A numerical method for TriGlobal (i.e., fully three-dimensional) adjoint stability analysis for compressible flows was developed and is presented in this paper. The developed method solves the adjoint stability problem using a matrix-free method based on Krylov-Schur method and a time-stepping approach. Because of the low memory (RAM) requirement of the matrix-free approach, the developed method can analyze fully three-dimensional flows that are difficult to analyze with conventional matrix-based methods. To perform time-stepping on the adjoint variables, the adjoint equations, including appropriate boundary conditions for the compressible Navier–Stokes equations, were derived. The equations were discretized using a finite compact difference method, and time integration was conducted using the three-step third-order Runge–Kutta method. A flow field around a two-dimensional square cylinder and a cubic cavity flow were analyzed using the developed method, and it was confirmed that the method reproduces the dominant instabilities reported in the literature. In addition, in the square cylinder flow analysis, the receptivity and sensitivity regions of the secondary wake mode, which corresponds to far wake instability, were clarified. Finally, the TriGlobal direct and adjoint stabilities of compressible flows over a finite width cavity were analyzed for the first time, and it was proven that large-scale adjoint stability analysis can be performed with the developed method. The results also show that instability phenomena similar to those obtained with BiGlobal stability analysis appear, but sidewall effects exist.

提出了一种用于可压缩流的TriGlobal（即全三维）伴随稳定性分析的数值方法，并在本文中进行了介绍。所开发的方法使用基于Krylov-Schur方法的无矩阵方法和时间步长方法解决了伴随稳定性问题。由于无矩阵方法对内存（RAM）的要求较低，因此开发的方法可以完全分析三维流，而传统的基于矩阵的方法很难分析这种三维流。为了对伴随变量执行时间步长，得出了伴随方程，包括可压缩Navier–Stokes方程的适当边界条件。使用有限紧致差分法将方程离散化，并使用三步三阶Runge-Kutta方法进行时间积分。使用开发的方法分析了二维方形圆柱体周围的流场和立方腔流，并证实该方法再现了文献中报道的主要不稳定性。另外，在方筒流动分析中，阐明了对应于远尾不稳定性的二次唤醒模式的接受度和灵敏度区域。最后，首次分析了有限宽度空腔上可压缩流动的TriGlobal直接和伴随稳定性，并证明了所开发的方法可以进行大规模伴随稳定性分析。结果还表明，出现了类似于通过BiGlobal稳定性分析获得的不稳定性现象，但是存在侧壁效应。