An Adaptive Harmonic Method for Unsteady Quasi-One-Dimensional Periodic Flow

A uniform formulation of linear harmonic method, nonlinear harmonic method and harmonic balance method, referred to as the uniform harmonic method, is first proposed for the quasi-one-dimensional Euler equations; and a modified adaptive technique is employed, by which the harmonic contents at each cell can be automatically augmented or diminished to efficiently capture the local flow details. Then the unsteady flows in a convergent-divergent nozzle are computed and analyzed for a test case with an oscillating shock wave in it. The harmonic contents, computational time and error in pressure are presented and compared for different harmonic interaction options, segment widths and thresholds, from which the adaption setups with excellent computational performance and high-level accuracy are determined. Finally, the adaptive harmonic method is extended to the multiple-perturbation case, which is verified by an example with pressure perturbations of two different fundamental frequencies. Compared to the non-adaptive harmonic balance method, the adaptive harmonic method produces accurate enough solutions with a 75.4% reduction in computational time and a 71.8% save in memory consumption for the single-perturbation case, while the drop rates are 42.0% and 62.8% respectively for the multiple-perturbation case.