Abstract
Abstract
Highlights
The resolvent is a linear operator that governs how harmonic forcing inputs are amplified by the linear dynamics of a system and mapped onto harmonic response outputs
Two decades before it was coined as such, resolvent analysis was first used in the seminal work of Trefethen et al (1993) to study the response of linearly stable flows to deterministic external disturbances, such as those coming from wall roughness, acoustic perturbations, body forces or free-stream turbulence
We show that, using an appropriate inner product, the resolvent of the matrix of Dynamic mode decomposition (DMD) eigenvalues is the resolvent of the system projected onto the span of the DMD eigenvectors, which are subsequently used to synthesize the resolvent modes in physical coordinates
Summary
The resolvent is a linear operator that governs how harmonic forcing inputs are amplified by the linear dynamics of a system and mapped onto harmonic response outputs. During the 1990s, non-modal stability theory emerged to provide a more complete picture of the linear perturbation dynamics for fluid flows using an initial-value problem formulation, as a complement to the eigenproblem from classic hydrodynamic stability theory (Schmid & Henningson 2001; Schmid 2007; Schmid & Brandt 2014) This formulation allowed the study of the response of fluid flows to initial conditions (Gustavsson 1991; Butler & Farrell 1992; Farrell & Ioannou 1993a; Reddy & Henningson 1993; Hwang & Cossu 2010b; Herrmann, Calderón-Muñoz & Soto 2018b), stochastic inputs (Farrell & Ioannou 1993b; Del Alamo & Jimenez 2006; Hwang & Cossu 2010a,b) and harmonic forcing (Jovanovic & Bamieh 2005; Hwang & Cossu 2010a,b; Herrmann et al 2018b). Gómez & Blackburn (2017) developed a promising data-driven method to identify sensitive spatial locations in unsteady flows and demonstrated its use for the design of passive flow control strategies This approach requires using time-resolved snapshots of both, the state (velocity field) and of the nonlinear forcing (Reynolds stress divergence).
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