Unsteady Lifting Line Theory Using the Wagner Function for the Aerodynamic and Aeroelastic Modeling of 3D Wings

Abstract

A method is presented to model the incompressible, attached, unsteady lift and pitching moment acting on a thin three-dimensional wing in the time domain. The model is based on the combination of Wagner theory and lifting line theory through the unsteady Kutta–Joukowski theorem. The results are a set of closed-form linear ordinary differential equations that can be solved analytically or using a Runge–Kutta–Fehlberg algorithm. The method is validated against numerical predictions from an unsteady vortex lattice method for rectangular and tapered wings undergoing step or oscillatory changes in plunge or pitch. Further validation is demonstrated on an aeroelastic test case of a rigid rectangular finite wing with pitch and plunge degrees of freedom.

Unsteady lifting-line theory and the influence of wake vorticity on aerodynamic loads

“…Since it is assumed that 2s c, and that changes in the flow happen on the span length-scale, the problem can be considered as 2D. The problem consists of a symmetric airfoil at a certain spanwise section undergoing small-amplitude pitch and plunge oscillations, h(y, t) = h * 0 (y)c(y)e iωt α(y, t) = α 0 (y)e i(ωt+ψ) (7) where h * 0 is plunge amplitude per unit chord, α 0 is pitch amplitude, ψ is the phase between plunge and pitch, and ω is the frequency of oscillation typically expressed as a chordwise or spanwise reduced frequency Figure 2 shows this problem, for which the solution is given by Theodorsen [60]. The results are expressed here in terms of a general unsteady thin-airfoil theory [36,52] which allows easy modification of the approach for various wing geometries and kinematics.…”

“…The first unsteady finite wing analysis in the time domain is due to Jones [35], although it is only applicable to elliptic planforms. A time-domain ULLT is presented by Boutet and Dimitriatis [7] using an inner solution based on Wagner's method [67], and a pseudosteady assumption. Ramesh et al [54] introduce a lifting-line theory for a geometrically nonlinear inner solution, again using a pseudosteady assumption.…”

“…Additionally, only limited validation of unsteady lifting-line theory has been undertaken. Sclavounos [57] and Boutet and Dimitriatis [7] compared their methods to the unsteady vortex lattice method. This method shares some assumptions with unsteady lifting-line theory and requires care in discretization to satisfy the Kutta condition [55].…”

A Fast Correction Method of Model Deformation Effects in Wind Tunnel Tests

“…Another method is based on the computational aeroelasiticity approach to determine the influence of model deformation [25][26][27]. Although the computational aeroelasticity method has been widely used in aircraft design [28,29], there are still some problems predicting the model deformation and variation of aerodynamic force coefficients accurately [19,30]. The third method is a combination of CFD and model deformation measurement (MDM) [31].…”

Special Issue: Aeroelasticity

“…The approach provides a minimal order realization with exact interpolation of the unsteady aerodynamic forces in tangential directions and overcomes certain drawbacks of the classical rational function approximation approach. Boutet et al [2] discussed the development of an unsteady aerodynamic model based on the combination of Wagner theory and lifting line theory through the unsteady Kutta-Joukowski theorem. The resulting set of closed-form linear ordinary differential equations are solved analytically or by using a Runge-Kutta-Fehlberg algorithm.…”