Four Decades of Studying Global Linear Instability: Progress and Challenges

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.

On linear instability mechanisms in incompressible open cavity flow

“…The work of Rowley, Colonius & Basu (2002) provided further insight regarding the onset of shear-layer oscillations from a steady flow and their nonlinear interactions, developing a criterion to predict the onset of disturbances from the steady flow. Later, Theofilis & Colonius (2003 revisited the open cavity and, applying the residuals algorithm (Theofilis 2000&;Gomez et al 2012;Gomez, Gomez & Theofilis 2014) recovered the same Tollmien-Schlichting eigenmode as well as an acoustic mode connecting hydrodynamic perturbations inside the cavity with pressure fluctuations having their origin at the downstream cavity corner. Recently, Yamouni, Sipp & Jacquin (2013) have given a detailed description of the effect of compressibility on the interaction of the acoustic feedback and resonance using global instability analysis in a two-dimensional cavity at a particular set of flow conditions and one Reynolds number.…”

Time-dependent modeling of oscillatory instability of three-dimensional natural convection of air in a laterally heated cubic box

“…As is known from well-established numerical solutions obtained for instability of convective air flow in a two-dimensional laterally heated square cavities, which is a TwoGlobal problem according to [ 4,5], the critical Grashof number for perfectly thermally conducting horizontal boundaries is of the order 10 6 , while for perfectly thermally insulated borders it becomes larger than 10 8 [ 2,20,22]. This makes the problem with perfectly thermally conducting boundaries computationally easier, so that 3D results obtained in [ 14,15] compare rather well with the experimental measurements of [ 13].…”

Spatial Stability Analysis of Subsonic Jets Modified for Low-Frequency Noise Reduction

“…The ratio of nonzero elements to total number of elements in the A and B matrices [see Eq. (15)] was found to be only 0.17444 and 0.0008607%, respectively. During the matrix factorization phase, even with a total processor core count of around 10 5 , the MUMPS solver was found to require an excessive amount of memory, which was not physically available on the parallel machine being used.…”

“…Spectral methods are applied for the spatial discretization in radial and azimuthal directions. Reviews of these and other related stability analysis techniques, alternative numerical solution methods, and applications to several other problems can be found in Theofilis [13], González et al [14], Gómez et al [15], and Juniper et al [16].…”