The paper proposes a novel numerical method called coupled time and passage spectral method for an efficient resolution of far upstream wakes in an unsteady analysis of flow field within generic multiple blade rows. The proposed method is a very simple and natural extension of the time spectral form harmonic balance method. By including inter blade phase angle and passage index in a truncated Fourier series, the proposed method is capable of circumventing the limitations of the harmonic balance method in dealing with zero frequency harmonics and time harmonics with same frequency but different inter blade phase angles. Different from the harmonic balance method, which seeks solution at different time instants of the same passage, the coupled spectral method seeks solutions at different passages and time instants. Same as the harmonic balance method, all these solutions are connected via a spectral operator. Coupled time and passage sampling is performed using the modified Gram Schmidt process to choose the time and passage pairs which give the most orthogonal rows of an inverse Fourier transform matrix. The coupled time and passage spectral method requires minimum change to an existing harmonic balance solver. A numerical case study has been provided in the paper to demonstrate the expected capability against the harmonic balance method.

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