Fast prediction of transonic aeroelastic stability and limit cycles

The exploitation of computational fluid dynamics for aeroelastic simulations is mainly based on time-domain simulations. There is an intense research effort to overcome the computational cost of this approach. Significant aeroelasticeffectsdrivenbynonlinearaerodynamicsincludethetransonic flutterdipandlimit-cycleoscillations.The paper describes the use of Hopf bifurcation and center manifold theory to compute flutter speeds and limit-cycle responses of wings in transonic flow when the aerodynamics are modeled by the Euler equations. The cost of the calculations is comparable to steady-state calculations based on computational fluid dynamics. The paper describes twomethodsfor findingstabilityboundariesandthenanapproachtoreducingthefull-ordersystemtotwodegreesof freedom in the critical mode. Details of the three methods are given, including the calculation of first, second, and thirdJacobiansandthesolutionofsparselinearsystems.ResultsfortheAGARDwing,asupercriticaltransporttype of wing, and the limit-cycle response of the Goland wing are given. Nomenclature A = Jacobian matrix of R with respect to w B, C = second and third Jacobian operators h = (scalar) increment for finite differences F = quadratic and higher terms in R G = Taylor coefficients of the residual restricted to the critical eigenspace H = Taylor coefficients of the residual restricted to the noncritical eigenspace f = convective flux discretization kij = coefficients in center manifold expansion of y P = right eigenvector of A, P1 � iP2 Q = left eigenvector of A, Q1 � iQ2 qs = constant scaling vector for the augmented system R = residual vector v = vector for the matrix-free product w = vector of unknowns y = part of w in the noncritical eigenspace z = part of w in the critical eigenspace � t = time step � i = sequence of eigenvalues in the inverse power method � = bifurcation parameter (dynamic pressure) ! = frequency of critical eigenvalue or shift for the inverse power method Subscripts