An alternative simulation approach for the ONERA aerodynamic model and its application in the nonlinear aeroelastic analysis of slender wings with pylon-store system
Introduction
Slender wings are usually adopted in high-altitude long-endurance aircrafts for achieving high aerodynamic efficiency. For this type of wing, the aeroelastic responses of wings are often in the dynamic stall regime. The interaction between the nonlinearities of aerodynamic loads associated with dynamic stall and the nonlinearities of wing structure will lead wing motions to be complicated significantly. Thus, to better understand the dynamic behaviors of slender wings, the used aerodynamic model should include the effects of dynamic stall.
For slender wings, the aerodynamic loads on the wing are usually determined by using the strip theory combined with the two-dimensional aerodynamic model. Among the two-dimensional aerodynamic models, the ONERA model is one of the mostly used models which include the effects of dynamic stall. The ONERA model is a semi-empirical, unsteady, nonlinear model developed by Tran and Petot [1] and Dat and Tran [2] of the Office National d’Etudes et de Recherches Aérospatiales (ONERA). But in their studies [[1], [2]], the ONERA model only can determine two-dimensional aerodynamic forces on an airfoil oscillating in pitch which experiences dynamic stall. The ONERA model was latter extended for both pitch and plunge motions by Peters [3]. Dunn and Dugundji [4] employed the ONERA model to investigate the nonlinear, stalled, aeroelastic behavior of rectangular, graphite/epoxy, cantilevered wings. The nonlinear aeroelastic responses of helicopter blades in the regime of dynamic stall were experimentally and theoretically studied by Tang and Dowell [[5], [6]] where the ONERA model is employed to determine the aerodynamic loads. Tang and Dowell [[7], [8]] also used the ONERA model to explore the nonlinear aeroelastic response and gust response of high-aspect-ratio wings. Depailler and Friedmann [9] adopted the ONERA model and a rational functional approximation to describe the aerodynamic loads of a helicopter blade during stall, and investigated the helicopter vibration reduction problem with this method. Ovesy et al. [10] employed the ONERA model to analyze the flutter behaviors of high-aspect-ratio wings, where only the linear parts of the ONERA model was considered since the wing motions were restricted in the unstalled regime.
For the ONERA model, the aerodynamic loads are given in terms of differential equations. Through coupling the differential equations in each aerodynamic elements and the wing structural equations, the governing equations of the aeroelastic system can be obtained. In order to calculate the aerodynamic loads accurately, a large number of aerodynamic elements are needed, which may lead to high computational cost. Thus, for improving the computational efficiency, a new simulation approach is proposed to simplify the calculation in the ONERA model. In the new approach, the circulatory part of the ONERA model is solved by using Duhamel integral method. And by introducing two additional variables, the circulatory part of the model can be directly expressed in the modal space of the wing motion. The aerodynamic elements are only used to express the nonlinear parts of the ONERA model. Hence the number of the state variables of the aeroelastic system can be reduced modestly.
In addition, the proposed approach is used to analyze the nonlinear aeroelastic responses of two models about slender wings in this paper. One is a slender wing without store, and the other is a slender wing with a pylon-store system which includes a free-play gap. For the store in the current study, the scale of it is smaller than the wing scale. If the scale of the store is similar to the wing scale, the system can be seen as the body with the wing mounted on it, which are widely used in the studies of micro-air vehicle. To this system, Fitzgerald et al. [11] examined two computational approaches which can be used to study it. The one is based on Navier–Stokes equations, the other is based on the unsteady vortex lattice method. In [[12], [13]], a computational co-simulation strategy was developed for modeling this system, where the structural model and aerodynamic model are combined and the information are exchanged in a strong way between the two models. Moreover, Hsu et al. [14] developed a method to construct a realistic estimation of the true configuration of this system in the presence of large rotation and deformation. On the other hand, to airfoils, delta wings and low-aspect-ratio wings, the effects of the free-play gap on the aeroelastic responses have been widely investigated [[15], [16], [17], [18], [19], [20]]. But to slender wings, the interaction between the nonlinearity of the free-play gap and the nonlinearities of wing structure and aerodynamic loads has not been studied sufficiently. Thus, the effects of the free-play gap on the dynamic behaviors of the slender wing are explored by using the new approach in the current work. Meanwhile, the classical approach in which the circulatory part of the ONERA model is expressed in terms of aerodynamic elements is also used in the analyses of the two wing models for comparison. Through simulation, some interesting nonlinear behaviors of the slender wing induced by the complex pylon-store system are observed. And the advantages of using the new approach for the complex aeroelastic system are presented.
Section snippets
The new simulation approach for the ONERA model
For a typical wing section in a slender wing shown in Fig. 1, the nonlinear stalled ONERA aerodynamic model provides time variation of unsteady lift coefficient and moment coefficient about the quarter-chord. In Fig. 1, for the wing section, the axes of the inertial reference coordinate system are , and the axes of the cross-sectional coordinate system are . The semi-chord of the section is is the distance between the elastic center and the mid-chord . is the ratio of this
Results of convergence studies
In this part, the convergence study about the proposed and classical approach is implemented for a slender wing without store. In the study, the parameters of the slender wing obtained from Ref. [10] are adopted as listed in Table 1. The air density is 0.0889 . In the simulation, the default unit of free-stream velocity U is m/s. For analyzing the responses of the system, the wing tip bending and torsional motion are concerned.
Results of response studies
In this part, both the proposed approach and the classical approach are employed to study the nonlinear aeroelastic problems of the slender wing without store and of the wing-pylon-store system. In the study, the parameters of the slender wing are the same as Table 1. The parameters of the pylon-store system are listed in Table 2. The air density is also 0.0889 . In the simulation, the default unit of free-stream velocity U is still m/s, the calculation will be ended if the divergent
Conclusions
In this paper, an alternative simulation approach has been proposed to simplify the calculations for the nonlinear ONERA aerodynamic model. For the proposed approach, the circulatory part of the ONERA model was solved analytically by using Duhamel integral method. By introducing two additional variables, the circulatory part of the ONERA model can be directly expressed by using the mode coordinates of the wing. The aerodynamic elements are only used to determine the nonlinear parts of the ONERA
Acknowledgment
This work is supported by the National Natural Science Foundation of China (No. 11472089).
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