Nonlinear interactions in turbulent jets
Akhil Nekkanti
∗
California Institute of Technology, Pasadena, CA USA
Ethan Pickering
Bayer Crop Science, Boston, MA, USA
Oliver T. Schmidt
†
University of California San Diego, La Jolla, CA, USA
Tim Colonius
‡
California Institute of Technology, Pasadena, CA, USA
Bispectral mode decomposition is used to investigate triadic interactions within a Mach
0.4 turbulent jet. We explore its potential to identify dominant triadic interactions and
their associated spatial structures in an unforced turbulent jet. The bispectral measure is
broadband in frequency for each azimuthal wavenumber triad. The [1,1,2] and [0,0,0] azimuthal
wavenumber triads are dominant, emphasizing the importance of the self-interactions of the
helical and axisymmetric components. Bispectral modes reveal that streaky structures are
fed by the interaction of a Kelvin-Helmholtz wavepacket with its conjugate. Streaks are also
observed in other frequency interactions, occurring in regions where the structures of these
frequencies are spatially active. Furthermore, integral interaction maps and nonlinear transfer
terms are computed to determine the direction of energy transfer and to pinpoint the spatial
regions where nonlinearity is most active. As the shear layer develops, small scales interact
nonlinearly, transferring energy to larger scales. Moving downstream, near the potential core
closure, larger scales become more active, resulting in a forward energy cascade.
I. Introduction
In turbulent jets, coherent structures are the primary sources of aft-angle noise [
1
]. The pioneering work by Crow
and Champagne
[2]
identified the presence of large-scale coherent structures in turbulent jets. Subsequent researchers
[
3
–
7
] have modeled these coherent structures as linear stability solutions of the turbulent mean flow. Most studies,
until now, have focused on characterizing these structures in terms of their statistical importance and have linked them
to linear mechanisms [
8
–
10
]. In this work, our emphasis is on identifying the nonlinear triadic interactions and their
associated spatial structures.
Quadratic nonlinearities arise from the convective term of the Navier-stokes equation, which leads to three-wave
coupling or triadic interactions. A triad is a set of three wavenumbers (or frequencies) that form a triangle, i.e., they
satisfy the following condition
k
̆
p
̆
q
“
0
,
(1)
푓
1
̆
푓
2
̆
푓
3
“
0
.
(2)
These triadic interactions play a vital role in the energy cascade of turbulent flows [
11
–
13
]. Researchers have employed
bispectral analysis to study these interactions [
14
–
17
]. The bispectrum, an extension of the power spectrum to third
order, is calculated through a double Fourier transform of third-order moments. It measures the quadratic phase coupling
only at a single spatial point, thereby characterizing triadic interaction locally. However, our objective extends beyond
this local analysis; we aim to identify coherent structures associated with triadic interactions. To this end, we employ
the bispectral mode decomposition (BMD) proposed by Schmidt
[18]
. BMD identifies the most dominant triads and
their corresponding coherent structures by maximizing the spatially integrated bispectrum. BMD has been used to
∗
Postdoctoral Scholar, AIAA Member
†
Associate Professor, Department of Mechanical and Aerospace Engineering, Senior Member AIAA.
‡
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: 10.2514/6.2024-3414
30th AIAA/CEAS Aeroacoustics Conference (2024)
June 4-7, 2024, Rome, Italy
10.2514/6.2024-3414
Copyright © 2024 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Aeroacoustics Conferences
investigate the triadic interactions in various flow configurations, such as laminar-turbulent transition on a flat plate
[19, 20], forced jets [21–23], swirling flows [24, 25], bluff body wakes [26], and wake of an airfoil [27].
Nonlinear dynamics in jets have predominantly been studied in two contexts: transitional flows and harmonically
forced jets. Transitional flows have been the focus of studies examining the nonlinear interaction between the fundamental
and subharmonic wave [
28
–
30
]. Meanwhile, studies on forced jets have examined nonlinear effects on mean flow
deformation [
31
,
32
], azimuthal symmetry breaking [
33
,
34
] and the formation of periodic and chaotic attractors [
35
,
36
].
Here, our focus is exclusively on non-resonant flows, intending to shed light on the nonlinear wave coupling that is still
present, albeit in lower magnitude. Furthermore, we will analyze the direction of energy transfer between different
azimuthal wavenumbers. The paper is organized as follows. In section II, the BMD methodology is discussed. Results
are presented in section III, and the paper is summarized in section IV.
II. Cross-Bispectral mode decomposition (C-BMD)
We employ the cross-bispectral mode decomposition, recently developed by [
18
], to identify the spatially coherent
structures arising from the triadic interactions in turbulent flows. Here, we provide a brief overview of the method. The
reader is referred to Schmidt [18] for further details of the derivation and mathematical properties of the method.
Given a statistically stationary flow field, let
q
푖
“
q
p
푡
푖
q
denote the mean subtracted snapshots, where
푖
“
1
,
2
,
̈ ̈ ̈
푛
푡
are
푛
푡
number of snapshots. For spectral estimation, using the Welch’s approach the dataset is first segmented into
푛
blk
overlapping blocks with
푛
fft
snapshots in each block. The neighbouring blocks overlap by
푛
ovlp
snapshots with
푛
ovlp
“
푛
fft
{
2
. The
푛
blk
blocks are then Fourier transformed in time and all Fourier realizations of the
푙
-th frequency,
q
p
푗
q
푙
, are arranged in a matrix,
ˆ
Q
푙
“
”
ˆ
q
p
1
q
푙
,
ˆ
q
p
2
q
푙
,
̈ ̈ ̈
,
ˆ
q
p
푛
blk
q
푙
ı
.
(3)
The auto-bispectral matrix is then computed as
B
“
1
푛
blk
ˆ
Q
퐻
푘
̋
푙
W
ˆ
Q
푘
`
푙
,
(4)
where
ˆ
Q
퐻
푘
̋
푙
“
ˆ
Q
̊
푘
̋
ˆ
Q
̊
푙
and
W
is the diagonal matrix containing the spatial quadrature weights. The auto-bispectral
density matrix measures the interactions between different frequenices at the same azimuhtal wavenumber. To estimate
the interactions between the azimuthal wavenumber triad, [
푚
1
,
푚
2
,
푚
3
], where
푚
1
`
푚
2
“
푚
3
, we construct the
cross-bispectral matrix
B
푐
“
1
푛
blk
`
ˆ
Q
̊
푘
̋
ˆ
R
̊
푙
̆
W
ˆ
S
푘
`
푙
.
(5)
Here,
ˆ
Q
푘
,
ˆ
R
푙
,
ˆ
S
푘
`
푙
, comprises of all the Fourier realizations at the
푘
-th frequency of the azimuthal wavenumber
푚
1
,
the
푙
-th frequency of the azimuthal wavenumber
푚
2
, and the
p
푘
`
푙
q
-th frequency of the azimuthal wavenumber
푚
3
,
respectively. Owing to the non-Hermitian nature of the bispectral matrix, the optimal expansion coefficients,
a
1
are
obtained by maximising the absolute value of the Rayleigh quotient of
B
푐
a
1
“
arg max
}
a
}“
1
ˇ
ˇ
ˇ
ˇ
a
̊
B
푐
a
a
̊
a
ˇ
ˇ
ˇ
ˇ
.
(6)
The complex mode bispectrum is then obtained as
휆
1
p
푓
푘
, 푓
푙
q “
ˇ
ˇ
ˇ
ˇ
a
̊
1
B
푐
a
1
a
̊
1
a
1
ˇ
ˇ
ˇ
ˇ
.
(7)
Finally, the leading-order bispectral modes and the cross-frequency fields are recovered as
흓
p
1
q
푘
`
푙
“
ˆ
S
푘
`
푙
a
1
,
and
(8)
흓
p
1
q
푘
̋
푙
“
`
ˆ
Q
푘
̋
ˆ
R
푙
̆
a
1
,
(9)
respectively. By construction, the bispectral modes and cross-frequency fields have the same set of expansion coefficients.
This explicitly ensures the causal relation between the resonant frequency triad, (
푓
푘
,
푓
푙
,
푓
푘
`
푓
푙
), where
ˆ
Q
푘
̋
ˆ
R
푙
is the
2
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