Localized resolvent-mode bases for turbulence statistics
Ethan Eichberger
∗
Liam Heidt
†
and Tim Colonius
‡
California Institute of Technology, Pasadena, CA USA
Modes from global resolvent analyses have been shown to accurately model the frequencies
and spatial structure of the dominant coherent structures in several turbulent flows. However,
resolvent-mode forcing models must be developed to predict the amplitude of the structures
or other flow statistics, including the radiated noise. The present research aims to apply
data-driven approaches to learn forcing coefficients from lower-order statistics available from
Reynolds-averaged Navier-Stokes (RANS) predictions. As a first step towards this goal, we
present a novel localized resolvent framework that reconstructs global quantities at low rank
through spatially restricting the resolvent forcing and response domains. To illustrate the
flexibility and robustness of the proposed framework, we initially utilize localized resolvent
modes to reconstruct the spectral proper orthogonal decomposition (SPOD) modes of an
isothermal Mach 0.4 jet at
푹풆
=
450
,
000
. The results showcase the flexibility localized resolvent
modes provide in the construction of global SPOD, while using 10 or fewer total localized
modes total across
푺풕
=
[
0
.
05
,
1
.
00
]
. Furthermore, we employ localized resolvent modes to
reconstruct second-order statistics, comparing their performance with that of global modes.
At low reconstruction error, it is shown that about twice as many global modes are needed to
achieve comparable errors.
I. Introduction
Coherent structures play a crucial role in the dynamics and acoustics of turbulent jets. Their identification from
data using spectral proper orthogonal decomposition (SPOD) [
1
] and their modeling using resolvent analysis [
2
] have
improved our understanding of these flows [3–7].
Resolvent analysis identifies the optimal forcing and response modes with the largest gain based on a linearization of
the governing equations about the turbulent mean flow [
2
]. The framework bears resemblance to the acoustic analogy
[
8
], and more specifically its generalization [
9
], wherein the predicted flow or acoustic field are governed by a linear
operator driven by nonlinear products of fluctuations, regarded as equivalent sources of the resulting motions/waves.
The resolvent differs from the traditional approach by replacing the Green’s function solution with a decomposition
of the operator into a set of orthonormal modes ordered by the
gain
between the forcing and response. Compare to
traditional acoustic analogy, this potentially lowers the modeling burden to the forcing coefficients of a few modes. This
approach has already enjoyed some success in jet noise modeling[7, 10–13].
Resolvent modes are likewise capable of predicting the structure of the near-field coherent vortical motions [
2
–
4
,
14
],
but, as for the far field, models for the sources are required to predict their amplitudes. While several data-driven
approaches have been proposed to estimate the sources from data [
15
–
17
], fully predictive frameworks have not yet been
established. In this research, we aim to develop resolvent forcing models that are informed by mean-flow quantities that
can reliably be predicted in RANS, ultimately in order to develop RANS-based jet-noise models. We are particularly
interested in resolvent modes that simultaneous capture near and far-field behaviors, so that solely near-field quantities
can be used as inputs to determine their magnitude.
However, a challenge stems from the nature of global modes, which span the entire spatial flow. In many flows,
the coherent structures differ in structure and frequency in different regions of the flow. For example, a turbulent
jet has a spatial domain that includes the near nozzle flow, the annular shear layer, the end of the potential core, the
fully-developed jet (with an increasingly thick shear layer), and the near and far acoustic fields. By focusing solely on
the global jet behavior, these modes are, at low rank, likely to be only weakly correlated with corresponding statistics
available from RANS.
To address this challenge, in this paper we develop what we term as a
localized resolvent basis
that computes modes
whose forcing and observation regions are broken down into a set of possibly overlapping spatial regions, while retaining
∗
PhD Student, Mechanical and Civil Engineering, AIAA Student Member
†
PhD Candidate, Graduate Aerospace Laboratories of the California Institute of Technology, AIAA Student Member
‡
Frank and Ora Lee Marble Professor of Mechanical Engineering, Mechanical and Civil Engineering, Associate Fellow AIAA
1
Downloaded by California Institute of Technology on September 23, 2024 | http://arc.aiaa.org | DOI: 10.2514/6.2024-3205
30th AIAA/CEAS Aeroacoustics Conference (2024)
June 4-7, 2024, Rome, Italy
10.2514/6.2024-3205
Copyright © 2024 by Ethan Eichberger, Liam Heidt, Tim Colonius. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Aeroacoustics Conferences