关键词: 稳定性分析, 特征值方法, 压气机, 旋转失速, 模态波

Abstract: Rotating stall limits the stable operating range of compressors; thus a deep understanding and accurate prediction of this phenomenon is key to stall prediction and control. Although there are a variety of models to explain the flow insta-bility mechanism of the stall onset, all are based on certain simplification of the compressor geometry and flow, and rarely consider the three-dimensional effect of the actual compressor and its complex flow. Therefore, great challenges are faced when directly applying these models to the stall onset prediction of actual compressors. Meanwhile, despite the progress in experimental measurements and flow simulations, most experiments and numerical simulations are mostly phenomenological research, and did not reveal the root cause of compressor flow instability. Moreover, due to the complexity of three-dimensional flow measurements and the high cost of unsteady simulations, most stall studies are conducted only for isolated working conditions of a particular compressor, as a systematic parametric study to iden-tify the key influencing factors is too costly. In order to circumvent the shortcomings of both measurements and un-steady simulations, a global stability analysis method based on the efficient eigenvalue solution of the three-dimensional flow governing equation is proposed. On the one hand, this method can obtain the spatial resolution that is difficult to achieve by experimental measurement, and on the other hand, it can obtain the same rich information of the three-dimensional flow perturbation development at a cost two to three orders smaller than the unsteady simulation, which provides an efficient analysis tool for accurately predicting the stall onset, exploring the flow destabilization mechanism, and systematically studying the key influencing parameters of flow instability. In this paper, for a typical transonic compressor annular cascade, the Newton-Krylov fully implicit time propulsion method is first used to solve the Reynolds-averaged Navier--Stokes (RANS) equations to obtain steady-state flow solutions near the stall condition. Then, the accurate Jacobian matrix obtained for the converged steady state is used to compute relevant eigenvalues and eigenvectors through efficient iterative algorithms. The compressor stall boundary can be quickly determined based on the eigenvalues, without resorting to computationally expensive unsteady simulations. Finally, the unstable modes obtained are studied in detail, and it is found that the rotating stall modes of the compressor are highly correlated with the phenomenon of shock boundary layer interaction. At the same time, the accuracy of the calculated modes is veri-fied via comparison with unsteady simulation results. In addition, based on the analysis of eigenmodes, for the first time, an explanation is given for the positive correlation between the circumferential wavenumber of stall modes and their rotation speed, which has been frequently seen in experiments but was never satisfactorily explained. The computa-tional cost of the efficient eigenvalue analysis method developed in this paper is only 28% of that for computing one steady speedline, and it is 155 times faster than the unsteady calculation. Therefore the proposed eigenvalue-based stability analysis method provides an important research tool for accurate and rapid prediction and mechanism re-search of rotational stall of three-dimensional highly-loaded compressor.

Key words: Global stability analysis, eigenvalue method, compressor, rotating stall, modal wave