Supplemental-frequency harmonic balance: A new approach for modeling aperiodic aerodynamic response

Highlights: • Previous works in solving aperiodic flow problems using harmonic-balance-based methods are reviewed. • Supplemental frequencies are introduced for a nearly-harmonic frequency array to stabilize the pseudo-spectral operator. • Two different methods are proposed to determine the base frequency. • The efficacy of the SF-HB method is demonstrated for a forced-pitching airfoil subjected to two excitation frequencies. • The entire time history of the dynamic response can be resolved through a single run of the SF-HB solver. Presented in this work is a novel and easy-to-implement supplemental-frequency harmonic balance (SF-HB) approach to efficiently compute dynamically aperiodic systems, which can be cumbersome to model using the nominal high-dimensional harmonic balance (HDHB) technique. The stability of the time-spectral operator and the SF-HB solver involving multiple excitation frequencies is ensured by introducing a group of supplemental frequencies based on the fact that aperiodic (or almost aperiodic) response would contain all possible frequencies. Together with the excitation (primary) frequencies, the frequency set forms a nearly-harmonic series leading to a small condition number for the Fourier transformation matrices directly impacting the stability of the HDHB solver. In contrast to the similar work reported in the literature where optimal unequally-spaced sub-time levels are determined through a rather complicated procedure, the current approach simply uses equally-spaced sub-time levels spanning a time period that depends on a predicted base frequency. At convergence, the original excitation frequency modes dominate the solution as desired. This new technique is verified for a forced pitching airfoil in the transonic inviscid flow regime subjected to two excitation frequencies. Results of both periodic and aperiodic cases show that with the help of Fourier interpolation, the entire time history of the dynamic response can be obtained through a single run of the SF-HB solver, for which the efficiency and robustness of the traditional HDHB method is preserved.