# Bulletin of the American Physical Society

# 76th Annual Meeting of the Division of Fluid Dynamics

## Sunday–Tuesday, November 19–21, 2023; Washington, DC

### Session T31: NLD Coherent Structures II

4:25 PM–6:09 PM,
Monday, November 20, 2023

Room: 156

Chair: Sutanu Sarkar, University of California, San Diego

### Abstract: T31.00008 : A time-domain preconditioner for the resolvent and harmonic resolvent analyses

5:56 PM–6:09 PM

#### Presenter:

Daniel J Bodony

(University of Illinois at Urbana-Champaign)

#### Authors:

Alberto Padovan

(University of Illinois at Urbana-Champaign)

Ricardo Frantz

(Arts et Métiers Institute of Technology)

Jean-Christophe Loiseau

(Arts et Métiers Institute of Technology)

Daniel J Bodony

(University of Illinois at Urbana-Champaign)

*post-transient*harmonic outputs. In practice, this involves solving large (or extremely large, in the case of the harmonic resolvent) frequency-domain algebraic systems of equations. Furthermore, solving these equations in the frequency domain usually requires computational functionality that is not readily avilable in in-house or open-source time-stepping CFD codes.

We begin to address these issues by observing that the post-transient

*T*-periodic response to a

*T*-periodic forcing input can be computed in the time domain by integrating the Navier-Stokes equations far enough into the future until transients have decayed. (This is a well-known fact.) In order to ``skip the transients'' and make the computation feasible, it is necessary to initialize the time-stepper with an appropriate initial condition

*q(0)*that satisfies

*q(0) = q(T)*, where

*q(T)*is the solution at time

*T*. This initial condition can be computed using Newton's method, which we precondition using a novel preconditioner based on a truncated eigendecomposition of the state transition matrix. We use the NekStab package (based on the open-source CFD solver Nek5000) to demonstrate that this approach has the potential to significantly speed up the computations required to perform the resolvent and harmonic resolvent analyses.

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