Matrix-free TriGlobal adjoint stability analysis of compressible Navier–Stokes equations

Abstract

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.