Implicit Harmonic Balance Solver for Transonic Flow with Forced Motions

Abstract

The computation of the aerodynamic forces arising from forced periodic motions is required for the generation of dynamic terms in models for flight simulation. The periodicity can be used to avoid using fully unsteady calculations by using the harmonic balance method. The current paper develops an implicit solver for the harmonic balance equations. The method is tested on two transonic test cases and evaluation is made against the unsteady simulation results. The first caseis for the pitching NACA 0012aerofoil. The second is for forced pitching of the F-5 wing with a wing tip launcher and missile. A reduction in computational time by one order of magnitude compared with the unsteady solver is obtained. Nomenclature A = matrix in frequency domain equation c = chord D = matrix in harmonic balance equation E = transformation matrix between frequency and time domains e = energy F, G, H = convective fluxes I = residual of semidiscrete system I = identity matrix k = reduced frequency nH = number of harmonics p = pressure R = residual vector T = period t = time u, v, w = Cartesian velocity components W = conserved variables � = angle of attack � t = pseudo time step