^{1}

^{2}

^{1}

^{2}

The objective of this research is to propose a new reduced-order modeling method. This approach is based on fluid eigenmodes and body vortices without using static correction. The vortex lattice method (VLM) is used to analyze unsteady flows over two-dimensional airfoil and three-dimensional wing. Eigenanalysis and reduced-order modeling are performed using a conventional method with static correction and an unconventional one without the static correction. Numerical examples are proposed to demonstrate the performance of the present method. The results show that the new method can be considered an alternative way to perform the reduced-order models of unsteady flow.

Reduced-order modeling (ROM) with a static correction technique is an efficient technique which has been used for analyzing unsteady incompressible aerodynamic flows [

Another method for constructing ROMs is proper orthogonal decomposition (POD), which has been used for investigating unsteady aerodynamic and aeroelastic models [

The purpose of this research is to derive an alternative ROM based on a vortex lattice method (VLM) in which the eigenvalue problem is defined only based on the unknown body vortices. The new ROM does not need the static correction as with the previous formulation, which is based on the wake vortices. The eigenvalues of the new eigensystem are nonzero or have no quasistatic eigenmodes. Eigenanalysis results show that the eigenvalues of the proposed method are equal to the corresponding nonzero values of the previous studies. Furthermore, the reduced-order models are constructed for steady flows over a two-dimensional airfoil and a three-dimensional wing. The results show that the present methods can accurately analyze unsteady flows and it is an alternative reduced-order model.

The vortex lattice method for unsteady flow (termed as unsteady vortex lattice method (UVLM)) is a relation of downwash and vortex of unsteady flow [

The dynamic behavior of the fluid as the sum of the individual eigenmodes can be expressed as

The reduced-order model can be carried out by retaining only a few

Previous work [

In this section, the authors proposed another way to remove zero eigenvalues by defining a new eigenvalue problem. From (

In this section, we validate unsteady vortex lattice model with two unsteady flow models. The first model is the lift acting on the airfoil due to a step change in airfoil downwash (Wagner problem). The result is shown in Figure

Lift acting on the airfoil due to step change in airfoil downwash.

Unsteady lift due to plunging motion of an isolated airfoil.

This section proposed the eigenanalysis of the conventional method, the method based on

Eigenvalues for two-dimensional airfoil.

Eigenvalues of vortex lattice model of unsteady flow about the two dimensional airfoil and wake are modeled with 100 vortex elements are shown in Figure

Eigenvalues for two-dimensional airfoil when increasing the number of element panels.

Eigenvalues of vortex lattice model of unsteady flow about the three-dimensional wing are plotted in Figure

Eigenvalues of vortex lattice model of unsteady flow about the three-dimensional wing— □ conventional method, + method based on

In this section, we used the eigenvalues, which have been studied in the previous section to construct a reduced-order aerodynamic model. The present ROM is called new ROM based on

Unsteady lift for two-dimensional airfoil predicted using ROM based on

Lift acting on the airfoil due to step change in airfoil downwash predicted using ROM based on

In the present work, a reduced-order model for unsteady flows is developed without a static correction technique. This method is proposed to construct the eigensystem based on body vortices, and the rank of the eigensystem is lower than the corresponding eigensystem of the conventional method and the method based on wake vortices. Furthermore, the eigensystem does not contain any zero eigenvalues, so the static correction is not needed. The results show that the proposed method can produce satisfactory results like the conventional method and the method based on wake vortices. For the future work, we plan to apply a pseudoinverse operator to solve nonsquare matrix

The authors are grateful for the financial support provided by Chiangrai College.