12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

TRIADIC NONLINEAR INTERACTIONS AND ACOUSTICS OF FORCED
VERSUS UNFORCED TURBULENT JETS

Akhil Nekkanti
Igor Maia
Department of Mechanical and Aerospace Engineering Département Fluides, Thermique, Combustion
University of California San Diego
Institut PPRIME, CNRS – Université de Poitiers
CA 92093, USA
86036 Poitiers, France
aknekkan@eng.ucsd.edu
igor.albuquerque.maia@univ-poitiers.fr

Peter JordanDépartement Fluides, Thermique, Combustion
Institut PPRIME, CNRS – Université de Poitiers
86036 Poitiers, France
peter.jordan@univ-poitiers.fr

Liam Heidt
Division of Engineering and Applied Science
California Institute of Technology
Pasadena, CA 91125, USA
lheidt@caltech.edu

Tim Colonius
Oliver T. Schmidt
Division of Engineering and Applied Science Department of Mechanical and Aerospace Engineering
California Institute of Technology
University of California San Diego
Pasadena, CA 91125, USA
CA 92093, USA
colonius@caltech.edu
oschmidt@eng.ucsd.edu

Abstract

2020, 2021b, 2022). Once a strategy is designed to control
the elevated wavepackets, it can be applied to a more realistic
scenario of an unforced jet to reduce jet noise.

We explore the potential of bispectral mode decomposition (BMD) for physical discovery in jet flows. BMD is amodal decomposition that is tailored to the extraction of flow
structures involved in triadic interactions. Large-eddy simulations (LES) of turbulent forced and unforced round jets
at Re = 50, 000 and M j = 0.4 are conducted and validatedwith the companion experiment at the Institut Pprime. The
comparative BMD analysis reveals that triadic interactions inboth forced and unforced jets are most prevalent in a welldefined region near and downstream of the closure of thepotential core. A BMD analysis of far-field pressure sheds
light on the previously reported observation that differenceinteractions are more efficient radiators of jet noise than thesum-interactions.

Following Crow & Champagne (1971), harmonicallyforced jets has been the subject of numerous studies. The
actuation of the jet has varied over a wide range of frequencies and forcing amplitudes. Most of these studies have focussed on two mechanisms that lead to the formation of coherent structures. One is the shear-layer mode or the KelvinHelmholtz instability, the frequency at which the shear layerrolls up into vortices. The other is the preferred mode or thejet column mode (Hussain & Zaman, 1981), characterized by
the frequency at which the jet exhibits the greatest response toexternal forcing. At low amplitudes of forcing, the jets exhibit
a linear response. As the forcing amplitude increases, it triggers nonlinear interactions in jets. High levels of forcing andthe resulting nonlinear interactions of coherent structures haverarely been explored. Raman & Rice (1991); Husain & Hussain (1995); Broze & Hussain (1994); Shaabani-Ardali et al.
(2019) have investigated the nonlinear response to high amplitude forcing in jets, and most have focussed on initiallylaminar jets. In this work, nonlinear interactions in unforced
and forced jets are examined and compared for turbulent jets.
Here, we perform LES of unforced and forced turbulent subsonic jets at a Reynolds number, Re = ρ jU j D/µ j = 50, 000,
and Mach number, M = U j /c∞ = 0.4, where ρ is the density,
U the mean flow velocity, µ the dynamic viscosity, c the speedof sound, and the subscripts j and ∞ denote the jet and freestream conditions. We validate our simulations with the experiments performed at Institut Pprime. Following the experiments, the jets are forced harmonically at a non-dimensionalfrequency of St = f D/U j = 0.4, near the nozzle lip.

Introduction
The reduction of jet noise is an important objective for theaviation community. The pioneering work by Crow & Champagne (1971) identified the presence of large-scale coherent
structures in turbulent jets, and these coherent structures or
wavepackets are the prime sources of aft-angle noise (Jordan& Colonius, 2013). A potential strategy to reduce jet noise is toperform closed-loop control that attenuates the acoustic radiation from the wavepackets. With this objective in mind, first,
we harmonically force the jet to gain an understanding of thealternation of dynamics in forced jets. The forcing increases
the amplitude of the wavepackets above the background turbulence. Recently, the experimental closed-loop control of
turbulent jets has demonstrated the attenuation of amplified
wavepackets over a broad range of frequencies (Maia et al.,1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Bispectral mode decomposition

by a carborundum strip located 2.5D upstream from the nozzle
exit. The turbulent jet is forced by eight loudspeakers that are
equally distributed around the nozzle lipline (r/D = 0.5). The
loudspeakers generate synthetic jets through an annular gap ofwidth 0.01D. For further details of the experimental setup, thereader is referred to Maia et al. (2020, 2021b, 2022).
We perform large-eddy simulations of subsonic jets using the compressible flow solver “Charles” developed at Cascade Technologies Brès et al. (2017, 2018). Charles solves thespatially filtered compressible Navier–Stokes equations on unstructured grids using a density-based finite-volume method.
The LES combines the Vreman sub-grid model (Vreman,2004) with the wall-model by Bordat and Larsson (Bodart &Larsson, 2012; Kawai & Larsson, 2012). The reader is referred
to Brès et al. (2017, 2018) for further details on the numericalmethod and validation on jet flows.
The validation of the LES case for the present case follows the previous studies. In particular, the mesh used by Brèset al. (2018) is modified to accommodate the new nozzle geometry and refined in the vicinity of the synthetic jet actuators.
The total grid size is 16.6 million control volumes. The LES
is conducted for the experimental Reynolds number, whereas
the Mach number is artificially increased to M j = 0.4. Effectsof compressibility are still small in this regime and the verysmall time steps associated with the incompressible limit are
avoided.
The LES results for the unforced jet are compared to theexperiments in figure 1. The mean and RMS streamwise velocities on the centerline and lipline are reported in figure 1(a,c)
and 1(b,d), respectively. In figure 1(a), the mean streamwise
velocities along the centerline of the experiment and the simulation are almost indistinguishable within the first ten jet diameters. The corresponding potential core length, indirectly
defined as ū(x = xc ) = 0.95U j is xc ≈ 6.2. The RMS velocity
along the centerline, shown in figure 1(c), matches well for the
first six jet diameters and is underpredicted by about 10% further downstream. The lipline RMS velocity in figure 1(d) isslightly overpredicted near the nozzle and underpredicted witha maximum deviation of also about 10% further downstream.
Clearly visible in the simulation results are grid transitions thatwere similarly observed by Brès et al. (2018). It was confirmedin the same work that increasing the resolution mitigates thesetransitions. More importantly, Brès et al. (2018) also showedthat both nozzle-exit turbulent statistics and far-field noise predictions were accurate for the lower resolution simulations thatexhibit grid transitions.
For the forced simulation, we follow Heidt et al. (2021)
and model the effect of the loudspeaker actuators as an acoustic forcing in the annular region along the lip line, 0.50 ≤ r ≤0.51, asIn this study, we use the bispectral mode decomposition(BMD), recently proposed by Schmidt (2020), to investigate
the triadic interactions triggered by the axisymmetric excitation of the jet. BMD is a technique that can be understoodas an extension of classical bispectral analysis to multidimensional and multivariate data. The bispectrum is defined as the
double Fourier transform of the third moment of a time signal.
For a time series, y(t) with zero mean, the bispectrum is
Z Z

Syyy ( f1 , f2 ) =Ryyy (t1 ,t2 )e−i2π( f1 τ! + f2 τ2 ) dτ1 dτ2 ,(1)where Ryyy (τ1 , τ2 ) = E[y(t)y(t − τ1 )y(t − τ2 )] is the third moment of y(t), and E[·] is the expectation operator. The bispectrum is a signal processing tool for one-dimensional time series which only measures the quadratic phase coupling locally.
On the contrary, BMD is a modal decomposition techniquethat identifies the spatially coherent structures associated withthe triadic interactions.
For a fluctuating flow field qi = q(x,ti ), where i =1, 2, · · · nt , BMD maximizes the integrated point-wise bispectrum
Zb( fk , fl ) = Eq̂∗ (x, fk ) ◦ q̂∗ (x, fl ) ◦ q̂(x, fk + fl )dx . (2)ΩHere, q̂ is the temporal Fourier transform of q computed using
the Welch approach (Welch, 1967), Ω is the spatial domain ofinterest, (·)∗ denotes the complex conjugate, and ◦ denotes the
Hadamard (or element-wise) product.
Next, all the Fourier realizations
at frequencyhi fk are ar(1) (2)(nblk )
ranged into a matrix, Q̂k = q̂k , q̂k , · · · , q̂k. The autobispectral matrix is then computed as

B=

1 H
Q̂ WQ̂k+l ,nblk k◦l(3)∗∗where Q̂Hk◦l = Q̂k ◦ Q̂l and W is the diagonal matrix containing
the spatial quadrature weights. Owing to the non-Hermitian
nature of the bispectral matrix, we obtain the complex mode
bispectrum by solving the optimization problem

λ1 ( fk , fl ) = arg max

a∗1 Ba1,a∗1 a1(4)

for the optimal expansion coefficients a1 , by determining thenumerical range of B. For further details on computing themode bispectrum, the reader is referred to Schmidt (2020). Finally, the bispectral modes are recovered asp(r) = 1 − 40000(r − 0.505)2 ,ux (r,t) = Ap(r) sin 2πSt f t ,ur = uθ = 0,
ρ = ρ∞ + ρ∞ ux /c∞ ,(1)
φk+l = Q̂k+l a1 .

p = p∞ + ρ∞ c∞ ux .

(5)

The amplitude, A, was manually adjusted to match the experimental observations. Here, St f = 0.4 is the forcing frequency.
Figure 2 shows the power spectral densities of the centerline
streamwise velocity at x = 2 for the unforced and forced jets.
The comparison between experiment and simulation are excellent for the unforced jet seen in figure 2(a). The best agreement
for the forced case was obtained for A = 0.4 and is shown in

Validation of LES
The experiments of the isothermal subsonic jets were performed at the “JET100” low Mach number jet facility at Institut PPRIME, Poitiers, France. The experiments were carriedout for a jet Mach number M j = 0.05, and Reynolds number of
Re = 50000. The boundary layer is tripped inside the nozzle

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3. Integrated PSD of first five azimuthal wavenumbers, m = 0, 1, 2, 3, 4 for the two jets: (a) unforced turbulent;(b) forced turbulent jet. PSD is integrated over the compressible energy norm.

Figure 1. Comparison of the experiment with the LES simulations: centerline (a, c) and lipline (b, d). Profiles of themean (a, b) and (c, d) RMS streamwise velocity. Black dashedline represents U/U j = 0.95 and its intersection with the mean
streamwise velocity indicates the length of the potential core.

Figure 4. BMD spectra: (a) unforced turbulent; (b) forced
turbulent jet.

Figure 2. Power spectral density of the streamwise velocity
at x = 2 and r = 0: (a) unforced jet; (b) forced jet. The experiment (blue line) is compared to the simulations (red line).
St = St f = 0.4 corresponds to the forcing frequency (red dotted line).

maxima at the intersection of these lines. These local maxima
represent the prominent triadic interactions. In particular thetwo most dominant triads are (0.4,0.4,0.8) and (0.8,-0.4,0.4).
In the case of unforced jets, a broadband behavior is observed,
with the highest values concentrated at lower frequencies.
Next, we examine the mode bispectrum along different
diagonals with St3 = const to identify the strongest non-linear
interaction that result in a triad at each frequency. Figure 5
shows the mode bispectrum along the constant frequencies of
St3 = 0.0, 0.4, 0.8, and 1.2 for the two cases. The spectrum of
the unforced jet is broadband for all St3 . The forced jet exhibits
distinct peaks at the forcing frequency and its harmonics. The
two most dominant triads are observed for St3 = 0.4 in figure5(b) and St3 = 0.8 in 5(c). The first of these two triads is thedifference interaction between St1 = 2St f and St2 = −St f , i.e.,
between the second harmonic and the conjugate forcing frequency. The second triad is the fundamental self-interaction,(St f ,St f ,2St f ), that creates the second harmonic. The dips ofthe forced mode bispectrum observed for St2 = 0 in figure 5(b),(c), and (d) correspond to the relative lower values of the mode
bispectrum previously seen along the abscissa in figure 4(b).
Note that the zero-frequency bin contains unresolved low frequency components due to the finite sampling frequency, andthe non-zero mode bispectrum should not be interpreted along
St2 = 0.
The BMD modes of the triad cascade including the mostsignificant interactions up to St1 ≤ 2St f are visualized in figure
6. The cascade starts with the mode at the forcing frequency,
φSt f +0 (left column middle row). The modes associated withthe sum interactions are shown in the first and second row, andthose associated with the difference interactions are shown infigure 2(b). Both the peaks at the forcing frequency and its
harmonics, as well as the number of active harmonics and theunderlying broadband spectrum, are well predicted.

Triadic interactions in forced and unforced jets
Figure 3 shows the area-integrated power spectral densities (PSD) for the five most energetic azimuthal wavenumbers
m = 0, 1, 2, 3, 4. The PSDs are integrated over the compressible energy norm. The azimuthal wavenumber m = 0 contains
most of the energy for the unforced and forced jets. For theforced jets, we observe large peaks at the forcing frequency
and its harmonics in the wavenumber m = 0. The peaks atthese harmonic frequencies indicates the presence of triadic
sum interactions. As the other wavenumbers are not affected,
we focus our analysis on the nonlinear interactions in the axisymmetric component m = 0.
The mode bispectra for the unforced and forced jets areshown in figure 4. The high-intensity regions (false red color)in the mode bispectra signify the dominant triads that arise
from the interactions of two frequencies. Different combinations of frequencies (St1 ,St2 ) interact to generate the same frequency (St1 + St2 = constant) along the diagonals of slope -1 inthe mode bispectrum. For the forced jet (figure 4 (b)), a gridlike pattern is observed with vertical, horizontal, and diagonal lines at the forcing frequency and its harmonics with local

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 7. Streamwise velocity component of the BMD
modes for the turbulent unforced (a,c,e,g) and forced jets
(b,d, f ,h) for different frequency triads: (a,b) (St f ,-St f ,0); (c,d)(St f ,0,St f ); (e, f ) (2St f ,−St f ,St f ); (g,h) (St f ,St f ,2St f ).
Figure 5. Mode bispectrum along diagonal of constant frequency: (a) St3 = St1 + St2 = 0.0; (b) St3 = St1 + St2 = 0.4;(c) St3 = St1 + St2 = 0.8; (d) St3 = St1 + St2 = 1.2. The red
dotted and dash-dotted lines denote the forcing, and conjugate
forcing frequency, respectively.

Figure 8. Integral interaction maps, ∑k,l λ1 (Stk , Stl )|φk◦l ◦φk+l |: (a) unforced turbulent jet; (b) forced turbulent jet.
White solid and dashed lines outline the potential core and thejet width defined as lines of constant ux = 0.95U j and 0.05U j ,respectively.

the streak-like structures with an increasing number of streamwise lobes are created by the destructive self-interactions of
the forcing mode and its harmonics.
Next, we compare the spatial structures of the BMD
modes associated with the triadic interactions for unforced
and forced turbulent jets. Figure 7 shows the streamwise
velocity components for the dominant triads at (St f ,-St f ,0),
(St f ,0,St f ), (2St f ,−St f ,St f ), and (St f ,St f ,2St f ), which correspond to the mean flow deformation, fundamental instability
mode, a harmonic-fundamental difference interaction and fundamental self-interaction, respectively. The BMD modes reveal that the structures contributing to the triadic interactions
are localized to the region near the end of the potential core forthe unforced jet, whereas, for the forced jet, they are locatedat a more upstream location and are concentrated in the shear
layer. This effect is more pronounced for the BMD mode of
the fundamental self-interaction triad (St f ,St f ,2St f ), which exhibits a KH type wavepacket near the nozzle exit in the forcedjet.
We next seek to identify the regions where triadic interactions are most active. This is achieved by summing
the weighted interaction maps over all frequency triads as∑k,l λ1 (Stk , Stl )|φk◦l ◦ φk+l |. The resulting integral interaction maps are presented in figure 8 and indicate the regions
where the triadic interactions are most active. Both jets exhibit
strong triadic nonlinear interactions downstream of the potential core near the centerline. This observation is in agreementwith previous findings by Cavalieri et al. (2013) and Tissot
et al. (2017), who, in the same region, observed that linear
wavepacket models differ significantly from experiments. In
addition to this ‘natural’ occurrence of nonlinear effects, theforced jet interaction map also reveals the presence of strong
nonlinearity in the shear layer near the nozzle exit that is directly associated with the forcing. Also observed is a shortening of the potential core and a thickening of the shear layer.
This is in agreement with the previous works by Crow &Figure 6. Bispectral modes forming a cascade of triads, starting from the forcing frequency (St1 ,St2 ,St3 ) = (0.4,0.0,0.4),
middle-left. The real part of the streamwise velocity component is shown.

the fourth and fifth row. The mode φSt f +0 self-interacts to generate the second harmonic, φSt f +St f . The fundamental forcing mode and its second harmonic then interact to create thethird harmonic, φ2St f +St f , and so on. In the negative St2 plane,
the destructive self interaction of the forcing mode creates amean flow distortion mode, φSt f +−St f , with St3 = 0. Inspecting the modes reveals that sum-interactions create modes withhigh wavenumbers and difference-interactions create modes
with lower streamwise wavenumbers. This behaviour is inagreement with the linear dispersion relation of these KelvinHelmholtz (KH) type instability waves.
Another interesting observation is that the spatial structures of the modes along the diagonals of constant frequencies
St3 = 0.4, 0.8, 1.2 and 1.6 are very similar. As shown forthe companion experiment by Maia et al. (2021a), these similar bispectral modes at constant St3 ̸= 0 resemble the leading
SPOD modes at each frequency. This indicates that the spatial
structures created by the triadic interactions are also the mostenergetic coherent structures. Along St3 = 0, on the contrary,4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
tegral interaction maps are defined to identify regions of activity of triadic interactions. Both unforced and forced jets
exhibit strong interactions in the closure region of the potential core. This ‘natural’ region of nonlinear activity extends
over ∼ 6 jet diameters in the unforced case and over ∼ 4 in
the forced case, respectively. We speculate that the nonlinear activity in that region is the reason for the discrepancyof SPOD modes and linear resolvent (Schmidt et al., 2018)
and global modes (Schmidt et al., 2017), and similarly the discrepancy between linear wavepacket models and experimental
data, also attested to nonlinearity by Cavalieri et al. (2013)
and Tissot et al. (2017). In the forced jet, strong interactions
that are associated with the large-amplitude forcing are identified in the vicinity of the actuator near the lip line. Lastly,
BMD was applied to the far-field pressure and identified thedifference-interaction between the first harmonic and the forcing frequency as the most significant triad. This observation isin agreement with the findings of Sandham et al. (2006).
Acknowledgements AN and OTS gratefully acknowledge
support from Office of Naval Research grant N00014-20-12311 and National Science Foundation grant CBET 2046311.

Figure 9. BMD modes of the pressure field for the turbulent
unforced (a,c,e,g) and forced jets (b,d, f ,h) for different frequency triads: (a,b) (2St f ,−St f ,St f ); (c,d) (St f ,St f ,2St f ); (e, f )(2St f ,St f ,3St f ); (g,h) (3St f ,St f ,4St f ).

Champagne (1971); Samimy et al. (2007), who argue that theenhanced entrainment due to forcing leads to a widening of
the shear layer, and consequently, a contraction of the potential core.
To identify the link between the triadic interactions in thenear-field and the acoustic far-field, we perform BMD on thepressure data using a weight matrix,Wac (x) =1 for 5 ≤ r ≤ 6, ∀x ∈ Ω
0 otherwise,

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Summary and conclusions
LES of unforced and forced turbulent jets were performed
and validated with companion experiments. The jet is axisymmetrically forced by blowing and suction at a frequency of
St = 0.4 and with an amplitude that correspond to the jet velocity, triggering strong nonlinear interactions. The forcing
produces a purely axisymmetric response and entails significant thickening of the shear layer and shortening of the potential core. The axisymmetric component, m = 0, exhibits
peaks at the forcing frequency and its harmonics due to triadic
interactions. BMD is employed to investigate the structuresassociates with these triads and their relative significance. In-

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Osaka, Japan (Online), July 19-22, 2022
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