Aeroelastic Solutions Using the Nonlinear Frequency-Domain Method

Three-Dimensional Aeroelastic Solutions via the Nonlinear Frequency-Domain Method

“…Accordingly, the accuracy of the solution is dependent on the number of harmonics used as well as the level of unsteadiness in the flow. Further work demonstrated thoroughly the increased efficiency and excellent accuracy provided by frequency-domain methods in comparison with typical time-accurate approaches [43,51,53,63]. In addition to the reduced number of time instances, the computation of the initial transient flow behavior that is usually necessary in a typical time-accurate framework is discarded due to the direct convergence of frequency-domain methods to the final periodic solution.…”

“…For generality, the equations for the temporal discretization detailed herein are an extension of those demonstrated by Kachra and Nadarajah [43], and yield a fully nonlinear structural solver. Assuming that both the flow solution and the structure behave in a periodic fashion, and considering that the mass and stiffness matrices are constant in time, the displacement and load vectors of equation (3.2) can be expressed using their DFT representation as 9) where N s is the number of harmonics employed in the discretization of the structural equations of motion, and is not necessarily equal to N , the number of harmonics used for the flow solution.…”

“…Therefore, this method imposes a convergence study over the employed number of mode shapes, which may become tedious when added to the traditional spatial and temporal accuracy studies. Kachra and Nadarajah [43] first coupled an NLFD flow solver to a fully nonlinear two-degree-of-freedom structural solver using multiple structural harmonics, in order to assess the aeroelastic behavior of a two-dimensional airfoil. The structural displacements and aerodynamic loads were decomposed via Fourier transforms between each period of the flow solution, and a 2 × 2 system of structural equations was obtained and solved for each harmonic separately.…”

“…The two-dimensional flow solver employed in section 6.1 is the single-block single-processor code used in the work of Kachra and Nadarajah [43]. The three-dimensional implementation is based on the work of Nadarajah et al [65] and Nadarajah and Jameson [64], and consists in a parallel multiblock finite-volume NLFD flow solver.…”

“…As the main advantage of the NLFD method is its direct convergence to a periodic steady-state solution, the fluid and structural solutions cannot be coupled at each physical time step, and are instead coupled for an entire period at once. Indeed, the coupling method employed in this work is based on the work of Kachra and Nadarajah [43]. The flow and structural solvers are coupled every N mg multigrid cycles, where N mg is a user-defined parameter chosen such that the convergence of the flow solution is satisfactory between each coupling.…”