Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis

Abstract

Identifying the convective/absolute instability nature of a local base flow requires an analysis of its linear impulse response. One must find the appropriate singularity in the eigenvalue problem with complex frequencies and wavenumbers and prove causality. One way to do so is to show that the appropriate integration contour of this response, a steepest decent path through the relevant singularity, exists. Due to the inherent difficulties of such a proof, one often verifies instead whether this singularity satisfies the collision criterion. In other words, one must show that the branches involved in the formation of this singularity come from distinct halves of the complex wavenumber plane. However, this graphical search is computationally intensive in a single plane and essentially prohibitive in two planes. A significant computational cost reduction can be achieved when root finding procedures are applied instead of graphical ones to search for singularities. They focus on locating these points, with causality being verified graphically a posteriori for a small parametric sample size. The use of root-finding procedures require auxiliary equations, often derived by applying the zero group velocity conditions to the dispersion relation. This relation, in turn, is derived by applying matrix forming to the differential eigenvalue problem and taking the determinant of the resulting system of algebraic equations. Taking the derivative of the dispersion relation with respect to the wavenumbers generates the auxiliary equations. If the algebraic system is decoupled, this derivation is straightforward. However, its computational cost is often prohibitive when the algebraic system is coupled. Other methods exist, but often they can also be too costly and/or not reliable for two wavenumber plane searches. This paper describes an alternative methodology based on sensitivity analysis and adjoints that allow the zero group velocity conditions to be applied directly to the differential eigenvalue problem. In doing so, the direct and auxiliary differential eigenvalue problems can be solved simultaneously using standard shooting methods to directly locate singularities. Auxiliary dispersion relations no longer have to be derived, although it is shown that they are the algebraic equivalent of the auxiliary differential eigenvalue problems obtained by this alternative methodology. Using the latter dramatically reduces computational costs. The search for arbitrary singularities is then not only accelerated in single wavenumber planes but it also becomes viable in two wavenumber planes. Finally, the new method also allows group velocity calculations, greatly facilitating the verification of causality. Several test cases are presented to illustrate the capabilities of this new method.

Linear disturbance growth induced by viscous dissipation in Darcy–Bénard convection with throughflow

“…The reader is referred to recent in depth reviews for more information about absolute instability calculations (Alves et al. 2019; Barletta 2019).…”

“…Identifying the transition to absolute instability can be computationally quite intensive when employing classical techniques, e.g. finding the steepest descent curve or verifying the collision criterion, unless saddle points can be cheaply calculated a priori (Alves et al 2019). In the present case, this can be done by applying the zero group velocity conditions to the dispersion relation, coupling it with auxiliary dispersion relations that can identify saddle points.…”

“…When the disturbance is no longer local, parabolised stability (Herbert 1997) and global stability (Theofilis 2011) concepts can be employed. Adjoint equations (Luchini & Bottaro 2014) also deserve special mention, since they have been connected to absolute instability in both discrete (Lesshafft & Marquet 2010) and continuous (Alves et al 2019) senses. Many of these techniques have been applied to both natural and mixed convection in porous media.…”

Spatial input-output analysis of large-scale structures in actuated turbulent boundary layers

“…We identify upstream and downstream modes in equations ( 7)-( 8) through the eigenshuffle [43] function, which tracks the variation of each eigenvalue numerically based on its continuity with varying parameter 𝜂. This numerical method is selected because analytical tracking is typically challenging; see e.g., Alves et al [44]. For results in this work, we use 60 logarithmically spaced values in the range 𝜂 + ∈ [10 −3 , 10] to approximate 𝜂 → ∞ in equations ( 7)- (8).…”

Onset of Thermal Instabilities in the Plane Poiseuille Flow of Weakly Elastic Fluids: Viscous Dissipation Effects

“…This eigenvalue problem was solved numerically by considering disturbances in the form of longitudinal rolls (k x = 0, k y = k), transverse rolls (k x = k, k y = 0) and oblique rolls (k x = 0, k y = 0) by means of a shooting method (please refer to [14,29] for details on the numerical method).…”