired. The drawback of this method is its high cost, especially when complex motions are tested. Furthermore, experimental work is complex, since setup and test implementation require a wide experience both in preparing the system setup and in understanding results. An extensive work has been done so far in this field; some references (Almosnino, 1994; Altun & Iyigun, 2004; Anderson & Newmann, 1999; Anderson et al., 1984; Guglieri & Quagliotti, 1993; Guglieri et al., 1993; Murphy & Klein, 2001) can give just an idea of the huge effort so far deployed. But in the latest years numerical approaches have been extensively developed and employed to fulfil the same results in a cheaper and faster way. The simplest algebraic way