Citation
Gómez, F. and Le Clainche Martínez, Soledad and Paredes Garcia, Pedro and Hermanns Navarro, Miguel and Theofilis, Vassilios
(2012).
Four Decades of Studying Global Linear Instability: Progress and Challenges.
"AIAA Journal", v. 50
(n. 12);
pp. 2731-2743.
ISSN 0001-1452.
https://doi.org/10.2514/1.J051527.
Abstract
Global linear instability theory is concerned with the temporal or spatial development of small-amplitude
perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two(and periodic in one)or all three spatial directions.After a brief exposition of the theory,some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton–Krylov time-stepping method into a standard
finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in
flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the
spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability
equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse
matrix treatment, all these problems may now be solved on standard desktop computers