# Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity
## 3 Numerical Investigation
### Validation of the Proposed LLT-like Models  
For the constant shape functions and \(l=5\) m, Figures 1 and 2 present the transfer functions \(E_{11}\) and \(E_{12}\) (namely the lift due to the heave motion and the lift due to the pitching motion), with S-T and U-T representing, respectively, the results obtained by the sectional loads described by the [PERSON] theory combined with the steady and unsteady versions of the Kutta-Joukowski theorem, and with S-G and U-G representing, respectively, the results obtained by the sectional loads described by the [PERSON] theory combined with the steady and unsteady versions of the Kutta-Joukowski theorem. For both transfer functions, the S-T and U-T predictions are practically coincident and in very good agreement with the three-dimensional BEM solution. Instead, the differences between the S-G and U-G results arise both in terms of the magnitude and phase, with the S-G predictions better predicting the transfer function magnitudes and U-G better predicting the phases, as related to the BEM outcomes. These results can be explained by the small three-dimensional effect contribution to the wake normalwash by the trailed vortices, for the considered value of the wing aspect ratio. Indeed, for the S-T and U-T solutions, the different versions of the [PERSON] theorem affect only the intensity of the trailed vortices and are practically coincident and in agreement with the three-dimensional BEM results. On the other hand, the S-G and U-G results are more evidently affected by the versions of the Kutta-Joukowski theorem applied because these contribute to the definition of the shed vortices that, in these cases, are present in the wake to take into account the wake effects related to the flow unsteadiness. These results demonstrate that, for this wing configuration, the analytical shed vorticity contribution included in the Theodorsen formulation is capable of capturing unsteady-flow effects much better than the simulations combining the steady-flow load description with the numerically evaluated shed vorticity effects. Next, Figure 3 shows the magnitude of the transfer functions \(E_{21}\) and \(E_{22}\) relating the pitching moment to the heave and pitching motion, respectively. In these cases, all the solvers considered provide almost identical results (note that the coincident null-phase predictions are omitted for the sake of conciseness). This is an expected result because the pitching motion is assumed to occur around the \(1/4\)-chord point, thus deleting the contribution of the circulatory lift (see Equation (4)) which, in turn, is the term affected by the application of the Kutta-Joukowski theorem (through the matrix **H**; see Equation (15)), and by the wake vorticity (through matrices **C** and **D**; see Equations (8) and (1)). The predictions from the LLT-like solvers and from the BEM tool are in very good agreement, with some discrepancies appearing at high values of the reduced frequency. Then, Figures 4 and 5 show the transfer functions related to the NMV shape functions. In this case, the S-T and U-T solutions are not coincident and present small discrepancies, particularly in terms of the magnitude, with respect to the BEM outcomes. These differences with respect to the results in Figures 1 and 2 are due to the fact that the NMV shape functions are such to increase the three-dimensional wing-tip effects, thus increasing, at the same time, the influence of both the trailed vorticity and the version of the Kutta-Joukowski theorem applied. Nevertheless, the S-T and U-T results are still very accurate. On the other hand, the S-G and U-G solutions present a behaviour similar to that shown in Figures 1 and 2.