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

MODAL DECOMPOSITION OF NON-LINEAR INTERACTIONS IN WALLTURBULENCE

Ugur Karban
Eduardo Martini
Department of Aerospace EngineeringDépartement Fluides, Thermique, Combustion
Middle East Technical University
Institut Pprime, CNRS-University of Poitiers-ENSMA
06800 Ankara Turkey
86073 Poitiers France
ukarban@metu.edu.tr
eduardo.martini@univ-poitiers.fr

André V. G. CavalieriPeter Jordan
Instituto Tecnológico de Aeronáutica
Département Fluides, Thermique, CombustionSão José dos Campos/SP, Brazil
Institut Pprime, CNRS-University of Poitiers-ENSMA
andre@ita.br
86073 Poitiers France
peter.jordan@univ-poitiers.fr

ABSTRACT

the spanwise wall shear is considered as the target observable. In similar minimal channel configurations, Bae et al.
(2021) investigated the triadic interactions contributing to the(α, β ) = (0, 2π/Lz ) mode, where α and β are streamwise andspanwise wavenumbers, respectively, and Lz is the domain sizein z-direction. We investigate here the triadic interactions systematically extracted using RESPOD for the same mode.
The remainder of the paper is structured as follows: themathematical framework to extract triadic interactions associated with a measured quantity is explained in §2. The details
about the DNS database of the minimal Couette flow are provided in §3. The results about identifying the relevant triadic
interactions and the energy transfer via these interactions in
the minimal Couette flow are discussed in §4. Finally, someconcluding remarks are provided in §5.

Coherent structures are found in many different turbulent
flows, and they are known to drive self-sustaining processes
in minimal-unit turbulence. Identifying the triadic interactions
between coherent structures can provide insights beyond what
is possible in the framework of linearised models. There areinfinite possible interactions that may generate a given structure, and thus a method to systematically study those, ranking
them in terms of their contribution, is of interest. We here
use the resolvent-based extended spectral proper orthogonal
decomposition (RESPOD) approach (Karban, U. et al. 2022
Self-similar mechanisms in wall turbulence studied using resolvent analysis. Journal of Fluid Mechanics 969, A36) to
identify the relevant triadic interactions for a minimal Couette flow at Reτ = 34, studying the interactions that give rise
to wall-attached structures, obtained by measuring the wallshear. Our analysis reveals that there are six triadic interactions
that dominate the most-energetic wall-attached structure.

1

2

EXTRACTING NONLINEAR INTERACTIONSUSING RESPOD

We consider the incompressible Navier-Stokes (N-S)
equations as,

INTRODUCTION

We investigate dominant nonlinear mechanisms in wallbounded turbulence. The complexity of all possible triadic interactions in a turbulent flow can be reduced by focusing ona certain quantity and eliminating all the non-relevant interactions. We use the resolvent-based extended spectral proper orthogonal decomposition (RESPOD) (Karban et al., 2022) for
this purpose. The should aid to design reduced-order models
appropriate for the description of a given observable in a turbulent flow.
We use the resolvent framework (McKeon & Sharma,2010), where the Navier-Stokes operator linearized around themean can be used to link the triadic interactions and a flow
quantity of interest, which appear, respectively, as forcing andresponse. Such a linear relation allows us to understand therole of various interactions that appear in interaction maps thatmeasure the dominance of a given interaction with respect to agiven observable.
The method is implemented using a direct numerical simulation (DNS) of minimal Couette flow with Reτ ≈ 34, whereM∂t q(x,t) = N (q(x,t)) ,(1)

where q = [u v w p]⊤ is the state vector, N denotes the nonlinear N-S operator for incompressible flows and the matrix M
is zero for the continuity equation and identity matrix for the
rest. Discretisation in space and linearisation around the mean,q(x), yields
M∂t q′ (x,t) − A(x)q′ (x,t) = Bf(x,t),(2)

where A(x) = ∂q N|q is the linear operator obtained from theJacobian of N and f(x,t) denotes all the remaining nonlinear
terms, interpreted as a forcing term in the above equation with
B = M for this particular case. Without loss of generality, we
focus on parallel flow, i.e., a flow that is homogeneous in two

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
directions, for instance, in x and z, with the mean flow varying
only in y. We can modify (2) to cast it in the resolvent form byapplying Fourier transforms in all homogeneous dimensions
and rearranging as,q̂(α̃, y, β̃ , ω̃) = R(α̃, y, β̃ , ω̃)f̂(α̃, y, β̃ , ω̃),Given the expansion in (4), χ k can be decomposed as,−1/2

Θk Λ kχ k = ∑ Γ i,k−i ≜ ∑ Ûi ∇Ûk−i Θ̂i(3)∥ξξ ∥2 = ε{ξξ H Wξξ },(4)where i = (αi , βi , ωi ), and k = (αk , βk , ωk ) denotewavenumber-frequency combinations and summation
over i implies nested summation over αi , βi and ωi . Note that(4) is valid assuming that the target triplet k contains at least
one non-zero element.
The RESPOD method, adapted from extended proper orthogonal decomposition (Borée, 2003; Hoarau et al., 2006),
finds, for a given ‘observed’ SPOD mode of state (response),
all correlated parts in a ‘target’ event. Here we choose the target event to be the non linear interactions, which give rise to
the forcing terms in the resolvent framework, as in Towne et al.
(2015) and Karban et al. (2022).
The SPOD mode of a measured quantity, ŷk , can be estimated using the ensemble matrix of realisations, through the
eigendecomposition,
H

3

−1/2

,

(5)

(6)

where Ŷk ≜ [ŷk (1) ŷk (2) · · · ŷk (P) ] denotes the ensemble matrix
for different realisations of ŷk with P being the total number ofrealisations, Ψ k and Λ k are SPOD modes and their associatedeigenvalues, respectively (see Towne et al. (2018)), and W is
a positive-definite matrix of quadrature gains along y, which is
discretised.
In Karban et al. (2022), it is shown that this coefficientΘk can be used to extract the part in the forcing that is
matrix Θ̂correlated with the observed SPOD mode as,−1/2Θk Λ kχ k = F̂k Θ̂

,(7)where, F̂k is the ensemble matrix of f̂k . The RESPOD forcing
mode χ k satisfies,
Ψ k = Rk χ k .

DETAILS OF THE DNS OF THE COUETTE
FLOW

The use of RESPOD for detection of ‘important’ nonlinear interactions associated with a specific measurement istested on a minimal Couette flow, similar to that investigatedin Nogueira et al. (2021). The simulations are performed
using ‘ChannelFlow’ code, a pseudo-spectral incompressible
flow solver using a Fourier-Chebyshev discretisation in thewall-parallel and wall-normal directions, respectively (seewww.channelflow.ch for details). The dimensions of the minimal box are (Lx , Ly , Lz ) = (1.75πh, 2h, 1.2πh), where the subscripts x, y and z denote the streamwise, wall-normal and spanwise directions, and h is the channel half-height. The domain
was discretised as (nx , ny , nz ) = (32, 65, 32) with a dealiasing
factor of 3/2 in the wall-parallel directions. The channel walls
move with wall velocity, ±Uw yielding a Reynolds number,
Re = 400 based on Uw and h, which corresponds to a friction Reynolds number, Reτ ≈ 34. The flow data was stored,
once the initial transients disappeared, for 300 convective units
with a sampling rate, ∆t = 0.1. Temporal data is transformed
into frequency space using blocks of 1024 time steps with
50% overlapping and using the exponential windowing function given in Martini et al. (2019) with n = 2.
Figure 1 presents the profiles for the mean and the rootmean-square (RMS) of the velocity components, u, v and w in
the streamwise, wall-normal and spanwise directions, respectively, along the wall-normal direction, y. We see that the mean
flow is deviated from the laminar solution given by −(y − 1)due to nonlinear interactions between turbulent fluctuations.
The RMS plots indicate that the fluctuations in u peak aroundy = 1.5 and y = 0.5. A similar but smaller double-peak structure is seen in the RMS of w with the peaks occurring at thesame wall-normal positions. The RMS of v peaks around thecenter at an amplitude slightly less than that of w.
We choose wall shear fluctuations in the spanwise direction, τz ≜ ∂z u′ |y={−1,1} at both upper and lower walls as ourobservable. Spanwise wall shear was used to extract selfsimilar wall-attached structures in a turbulent channel in Karban et al. (2022). Here, we use it simply to have a low-rank
representation of the flow, associated with this quantity.

Θk as,and the SPOD modes are obtained from Θ̂Θk Λ kΨ k = Ŷk Θ̂

(10)where the superscript H indicates Hermitian transpose, andε{·} denotes the expectation operator. In what follows, ε{·}corresponds to time-averaging for time-dependent structures,and to ensemble averaging for Fourier realisations in freΓi,k−i ∥2 ,quency space. The energy of Γ i,k−i , denoted by ∥Γfor all i shows the correlation map of the nonlinear interactions related to the observed SPOD mode, Ψ k . This includes
the ‘silent’ correlations (Karban et al., 2022). One can exclude the effect of the silent-but-correlated part by investigating ∥Rk Γ i,k−i ∥2 . Equation (8) implies that such a map indicates the contribution of different nonlinear interactions to agiven SPOD mode of the measured state.

iΘk Λ k Θ̂Θk ,ŶHk WŶk = Θ̂(9)where Û denotes the ensemble matrix of û. We define the overall energy of a flow structure, ξ , as,where α̃ and β̃ are the streamwise and spanwise wavenumbers,
respectively, and ω̃ is the angular frequency, the hat indicatesa Fourier transformed quantity and R(α̃, y, β̃ , ω̃) = (−iω̃M −A(α̃, y, β̃ ))−1 B is the resolvent operator. For brevity, we drop
the notation showing dependence on wavenumber and frequency in what follows.
For the incompressible N-S equations, the forcing termin (2) is given as f = u′ · ∇u′ − u′ · ∇u′ . The forcing in thewavenumber-frequency space, f̂k , is then obtained via a convolution,f̂k = ∑ ûi · ∇ûk−i ,

,

i

(8)

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

Figure 3. PSDs of q̂(0,1) (blue), q̂(0,2) (orange), q̂(1,0) (yellow) and q̂(1,1) (violet) integrated over the wall-normal direction.
Figure 1. Mean (a) and the RMS (b) profiles of the velocity
components, u (black solid), v (red dashed) and w (blue dashdotted) along the wall-normal direction.

Figure 4. PSDs of q̂(0,{1,3}) (black; solid, dashed, dashdotted, respectively), and q̂(1,{0,3}) (red; solid, dashed, dashdotted, dotted, respectively) integrated over the wall-normal
direction and normalised with respect to the peak value of each
mode.
Figure 3 shows the power spectral densities (PSD) integrated along the wall-normal direction for the modes (α, β ) =(0, 1), (0,2), (1,0) and (1,1). Although the mode (α, β ) = (1, 1)
is energy-wise insignificant, it plays a critical role for transfer
of energy to (α, β ) = (0, 1) mode, as will be shown later, andhence is included here. We see that the streamwise-constantmodes peak around the zero frequency while the (1,0) and(1,1) modes have their peak around ω̃ ≈ 0.1 (ω = 8). The
shape of the spectra is observed to be similar for the modes
that have the same streamwise wavenumber. This trend can be
more clearly seen in figure 4, where the integrated PSDs normalised with respect to the peak value are plotted for different
modes. We see two different families of PSD distributions forthe two streamwise wavenumbers, α = 0 and α = 1, respectively.
We now focus on the most energetic mode (α, β ) = (0, 1)
at its peak-energy frequency, ω = 0. The wall-attached response and forcing modes are shown in figure 5. The response field consists of streaks and streamwise vortices. Given
that the upper and lower walls have positive and negative velocities, respectively, the phase relation between streaks andthe streamwise vortices is reminiscent of lift-up mechanism(Brandt, 2014). This is further supported regarding the associated forcing mode. At the spanwise positions where thestreamwise vortices are located, the forcing is mainly locatednear the walls aligned with the y-direction, causing a momentto generate the streamwise vortices. These vortices then generate streaks by carrying the high- and low-velocity structurenear the upper and lower walls, respectively, towards the channel center. Note that the forcing component in the streamwise
direction is indeed in opposite phase to the streaks seen in

Figure 2. a) Energy of flow structures at different wavenumber pairs averaged over time. b) Ratio of the average energyof the wall-attached structures to that of the wall-attached and-detached structures added together.

For simpler notation, wavenumbers will be presented in
integers defined as α = α̃Lx /2π and β = β̃ Lz /2π. Similarly, mode frequencies will be presented in integer bins denoted by ω = ω̃NF / fs , ranging in [−NF /2, NF /2 − 1], where
NF = 1024 is the number of temporal points used for taking
the Fourier transform (FT) and fs ≜ 1/∆t is the sampling rateof the database.
Figure 2 shows the time-averaged energy contained ineach wavenumber pair together with the ratio of the timeaveraged energy of the wall-attached structures to the entire
turbulent fluctuations at each wavenumber pair. We see thatthe mode pair (α, β ) = (0, 1), related to streaks and rolls spanning the entire computational domain, contains the most of
the fluctuation energy (∼75%) while the modes (±1, 0) and(0, 2) contains slightly less than 5% of the total energy andall the other mode pairs contain less than 2%. The energy of
the wall-attached part of the state, denoted by qa , of the mode(0,1) is around 80% of its total energy. Therefore, the coherent structures correlated with the spanwise wall-shear can besaid to constitute a good low-rank representative of the flowat this wavenumber pair. A similar case is observed for themodes (0, ±1) while for the mode (0,2), the energy ratio of thewall-attached part is around 15%.

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

Γi,k−i ∥2
Figure 6. Amplitude maps of ∥ui ∇uk−i ∥2 (top), ∥Γ2Γi,k−i ∥ (bottom) obtained at ωi = 0 (left),(middle), and ∥RΓωi = 4 (center) and ωi = 8 (right), for the target mode k =(αk , βk , ωk ) = (0, 1, 0). Only the modes i are shown while thecomplementary modes k − i are selected to yield k = (0, 1, 0).

Figure 5. Wall-attached part of the velocity (a) and the associated forcing (b) reconstructed in the y-z plane for the mode(α, β , ω) = (0, 1, 0). The color plot indicates the streamwise
component and the arrows show the spanwise and wall-normalcomponents.

the response. This indicates that the streaks are generated by
the lift-up mechanism despite the counteracting effect of thestreamwise forcing, as previously reported by Nogueira et al.
(2021). The response generation at this triplet can therefore beconsidered suboptimal.

4

NONLINEAR INTERACTIONS IN THE MINIMAL COUETTE FLOW
4.1 Extracting important triadic interactions

Figure 7. a) Amplitude map of the response generated by
the wall-correlated interactions at all frequencies added together. b) The map of normalised inner product between theoverall response and the response with contribution of a single interaction masked, computed at different wavenumber
pairs used for masking. The target mode for both maps is(αk , βk , ωk ) = (0, 1, 0).

The interaction maps obtained by computing via (10) the
energy associated with a nonlinear interaction contributing to
the target mode, k = (αk , βk , ωk ) = (0, 1, 0) are shown in figure 6. Different columns compares the maps ∥ûi ∇ûk−i ∥2 ,Γi,k−i ∥2 and ∥Rk Γ i,k−i ∥2 , which correspond to the direct tri∥Γadic interactions, the wall-correlated triadic interactions, andthe wall-correlated triadic interactions filtered by the resolvent operator, respectively. Note that only the triplet i =(αi , βi , ωi ) is shown, where for each i, there exists a k − iyielding k = (0, 1, 0). We see that the interaction between(0, −1, 0) and its complementary, (0, 2, 0), is dominant in all
three maps, indicating that the interaction is large in amplitude, highly correlated to the target mode, and generates the
response with the largest amplitude. We also observe large amplitude for the interaction (0, 2, 0) + (0, −1, 0), which involves
the same structures with the previous one, but with the gradient operator acting on the (0, −1, 0) mode. The interactions(±1, 1, 0) + (∓1, 0, 0), although not yielding a large forcing
component, are seen to be present in the response map, implying the efficiency of these modes in driving the observable.
To investigate the overall contribution to the target mode(0, 1, 0) via all the triadic interactions associated with agiven wavenumber pair (αi , βi ), we define the forcing mode,
Γi,k−i obtained by integrating Γ i,k−i over the frequency inΓ̌
Γi,k−i . The interdex, ωi , and compute its response via Rk Γ̌
Γi,k−i ∥2 is plotted in figure 7, which shows
action map ∥Rk Γ̌that the response generation is dominated by six interac-tions: two streamwise-constant, which are (0, {−1, 2}) +(0, {2, −1}), and four streamwise-periodic over Lx , which are(±1, {0, 1}) + (∓1, {1, 0}). Note that here and in what follows, we use curly brackets for short hand notation of multiple
modes. For instance, (0, {−1, 2}) denotes the modes (0, −1)and (0, 2).
To understand whether a given interaction has a destructive or constructive effect on the response, instead of computing its individual response, we calculate the overall responsewith that particular interaction masked, i.e., its contribution is
not included in the response. We denote this masked wallattached response as q̂−a and investigate the change in the response energy due to this masking by computing the innerproduct,H−(q̂−a , q̂a ) ≜ ε{q̂a Wq̂a }.

(11)

An interaction map is obtained obtained by calculating (11)4

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

Figure 9. Production (left), dissipation (center), and nonlinear transfer (right) of the spectral turbulent kinetic energy for
different wavenumber pairs. Both the size and the color intensity of the markers indicate amplitude.

Figure 8. Velocity field corresponding to the wall-attachedstructure (αk , βk , ωk ) = (0, 1, 0) in the y-z plane. Topleft: the entire response; top-right: the response obtainedby masking the interactions between the modes (αi , βi ) =(0, {−1, 2}) and their complementary modes; bottom-left: theresponse obtained by masking the interactions between themodes (αi , βi ) = (±1, {0, 1}) and their complementary modes;
bottom-right: the response obtained by masking the interactions between the modes (αi , βi ) = (0, {−1, 2}) and theircomplementary modes as well as the interactions between themodes (αi , βi ) = (±1, {0, 1}) and their complementary modes.

4.2

Energy transfer via triadic interactions

The interaction map shown in figure 7-b can also be interpreted in terms of energy exchange between different modes
via nonlinear interactions. Symon et al. (2021) investigated,by employing the spectral form of the transport equation of turbulent kinetic energy (TKE), the overall relation between production, dissipation and the transfer of energy for individual
wavenumber pairs in parallel stationary turbulent flows. The
spectral TKE equation is given in indicial notation as,
2 ∂u
1
∂ Ê
û∗ v̂dy −
=−
∂t
Re
0 ∂y

Z

by individually masking each wavenumber pair and normalising the result with ∥q̂a ∥2 as shown in figure 7-b. The analysis reveals that masking the contributions from the interactions (0, {−1, 2}) + (0, {2, −1}) increases the response energy, implying a destructive interference between these interactions and the remaining ones. Masking the contributions
from the interactions (±1, {0, 1}) + (∓1, {1, 0}), on the other
hand, causes the response energy to decrease, implying thatthese interactions have a constructive effect on the response.

2∂ ûm ∂ û∗m
û∗m fˆm dy,dy −0 ∂ xn ∂ xn
0
(12)

Z 2

Z

where the hat in this equation denotes, by abuse of notation,
Fourier transformed quantities in the stremwise and spanwise
directions, Ê is the spectral TKE, the superscript * denotes
complex conjugate, m and n denote that the vector indices, fˆmis the mth component of the forcing vector (see Symon et al.
(2021) for derivation of (12)). Here, we assume that the Couette flow is stationary in the time interval we investigate, which
renders

The effect of these masking operations on the response
field is shown in figure 8. We see that masking the interactions (0, {−1, 2}) + (0, {2, −1}) mainly affects the streaks
causing an increase in their amplitude, while the streamwise vortex remains nearly unchanged. Masking the interactions (±1, {0, 1}) + (∓1, {1, 0}), almost completely eliminates the streamwise vortices, which also causes the lift-up
effect to be eliminated. This results in streaks with smaller
amplitude and reversed phase. This result is consistent with
the idea of a self-sustaining process in wall turbulence, withstreamwise vortices excited by non-linear interactions involving waves with non-zero α (Hamilton et al., 1995; Hall &Sherwin, 2010). Remember that in the RESPOD forcing
mode shown in figure 5, the streamwise component counteracts the lift-up mechanism forced by the spanwise components. These results, when combined, imply that the streamwise and spanwise components in the RESPOD forcing mode,χ k , are mainly constructed by the nonlinear interaction groups(0, {−1, 2}) + (0, {2, −1}) and (±1, {0, 1}) + (∓1, {1, 0}), respectively. Masking (±1, {0, 1}) + (∓1, {1, 0}) causes thelift-up mechanism, which is an efficient means to generatestreaks via streamwise vortices, to disappear. The remaining streamwise component in χ k is mainly constructed by(0, {−1, 2}) + (0, {2, −1}) and generates streaks with negative
phase, reducing the amplitude of the streaks generated by thelift-up mechanism. Masking all six modes eliminates almost
entirely the response as seen in figure 8.

∂ Ê
= 0.
∂t

(13)

The three terms on the right hand side of (12) correspond to
the production, dissipation and nonlinear transfer of the turbulent kinetic energy, which, thanks to (13), sum up to zerofor a given wavenumber pair. The contributions of production,
dissipation and nonlinear transfer to the energy balance for different wavenumber pairs are illustrated in figure 9. We see thatthe (0,1) mode draws the most energy from the mean flow to
produce TKE, and is the only mode to transfer this energy toother modes via nonlinear transfer. All the modes are seen to
lose energy via dissipation as expected.
The analysis done by computing (11), as a complementary tool to the energy balance analysis discussed above, provides information about the transfer of energy between individual modes. A red square in the interaction map shownin figure 7-b implies that the target mode is losing energy
via the associated nonlinear interaction, while a blue spot indicates the opposite. One can investigate interaction maps
for different target modes to understand the paths of energy
transfer among different modes. In figure 10, we show thesame interaction map given in figure 7-b for the target modes5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
shown by the energy balance analysis we conducted following Symon et al. (2021). The relation between (0,1) and thestreamwise-constant modes (0, {−1, 2}) is shown to be a oneway energy transfer which causes the (0,1) mode to lose its
energy. The relation between (0,1) and the streamwise periodic modes (±1, 0, 1), on the other hand, is found to be in twoways, with energy transfers in both directions.
The method we discuss provides a systematic means bywhich to understand mechanisms active in the generation ofa given observable in a turbulent flow, which can be freely
determined via a measurement matrix, thanks to the flexibilityof the resolvent framework. This makes application of thisapproach to more complicated flows possible.

Figure 10. The same map with figure 7-b obtained for the target modes (αk , βk , ωk ) = (0, 2, 0) (a), (1, 0, 8) (b) and (1, 1, 8)(c).

(αk , βk , ωk ) = (0, 2, 0), (1,0,8) and (1,1,8). Note that the frequency index for each target mode is selected as the peakenergy containing frequency for the interaction maps be a better representative of the energy transfer in the corresponding
wavenumber pair. We seen in figure 10 that all three target modes receive energy from the interactions involving the(0,1) mode. This is expected since the (0,1) mode receives thelargest energy from the mean flow, similar to the cases shownin Symon et al. (2021), and distributes them into other modes
via nonlinear transfer. Comparing figures 7-b and 10-a reveals
that the energy transfer between the modes (0,1) and (0,2) is
one way, going from (0,1) to (0,2), which, when blocked, increases the energy in the mode (0,1) and decreases that in (0,2).
The energy transfer between (0,1) and (1,1), on the other hand,is two way, i.e., the two modes transfer energy to each other via
different triadic interactions. A similar relation exists between(0,1) and (1,0).

5

REFERENCES
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CONCLUSIONS

We have discussed a method to investigate the triadic interactions that underpin the generation of flow structures associated with a given observable. The method is based on the
resolvent-based extended spectral proper orthogonal decomposition (RESPOD), used in Karban et al. (2022) to identify
self-similar structures in a turbulent channel flow. A minimal
Couette flow is here chosen as the test case, where the triadic
interactions associated with spanwise wall-shear are investigated.
The analysis reveals that the most energetic mode,(α, β ) = 0, at its peak-energy frequency, ω = 0, was mainly
driven by six triadic interactions: four interactions involving
modes periodic over Lx in the streamwise direction, that generate small-in-amplitude but efficient forcing, and two interactions involving streamwise-constant modes that generate forcing structures large in amplitude, and thus effective in driving
the observable. The streamwise-periodic interactions generate
a combined streak-streamwise vortex structure via the lift-up
mechanism, while the streamwise-constant interactions counteract the streak generation by generating a streamwise forcing component in phase opposition to the lift-up mechanism.
This explains in physical terms the destructive interference of
forcing observed by Nogueira et al. (2021) forcing comprises
different triadic interactions, with opposing effects in exciting
streamwise vortices and streaks.
Our framework also allows us to investigate energy transfer between different modes via triadic interactions. We observe that the triadic interactions involving the (0,1) mode provides a constructive contribution to all the modes we investigated. This is an expected result since it is the only mode withnegative nonlinear transfer rate of turbulent kinetic energy as

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