Hopf Bifurcation Calculations for a Symmetric Airfoil in Transonic Flow

Abstract

The application of a sparse matrix solver for the direct calculation of Hopf bifurcation points arising for an airfoil moving in pitch and plunge in a transonic flow is considered. The iteration scheme for solving the Hopf equations is based on a modified Newton’s method. Direct solution of the linear system for the updates has previously been restrictive for application of the method, and the sparse solver overcomes this limitation. Results of experiments with the approximation to the Jacobian matrix driving the iteration to convergence are presented. Finally, it is shown that an entire flutter boundary for the NACA0012 airfoil can be traced out in a time comparable to that required for a small number of time-response calculations. I. Introduction C OMPUTATIONAL fluid dynamics (CFD) has matured to the point where it is being applied to complex problems in external aerodynamics. Aeroelastic analysis relies on high-fidelity predictions of aerodynamics, particularly for phenomena associated with shock motions or separation. These two observations have motivated the development of CFD-based aeroelastic simulation, a field now being called computational aeroelasticity. Developments in computational aeroelasticity have mainly been focused on time-marching calculations, where the temporal response of a system to an initial perturbation is calculated to determine growth or decay, and from this to infer stability. This type of simulation has developed significantly in the past decade, with efforts concentrating on mesh movement, load and displacement transfer between the aerodynamic and structural grids, 1 and sequencing of solutions. 2,3 Recent and impressive example calculations have been made for complete aircraft configurations. 4,5 The time-marching method will remain a powerful tool in computational aeroelasticity because of its generality. However, the cost of these calculations motivates attempts to find quicker ways of evaluating stability while still retaining the detailed aerodynamic predictions given by CFD. One way of doing this is to boil down the CFD into a reduced-order model that still retains the essence of the aerodynamics. Various approaches have been proposed, with an expansion of the flowfield in a truncated series of modes derived from proper orthogonal decomposition currently receiving much attention. 6 A second approach proposed by Morton and Beran from the U.S. Air Force is to use dynamic systems theory to characterize the nature of the aeroelastic instability and then to use this additional information to concentrate the use of the CFD. Aeroelastic instabilities that are commonly termed flutter are of the Hopf type, where an eigenvalue of the system Jacobian matrix crosses the imaginary axis at the flutter point. A model problem was used to evaluate the approach 7 in which the main difficulties associated with the method (calculation of the Jacobian matrix, solution of the augmented system by Newton’s method, solution of a large sparse linear system) were considered. The method was applied to an aeroelastic system consisting of an airfoil moving in pitch and plunge in Ref. 8. The