The paper presents an efficient method for analyzing complex periodic unsteady flows within turbomachinery. The method transforms the original initial value problem of unsteady flows to a boundary value problem by taking advantage of temporal periodicity known a priori. This transform allows a pseudotime marching at all physical time instants to be performed simultaneously, for which the method is called the time vector marching (TVM) method. Time direction discretization can be achieved using finite difference schemes. A second-order backward scheme and a ninth-order mixed scheme are investigated. To demonstrate the stability, accuracy, and correctness of the TVM method, the one-dimensional (1-D) advection equation and a quasi-1-D Laval nozzle with time-varying inlet total pressures are used as the test cases. The unsteady flowfield within it was also analyzed using a time-domain dual time-stepping (DTS) method and the harmonic balance (HB) method to obtain the results as references. Further verification was carried out using a transonic compressor rotor with inlet absolute total pressure distortion. The phase-shift boundary condition was applied at geometric periodic boundaries to facilitate the use of a computational domain of one blade passage for a blade row. The time-averaged and instantaneous results of the DTS, HB, and TVM methods and associated time cost were compared and discussed.

本文提出了一种有效的方法来分析涡轮机械内的复杂周期性非定常流动。该方法通过利用先验已知的时间周期性将不稳定流的原始初始值问题转换为边界值问题。此变换允许在所有物理时刻同时执行伪时间行进，对此方法称为时间向量行进（TVM）方法。时间方向离散化可以使用有限差分方案来实现。研究了二阶后向方案和九阶混合方案。为了证明TVM方法的稳定性，准确性和正确性，将一维（1-D）对流方程和准时一维Laval喷嘴随进口总压力的变化作为测试案例。还使用时域双时步（DTS）方法和谐波平衡（HB）方法分析了其中的非恒定流场，以作为参考。使用具有入口绝对总压力畸变的跨音速压缩机转子进行了进一步验证。在几何周期边界处应用相移边界条件，以促进对一个叶片行使用一个叶片通道的计算域。比较和讨论了DTS，HB和TVM方法的时间平均和瞬时结果以及相关的时间成本。在几何周期边界处应用相移边界条件，以促进对一个叶片行使用一个叶片通道的计算域。比较和讨论了DTS，HB和TVM方法的时间平均和瞬时结果以及相关的时间成本。在几何周期边界处应用相移边界条件，以促进对一个叶片行使用一个叶片通道的计算域。比较和讨论了DTS，HB和TVM方法的时间平均和瞬时结果以及相关的时间成本。