11institutetext: Rongjiao Ji, Alessandra Micheletti 22institutetext:
Universitá degli Studi di Milano, 22email<EMAIL_ADDRESS><EMAIL_ADDRESS>33institutetext: Natasa Krklec Jerinkic
44institutetext: University of Novi Sad, 44email<EMAIL_ADDRESS>55institutetext: Zoranka Desnica 66institutetext: 3Lateral DOO, 66email:
<EMAIL_ADDRESS>
# Emotion pattern detection on facial videos using functional statistics

Rongjiao Ji  Alessandra Micheletti  Natasa Krklec Jerinkic  Zoranka Desnica

###### Abstract

There is an increasing scientific interest in automatically analysing and
understanding human behavior, with particular reference to the evolution of
facial expressions and the recognition of the corresponding emotions. In this
paper we propose a technique based on Functional ANOVA to extract significant
patterns of face muscles movements, in order to identify the emotions
expressed by actors in recorded videos. We determine if there are time-related
differences on expressions among emotional groups by using a functional
F-test. Such results are the first step towards the construction of a reliable
automatic emotion recognition system 111This work was funded by European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska Curie grant agreement No 812912 for the project BIGMATH.

###### Abstract

_C’è un crescente interesse scientifico nell’analizzare e intepretare
automaticamente il comportamento umano, soprattutto rispetto all’evoluzione
delle espressioni del volto e al riconoscimento delle corrispondenti emozioni
espresse. In questo lavoro proponiamo una tecnica, basata sull’ANOVA
Funzionale per estrarre pattern significativi dei movimenti dei muscoli
facciali, al fine di identificare le emozioni espresse da alcuni attori in
video registrati. In particolare determiniamo se, in istanti specifici, ci
siano differenze nell’evoluzione delle espressioni fra diversi gruppi di
emozioni, applicando un F-test funzionale. Questi risultati sono il primo
passo verso la costruzione di un sistema affidabile per il riconoscimento
automatico delle emozioni._

###### Keywords:

functional ANOVA, emotion, expression evolution, action units

##  1 Introduction

The study of human facial expressions and emotions never stops in our daily
life while we communicate with others. Following the increased interest in
automatic facial behavior analysis and understanding, the need of a semantic
interpretation of the evolution of facial expressions and of human emotions
has become of interest in recent years Fridlun2014 . In this paper, based on a
work cooperated with the Serbian company 3Lateral, which has special expertise
on building visual styles and designs in animation movies, we want to explore
functional statistical instruments to identify the emotions while analyzing
the expressions through recorded videos of human faces. The final aim of this
research is to use this information to better and more realistically establish
virtual digital characters, able to interact autonomously with real humans.

The data that we consider are multivariate longitudinal data, showing the
evolution in time of different face muscles contraction. Functional Data
Analysis (FDA) offers the possibility to analyze the entire expression
evolution process over time and to gain detailed and in-depth insight into the
analysis of emotion patterns. The basic idea in functional data analysis is
that the measured data are noisy observations coming from a smooth function.
Ramsay and Silverman ramsay1997functional  describe the main features of FDA,
that can be used to perform exploratory, confirmatory or predictive data
analysis. Ullah and Finch UllahShahid2013  published a systematic review on
the applications of functional data analysis, where they included all areas
where FDA was applied.

In our application, Functional ANOVA can be used to determine if there are
time-related differences between emotion groups by using a functional F-test
Dannenmaier2020 . Functional ANOVA yields the possibility to determine if a
functional response can be described by scalar or functional variables.

The structure of this paper goes as follows. In Section 2 we briefly describe
the RAVDESS dataset from where the expression data of interest is extracted.
Section 3 includes some methods of functional data analysis that we
implemented in our application, and in Section 4 our results are presented.

##  2 The RAVDESS Dataset

RAVDESS (Ryerson Audio-Visual Database of Emotional Speech and Song)
livingstone2018ryerson  fits our needs for studying the human expression
evolution and emotion identification, as it contains 24 professional actors
(12 female, 12 male) to offer the performance with good quality and natural
behavior under the emotions: calm, happy, sad, angry, fearful, disgusted and
surprised. Also a neutral performance is available for each actor. The actors
are vocalizing one lexically-matched statement in a neutral North American
accent (“Kids are talking by the door”).

To avoid being lost in the difference of individual facial appearances, when
analyzing the expressions and emotions, researchers mostly focus on the
movements of individual facial muscles which are encoded by the Facial Action
Coding System (FACS) ekman1997face . FACS is a common standard to
systematically categorize the physical expression of emotions, extracting the
geometrical features of the faces and then producing temporal profiles of each
facial movement. Such movements, corresponding to contraction of specific
muscles of the face, are called _action unit (AU)_. As action units are
independent of any interpretation, they can be used for any higher-order
decision-making process including recognition of basic emotions. Following the
FACS rules, OpenFace amos2016openface , an open-source software, is capable of
recognizing and extracting facial action unit from facial images or videos. We
applied OpenFace to extract the engagement degrees of action units for the
videos in RAVDESS. The extracted action units include 17 functions for each
video, taking values in $[0,5]$, sampled in about 110 time points (which is
also the number of frames in each video) varying around 110.

##  3 Functional Statistical Methods

We will represent the action units evolution recorded on each video as a
multivariate time series
$\mathbf{Y}(t)={(Y_{1}(t),\dots,Y_{d}(t),\dots,Y_{D}(t)),t\in[0,T]}$
containing a set of $D$ univariate longitudinal functions ($D=17$ in our
case), each defined on the finite interval $[0,T],0<T<+\infty.$ The
observation of $\mathbf{Y}$ on our sample of videos provides the set
${\mathbf{Y_{1}},\dots,\mathbf{Y_{n}}}$ of multivariate curves, that we
represent as multivariate functional data.

It is essential to align the action units functions into a common registered
internal timeline that follows the same pronunciation speed, to control the
influence of the specific pronounced sentence and to detangle it from the
influence of the emotions. Therefore, we need to isolate the phase variability
of the action units curves, but keeping, at the same time, the amplitude-phase
unchanged to maintain the information of the intensity level of the action
units.

The phase variation is normally represented by a random change of time scale,
which is mostly a non-linear transformation. We use the warping functions
$T_{i}:[0,T]\rightarrow[0,T],i=1,\dots,n$, assuming that they are increasing
functions independent of amplitude variation. They map unregistered
chronological time $t_{i}^{*}$ to registered internal time $t$ so that
$T_{i}^{-1}(t_{i}^{*})=t$, with $E[T_{i}(t)]=t$. The observed time-warped
curves, represented through a Karhunen-Loeve expansion based on a functional
basis $\mathbf{f}_{j,d}$, are

$\tilde{Y}_{i,d}(t)=Y_{i,d}(T_{i}^{-1}(t_{i}^{*}))=\mu_{d}(T_{i}^{-1}(t_{i}^{*}))+\sum\limits_{j\geq
1}C_{j}\mathbf{f}_{j,d}(T_{i}^{-1}(t_{i}^{*})),$

We used a spline basis and followed the principal components based
registration method with a generative process wrobel2019register , whose codes
are available in the R package ”registr” wrobel2018register .

Using the registered curves representing the AUs evolution in each video, we
then investigated if there exist patterns which could discriminate the
different emotions, using a Functional ANOVA model.

Let $y_{k,g}(t)$ be the evolution of one specific action unit in the video
$k\in\\{1,\dots,K\\}$ (in our case $K=48$) for emotion $g\in\\{1,\dots,7\\}$.
We can assume that

$y_{k,g}(t)=\mu_{0}(t)+\alpha_{g}(t)+\epsilon_{k,g}(t),$ (1)

where $\mu_{0}(t)$ is the grand mean function due to the pronounced sentence
and to the actor, independent from all emotions. The term $\alpha_{g}(t)$ is
the specific effect on the considered action unit of emotion $g$, while
$\epsilon_{k,g}(t)$ represents the unexplained zero mean variation, specific
of the $k$-th video within emotion group $g$. To be able to identify them
uniquely, we require that they satisfy the constraint
$\sum\limits_{g=1}^{7}\alpha_{g}(t)=0,\forall t.$

By grouping the videos representing the same emotion, we can define a
$8K\times 8$ design matrix $\mathbf{Z}$ for this model, with suitable 0 and 1
entries, as described in (ramsay1997functional, , Section 9.2), and rewrite
Equation 1 in matrix form:
$\mathbf{y}=\mathbf{Z}\mathbf{\beta}+\mathbf{\epsilon},$ where
$\mathbf{\beta}=[\mu_{0}(t),\alpha_{1}(t),\dots,\alpha_{7}(t)]^{T}$.

To estimate the parameters we use the functional least squares fitting
criterion

$\hat{\mathbf{\beta}}(t)=\arg\min\limits_{\mathbf{\beta}}\sum\limits_{g=1}^{8}\sum\limits_{k=1}^{K}\int_{0}^{T}[y_{k,g}(t)-\langle
z_{k,g},\mathbf{\beta}(t)\rangle]^{2}dt,$ (2)

subject to the constraint
$0=\sum\limits_{i=1}^{7}\alpha_{i}(t)=\sum\limits_{j=2}^{8}\beta_{j}(t),\ \
\forall t$.

In order to investigate which emotions are significantly influencing the
change of the action units patterns, for each emotion $\tilde{g}$ and for each
action unit we test the null hypothesis $H_{0}:\alpha_{\tilde{g}}(t)=0.$

Similarly to the classical univariate ANOVA model, the statistics used to test
$H_{0}$ is

$FRATIO(t)=\frac{MSR(t)}{MSE(t)}$

that under $H_{0}$ has an F distribution with suitable degrees of freedom.

##  4 Results

As mentioned before, we first aligned the curves by separating the amplitude
and phase variability. We choose to align the curves by AU25, which represents
the lip movement, and then we adjusted the time frames of the other AUs
according to this rescaling.

We then applied the F-test described in the previous section to detect, for
each emotion, which AUs have a mean behaviour significantly different from the
neutral performance and in which time period during the videos. In Figure 1 we
illustrate the results for emotion angry, as an example.

(a) AU07.

(b) AU10.

(c) AU26.

(d) AU07.

(e) AU10.

(f) AU26.

Figure 1: The functional coefficients of action units 07 (Lid Tightener), 10
(Upper Lip Raiser) and 26 (Jaw Drop) under neutral and angry emotion and the
corresponding F-test results Figure 2: Which and where AU values are affected
significantly by angry emotion

The first row of Figure 1 illustrates the estimated mean $\mu_{0}(t)$ (neutral
emotion) and the angry emotion effects for three action units. The second row
displays the observed F-statistics curves together with the pointwise and
maximum 95% significance level for the F-distribution in the dashed and
horizontal dotted lines respectively. Thus when the observed F-statistics is
higher than the critical level lines, the emotion has a significant effect on
the AU’s pattern. We found in general three main situations of influence of
one emotion on expression evolution: 1. locally strengthening (Figure 1(d):
AU07 in frame range 45 to 55) 2. locally inhibiting (Figure 1(f): AU26 in
frame range 70 to 90) 3. globally strengthening (Figure 1(e): AU10 in almost
the whole time). Further, we pointed out the time zones of significant effects
of the angry emotion on the action units in Figure 2, which is beneficial to
understand and detect dynamically when and how the facial muscles contractions
differ from the baseline.

Table 1: Emotions with corresponding significant action units Emotions | Related Action Units
---|---
Calm | $06,07,10,12,14,23$
Happy | $01,06,07,10,12,14,17,23,25,26$
Sad | $04,06,10,14,17,20,23,25$
Angry | $04,06,07,09,10,12,14,17,23,26$
Fearful | $04,09,10,12,14,15,17,23,25,26$
Disgust | $04,06,07,09,10,12,14,17,23,25,26$
Surprised | $06,09,10,12,14,15,17,23,25,26,45$

Table 1 summarizes for each emotion of interest the related action units that
show significant changes from the neutral case for our videos dataset.
Similarly to the example of angry, we found that for happy and disgust
emotions more action units have the globally strengthening effect on a large
time range. Sad emotion sometimes affects the action units to be more constant
than in neutral case. Emotion Fearful has more influence on upper half face
(brows, eye lids and nose), while emotion calm is more related with the center
of the face (Cheek Raiser, Lid Tightener and Lip Corner Puller). Surprised
emotion is the only emotion where AU45 is significantly influenced.

As a conclusion, our results can be joined in a multivariate setting and
exploited to build a classifier able to automatically recognize the emotions.
This task is left to subsequent works.

## References

  * [1] B. Amos, L. Bartosz, and M. Satyanarayanan.  Openface: A general-purpose face recognition library with mobile applications.  Technical report, CMU-CS-16-118, CMU School of Computer Science, 2016\.  https://cmusatyalab.github.io/openface/.
  * [2] J. Dannenmaier, C. Kaltenbach, T Kölle, and G. Krischak.  Application of functional data analysis to explore movements: walking, running and jumping-a systematic review.  Gait & postureh, pages 182–189, 2020.
  * [3] R. Ekman.  What the face reveals: Basic and applied studies of spontaneous expression using the Facial Action Coding System (FACS).  Oxford University Press, USA, 1997.
  * [4] A.J. Fridlund.  Human facial expression: An evolutionary view.  Academic Press, 2014.
  * [5] S.R. Livingstone and F.A. Russo.  The ryerson audio-visual database of emotional speech and song (ravdess): A dynamic, multimodal set of facial and vocal expressions in north american english.  PloS one, 13(5):e0196391, 2018.  https://smartlaboratory.org/ravdess/.
  * [6] J. Ramsay and B.W. Silverman.  Functional data analysis.  Springer, 1997.
  * [7] S. Ullah and F. F. Caroline.  Applications of functional data analysis: A systematic review.  BMC medical research methodology, 2013.
  * [8] J. Wrobel.  Register: Registration for exponential family functional data.  Journal of Open Source Software, 3(22):557, 2018.
  * [9] J. Wrobel, V. Zipunnikov, J. Schrack, and J. Goldsmith.  Registration for exponential family functional data.  Biometrics, 75(1):48–57, 2019.


# Composite active drag control in turbulent channel flows

Jie Yao<EMAIL_ADDRESS>Texas Tech University, Department of Mechanical
Engineering, Lubbock, Texas, USA, 79409  Xi Chen  Key Laboratory of Fluid
Mechanics of Ministry of Education, Beihang University, Beijing, People’s
Republic of China, 100191  Fazle Hussain  Texas Tech University, Department of
Mechanical Engineering, Lubbock, Texas, USA, 79409

###### Abstract

A composite drag control (CDC) combining the opposition (OC) and spanwise
opposed wall-jet forcing (SOJF) methods is studied in a turbulent channel flow
via direct numerical simulation of the incompressible Navier–Stokes equations.
A maximum drag reduction of about $33\%$ is obtained for CDC – much higher
than that produced by either individual method (namely, 19% for SOJF and 23%
for OC). Due to the small power input required for both OC and SOJF methods, a
significant net power saving (about 32%) is achieved via CDC. Flow analysis
shows that CDC can take advantage of both OC and SOJF methods to better
suppress drag producing, near-wall turbulent structures – vortices and
streaks. In particular, due to the presence of the large-scale coherent swirls
generated by SOJF, it is more effective than OC in suppressing the random
turbulence. Moreover, due to the OC’s role in suppressing random small-scale
turbulence, CDC requires weaker large-scale coherent swirls than those using
SOJF only – hence decreasing the drag contribution associated with large-scale
swirls. In summary, our results suggest prospects of employing composite
control strategy for effective skin friction drag reduction, particularly at
very high Reynolds numbers.

##  I Introduction

As the skin friction drag of aircraft and watercraft constitutes a large
fraction of the total aerodynamic drag (e.g., approximately 50% of aircraft
and about $90\%$ of ships and submarines), its reduction is of great
significance for energy saving and environmental protection. Benefited from
the continuing improved understanding of flow dynamics [1, 2, 3], especially
the role of near-wall organized coherent structures, various successful drag
control strategies have been proposed. In general, depending on whether
external energy is required to drive the control, these strategies fall into
two categories [4]: passive and active. For the passive method, riblets are
one of the highly investigated approaches and have been shown to yield about
$5-7\%$ drag reduction in the full-size trials [5]. However, due to the
specific requirement, sustenance of the tiny riblets against erosion by
atmospheric dust in practical applications is still a challenge. Recently,
drag control using superhydrophobic surfaces has been conceived and developed.
It has been reported to yield up to 50% and 75% drag reduction in laboratory
turbulent channel [6] and boundary layer [7] experiments, respectively.
However, as the superhydrophobic surfaces are severely vulnerable to high
pressure and high shear rate [8, 9], they have not yet evolved into a
practical means.

As for active control methods, they can be further grouped into open- and
closed-loop techniques, based on whether sensing is required or not. As one of
the simplest open-loop techniques, uniform blowing and suction has been
extensively investigated, where uniform blowing is found to reduce the skin
friction drag, while uniform suction increases it [10, 11]. The streamwise
traveling wave-based (also including spanwise wall oscillations) control
strategies are fascinating [12, 13, 14, 15, 16, 17], where significant drag
reductions are achieved [18, 19, 20, 21, 22]. However, considering the
typically large amount of power required for actuation, whether effective drag
reduction can be obtained at a reasonably high Reynolds number ($Re$) remains
unclear [23, 24]. Streamwise traveling wave-like wall blowing and suction is
another typical example of open-loop control techniques. With proper selection
of the traveling wave parameters, significant drag reduction and even sub-
laminar drag reduction were found in turbulent channel flows [25, 26, 22, 27].

One of the promising closed-loop control techniques is the opposition control
(OC), which was first studied by Choi  _et al._ [18]. Based on the
measurements in a plane parallel to the wall (called “detection plane”), OC
employs local wall blowing and suction to counteract the sweep and ejection
motions induced by the energetic near-wall streamwise vortices. Approximately
25% drag reduction was achieved in turbulent channel flow at friction Reynolds
number $Re_{\tau}(\equiv u_{\tau}h/\nu)\approx 110$ with the detection plane
located at $y^{+}_{d}=10$ [18]. [$u_{\tau}$ is the friction velocity, $\nu$ is
the kinematic viscosity, and $h$ is the half channel height; the superscript
$+$ indicates that the quantity is scaled by the viscous wall units, namely,
$y^{+}=yu_{\tau}/\nu$.] Hammond  _et al._ [28] later found that the maximum
drag reduction is obtained when $y^{+}_{d}=15$. For a practical application,
sensors and actuators need to be wall-based and flush-mounted to avoid
parasitic drag. Lee  _et al._ [29] developed the suboptimal control that
employs information measurable at the wall, e.g., using streamwise wall shear
stress, spanwise wall shear stress, and wall pressure. As machine learning is
becoming one of today’s most rapidly growing technique, Han and Huang [30] and
Park and Choi [31] recently pursued the feasibility of employing the
convolutional neural network (CNN) to predict the wall-normal velocities on
the detection plane using either spanwise or streamwise wall shear stress
obtained from the direct numerical simulation of channel flow. Applying the
trained CNNs to turbulent channel flows at low $Re$, significant amounts of
drag reduction can be achieved. In addition, Park and Choi [31] also found
that $15\%$ drag reduction can still be obtained at $Re_{\tau}=578$ by
applying the CNN trained at $Re_{\tau}=180$.

Currently, most of the above-mentioned drag control strategies are targeted at
interrupting the self-sustaining cycle [1, 3], where the low-speed streaks are
generated via the lift-up effect of faster advecting streamwise vortices
leaving streaks as their trails, while the instability/transient growth of
low-speed streaks – if sufficiently strong – regenerates the streamwise
vortices. At practical, high $Re$’s, these near-wall features are physically
very small; hence effective control of them becomes rather challenging due to
the tiny size of the appropriate sensors and actuators. In addition, as $Re$
increases, the role of outer large-scale structures becomes more important due
to their larger contribution to skin friction [32]. Hence, the efficacy of
these near-wall targeted methods decreases with increasing $Re$ [15, 23, 20,
33]. Samie  _et al._ [34] examined the coherence between a measurable wall
quantity (e.g., the wall-shear stress fluctuations) and the streamwise and
wall-normal velocity fluctuations in a turbulent boundary layer. They found
that the closed-loop drag reduction scheme targeting near-wall cycle of
streaks alone will be of limited success in practice as $Re$ grows. Therefore,
attempts have also been made to reduce drag by manipulating large-scale
coherent structures in the logarithmic and outer regions of turbulent boundary
layers. For example, Schoppa and Hussain [35] developed a conceptually simple
open-loop large-scale control strategy. By imposing large-scale counter-
rotating streamwise swirls with a relatively small excitation amplitude,
$20\%$ drag reduction was obtained in a turbulent channel at $Re_{\tau}\approx
100$. However, there is an ongoing debate over the effectiveness of the large-
scale control method with increasing $Re$. Canton  _et al._[36] found that the
drag reduction decreases with increasing $Re$ and becomes nearly zero at
$Re_{\tau}=550$. However, Yao  _et al._ [37] showed that the negative results
found by Canton  _et al._ [36] arose from their selection of the channel
center height as the fixed value for the location of the control swirl
centers. In addition, Yao  _et al._ [37] validated the large-scale drag
control concept for high $Re$’s using the near-wall spanwise opposed wall-jet
forcing (SOJF), and a detailed physical explanation on the drag reduction
mechanism is furnished in Ref. Note that SOJF introduce weak spanwise friction
and its mean is zero. [38]. Besides these numerical works, there are several
experimental studies on large-scale drag controls using either jet [39, 40] or
plasma actuation [41, 42].

Caution should be given to controlling large-scale (LSM) and very large-scale
(VLSM) motions, as recent studies show that suppression of these structures
does not yield significant drag reduction. For example, Abbassi  _et al._ [43]
performed an experimental investigation of closed-loop control of manipulating
LSMs and VLSMs, and only 3.2% drag reduction was achieved. This is because,
when suppressing these large-scale structures, smaller scales may also be
concomitantly altered due to nonlinear scale interaction [44, 32]. Hence, at
high $Re$, the simultaneous control of small- and large-scale structures may
be required to obtain effective drag reduction. Therefore, we propose a
composite drag control (CDC) strategy, where two or more different control
methods are combined together and applied simultaneously with the hope of
producing better performance. Through energy budget analysis, Chen  _et al._
[45] showed that both OC and SOJF can yield notable drag reductions, but with
different mechanisms. Their combination is potentially additive and may even
be synergistic – thus of interest and is examined in the present work.
Moreover, there are two additional considerations for performing such a study.
First, as OC and SOJF are effective in suppressing small- and large-scale
structures, respectively, their combination may yield a significant drag
reduction, even at high $Re$’s. Second, since both of the two methods require
small power inputs to drive the control, the net power saving might become
promising for CDC.

The rest of the paper is organized as follows. In the following section, the
governing equation, numerical scheme, and control method are described. The
drag reduction results are presented in section III. Flow statistics and
coherent structures are examined in section IV to elucidate the underlying
mechanism on the variation of skin friction under control. Finally, the
concluding remarks are provided in section V.

##  II Computational approach

###  II.1 Numerical method

Direct numerical simulations are performed in turbulent channel flows using
the code “POONPACK” developed by Lee and Moser [46]. In this study, $x$, $y$,
and $z$ denote the streamwise, wall-normal, and spanwise coordinates,
respectively; the corresponding velocity components are $u$, $v$, and $w$. The
incompressible Navier-Stokes equations are solved using the method of Kim  _et
al._ [47], in which equations for the wall-normal vorticity and the Laplacian
of the wall-normal velocity are time-advanced. This formulation has the
advantage of satisfying the continuity constraint exactly while eliminating
the pressure. A Fourier-Galerkin method is used in the streamwise and spanwise
directions, while the wall-normal direction is represented using a $7$th-order
B-spline collocation method. A low-storage implicit-explicit scheme based on
third-order Runge-Kutta for the nonlinear terms and Crank-Nicolson for the
viscous terms are used for time advance. The flow is driven by a pressure
gradient, which varies in time to ensure that the mass flux through the
channel remains constant. For more details about the code, refer to Lee and
Moser [46].

###  II.2 Control Schemes

Figure 1: (a) Schematic of composite drag control (CDC), which combines the
spanwise opposite wall jet forcing (SOJF) and the opposition control (OC); (b)
the detailed sketch of OC in the near-wall region [dashed box in (a)].

As mentioned in the previous section, CDC consists of OC and SOJF (Fig. 1).
For OC, the wall-normal velocity at the wall is set as

$\displaystyle v_{w}(x,z,t)=-A_{o}v(x,y_{d},z,t),$ (1)

where $A_{o}$ is the control amplitude and $v(y_{d})$ is the wall-normal
velocity at the detection plane $y_{d}$ (hereinafter, subscript $w$ represents
the value on the wall). Such a wall-velocity setting is imposed on both the
bottom and the top walls.

Following Yao  _et al._ [38], a spanwise forcing is applied for SOJF:

$\displaystyle F_{z}=A_{s}\sin(\beta z)g(y),$ (2)

where $A_{s}$ is the forcing amplitude, $\beta$ ($=2\pi/\lambda$) is the
spanwise wavenumber, and $g(y^{+})$ is a dimensionless forcing as a function
of $y$. Throughout the paper, all control parameters are normalized in viscous
units based on uncontrolled flow (indicated by the subscript $0$): a)
$A^{+}_{s}=A_{s}\nu/u^{3}_{\tau,0}$; b) $\lambda^{+}=2\pi/\beta^{+}$
($\beta^{+}=\beta\nu/u_{\tau,0}$); and c) $y^{+}_{c}=y_{c}u_{\tau,0}/\nu$. The
forcing function $g(y^{+})$ in Eq. (2) is $g(y^{+})=y^{+}\exp(-\eta y^{+2})$,
where the decay factor $\eta$ in the wall-normal direction specifies the wall-
jet velocity profile. The function $g$ has its maximum at
$y^{+}=y^{+}_{c}=1/\sqrt{2\eta}$, and $y^{+}_{c}$ represents the height of the
spanwise wall-jet maximum velocity. In Eq. (2), $g$ is normalized by its
maximum value $g(y^{+}_{c})=y^{+}_{c}\exp(-1/2)$. Note that SOJF is only
applied near the bottom wall, but due to its large-scale feature, it is also
found to be effective in reducing drag at the top wall.

###  II.3 Simulation parameters

Table 1: Details of the numerical discretization employed for the present simulations. The computational box size is $4\pi h\times 2h\times 2\pi h$ for all cases, and $N_{x}$, $N_{y}$ and $N_{z}$ are the number of grids in x, y, and z, respectively.  $Re_{\tau}$ | $Re_{b}$ | $N_{x}\times N_{y}\times N_{z}$ | $\Delta x^{+}$ | $\Delta y^{+}$ | $\Delta z^{+}$
---|---|---|---|---|---
$180$ | $2857$ | $256\times 192\times 256$ | $9.8$ | $0.11-4.7$ | $4.9$

DNS is conducted at a fixed bulk Reynolds number $Re_{b}=U_{b}h/\nu$, where
$U_{b}$ is the bulk velocity, and the corresponding $Re_{\tau}\approx 180$.
The domain sizes, grid sizes, and resolutions used in the simulations are
shown in Table 1. The adequacy of the domain size and resolution is confirmed
in Refs. [38, 45]. There are, respectively, three and two control parameters
for SOJF and OC – a total of five parameters, which make the search for
optimal control rather challenging. For simplicity, here we consider only one
critical control parameter for each method, namely, $A_{s}$ for SOJF, and
$y_{d}$ for OC. For other control parameters, we choose the optimal values
reported in previous works: namely, $\lambda=2\pi$ (or, equivalently,
$\lambda^{+}\approx 1000$) for simultaneous control of about ten streaks and
$y^{+}_{c}=30$ corresponding to the peak of Reynolds shear stress [38], and
$A_{o}=1$ [18, 28]. In the current study, $A^{+}_{s}$ varies between
$[0,2/3,4/3,2,8/3]\times 10^{-3}$ and $y^{+}_{d}=[0,5,10,15,20]$. All
computations start with the same initial condition of a fully developed
turbulent channel flow without control. Time averaging for statistics is taken
over at least 25 eddy turnover time (i.e., $tu_{\tau}/h\geq 25$) after the
initial transition.

Due to the presence of large-scale coherent swirls induced by the SOJF, the
total flow field can be divided into three components by using the triple
decomposition [48]:

$\displaystyle\mathbf{u}(x,y,z,t)=\mathbf{U}(y)+\underbrace{\tilde{\mathbf{u}}(y,z)+\mathbf{u}^{\prime\prime}(x,y,z,t)}_{\mathbf{u}^{\prime}}.$
(3)

Here, $\mathbf{u}$ is the instantaneous total velocity field;
$\mathbf{U}=\langle\overline{\mathbf{u}}\rangle$ is the mean velocity, where
the overbar indicates averaging in time and in the streamwise direction; and
the bracket $\langle\cdot\rangle$ indicates spanwise averaging over one
wavelength period; $\mathbf{u}^{\prime}=\mathbf{u}-\mathbf{U}$ is the total
fluctuation; $\tilde{\mathbf{u}}=\overline{\mathbf{u}}-\mathbf{U}$ is the so-
called “organized field” representing the coherent motion induced by the
swirls of SOJF, and
$\mathbf{u}^{\prime\prime}=\mathbf{u}-\overline{\mathbf{u}}$ is the random
turbulent fluctuation. Note that for OC only, $\tilde{\mathbf{u}}$ is absent,
and $\mathbf{u}^{\prime\prime}=\mathbf{u}^{\prime}$.

##  III Drag reduction and control efficiency

We define drag reduction as the relative change in the skin-friction drag
coefficient:

$\displaystyle\mathcal{R}=1-C_{f}/C_{f,0},$ (4)

where $C_{f}=2\tau_{w}/(\rho_{b}U^{2}_{b})$ and
$C_{f,0}=2\tau_{w,0}/(\rho_{b}U^{2}_{b})$ are the drag coefficients of the
controlled and reference (uncontrolled) cases, respectively.

Net power saving can be defined as a relative change in the power spent:

$\displaystyle\mathcal{N}=1-(P+W)/P_{0},$ (5)

where $P=u^{2}_{\tau}U_{b}/h$ and $P_{0}=u^{2}_{\tau,0}U_{b}/h$ are the
pumping power of the controlled and reference (uncontrolled) cases,
respectively; and $W$ is the power input due to control, which is given as
[45]

$W=\underbrace{\langle{\overline{wF_{z}}}\rangle}_{W_{s}}+\underbrace{\langle\overline{\rho
v^{3}_{w}/2}\rangle+\langle{\overline{p_{w}v_{w}}}\rangle}_{W_{o}}.$ (6)

Here, $F_{z}$ is the spanwise force given in Eq. 2, and $p_{w}$ and $v_{w}$
represent pressure and wall-normal velocity at the wall. $W_{s}$ and $W_{o}$
denote the power inputs to the fluid system by SOJF and OC, respectively. For
statistical equilibrium or steady state, the total power input is balanced by
total dissipation (i.e.,
$\epsilon=\nu\langle\overline{\partial_{j}u_{i}\partial_{j}u_{i}}\rangle$).
Hence, the net power saving rate can also be estimated as

$\mathcal{N}=1-\epsilon/\epsilon_{0},$ (7)

where $\epsilon$ and $\epsilon_{0}$ denote the dissipation of the controlled
and reference (uncontrolled) cases, respectively.

Note that Eq. 6 is the exact mathematical form of the control power input
[45]. Defined this way, $W$ can have locally negative values. To ensure that
any power input is always associated with energy consumption, several works
[49, 50] employ a more conservative form to estimate the power input by taking
the absolute values of these quantities before averaging. However, in this
work, we use this exact form, as it reflects the true energy flux in the flow
systems (see Sec.IV.4).

(a)

(b)

Figure 2: (a) Drag reduction $\mathcal{R}$ ($\%$) and (b) net power saving
$\mathcal{N}(\%)$ as functions of $A^{+}_{s}$ and $y^{+}_{d}$ for CDC at
$Re_{\tau}=180$. The symbol $+$ indicates simulation data points used for
interpolation.

Figures 2 (a,b) show the contour plots of drag reduction $\mathcal{R}$ and net
power saving $\mathcal{N}$ as functions of $A^{+}_{s}$ and $y^{+}_{d}$ at
$Re_{\tau}=180$. The contours are computed from a linear interpolation of
discrete simulated points (marked as $+$). Note that cases with $y^{+}_{d}=0$
($A^{+}_{s}=0$) corresponds to the SOJF (OC) only. In addition, figures 3(a,b)
show the variation of $\mathcal{R}$ and $\mathcal{N}$ as functions of
$A^{+}_{s}$ and $y^{+}_{d}$, respectively. It is clear from Fig. 2 (a) that
significant $\mathcal{R}$ can be achieved in broad ranges of $A^{+}_{s}$ and
$y^{+}_{d}$. For a given $A^{+}_{s}$, $\mathcal{R}$ first increases then
decreases with $y^{+}_{d}$, and the maximum $\mathcal{R}$ is always obtained
at $y^{+}_{d}=15$ – the optimal value for OC only. For a given $y^{+}_{d}$,
$\mathcal{R}$ also first increases then decreases with $A^{+}_{s}$. However,
different from $y^{+}_{d}$, the optimal $A^{+}_{s}$ changes and depends on
$y^{+}_{d}$. When $y^{+}_{d}=0$, the maximum $\mathcal{R}$ is achieved at
$A^{+}_{s}=0.0013$, which is very close to that reported in Ref. [38]. As
$y^{+}_{d}$ increases, the value of $A^{+}_{s}$ that produces maximum
$\mathcal{R}$ first decreases and then increases. Within the parameters
considered, the maximum $\mathcal{R}$ is $32.5\%$, with $y^{+}_{d}=15$ and
$A^{+}_{s}=0.0067$, and the uncertainty, estimated by following the procedure
in [23, 38], is approximately $0.5\%$. Note that as only the variations of
$y^{+}_{d}$ and $A^{+}_{s}$ are considered here, a higher $\mathcal{R}$ could
possibly be found by varying other control parameters. This is not pursued
here due to limited computational resources. It is also worth mentioning that
the current studies are performed under the constant flow rate assumption,
and, at this low $Re$, the actually drag reduction amount would be slightly
different if the constant pressure gradient assumption is employed [23].
However, the overall behavior of $\mathcal{R}$ as shown in Fig. 2a should
remain the same.

(a)

(b)

Figure 3: Drag reduction $\mathcal{R}(\%)$ (solid lines) and net power saving
$\mathcal{N}(\%)$ (dashed lines) as a function of (a) $y^{+}_{d}$ and (b)
$A^{+}_{s}$.

Chen  _et al._ [45] showed that compared to other active control techniques,
such as spanwise wall oscillation, both OC and SOJF are very promising for net
power saving. For OC, the power input $W_{o}$ increases as $y^{+}_{d}$
increases due to high intensity of wall-normal velocity fluctuations [50]; for
SOJF, $W_{s}$ increases with $A^{+}_{s}$ due to stronger large-scale swirls
[38]. However, as both SOJF and OC controls require little power input,
notable net power saving can be achieved when combining them together. As
shown in Fig. 2, the contour plot of $\mathcal{N}$ is rather similar to those
of $\mathcal{R}$, particularly for small $y^{+}_{d}$ and $A^{+}_{s}$. Fig. 3
further shows that for a given $A^{+}_{s}$, $\mathcal{N}$ is only slightly
smaller than $\mathcal{R}$, and for a given $y^{+}_{d}$, $\mathcal{N}$ slowly
deviates from $\mathcal{R}$ with increasing $A^{+}_{s}$. For the optimally
controlled case (i.e., $y^{+}_{d}=15$ and $A^{+}_{s}=0.0067$), approximately
1% of pumping power is needed to actuate the flow control, which results in a
maximum $\mathcal{N}$ of $31.5\%$. Although this high net power saving is
impressive, some caution is in order as this value is based only on a simple
estimation of the actuation, where the actuator’s energy transfer efficiency
(e.g., that of plasma actuators) has not been taken into consideration.

In summary, with the limited parameter search, we demonstrate that the CDC
technique could yield a maximum of $\mathcal{R}=32.5\%$ and
$\mathcal{N}=31.5\%$ – much higher than that can be achieved by either OC or
SOJF. This is indeed a very promising finding.

##  IV Flow analysis

To understand the mechanisms of drag reduction, flow statistics as well as the
dynamics are examined in detail below for four cases: Case I, $A^{+}_{s}=0$
and $y^{+}_{d}=0$ (uncontrolled); Case II, $A^{+}_{s}=0$ and $y^{+}_{d}=15$
(optimal for OC only); Case III, $A^{+}_{s}=0.0133$ and $y^{+}_{d}=0$ (optimal
for SOJF only); and Case IV, $A^{+}_{s}=0.0067$ and $y^{+}_{d}=15$ (optimal
for CDC).

###  IV.1 Near-wall streaks and vortical structures

Figure 4 shows the $\lambda_{2}$ vortices [51] along with the low-speed
streaks at $y^{+}=15$. For Case I, the typical meandering low-speed streaky
structures are observed aligned along the streamwise direction, and numerous
slender vortices are distributed throughout the wall region, lying around the
interface between low- and high-speed streaks [figure 4(a)]. For Case II,
although the streaks still have typical meandering shapes, their strengths are
significantly weakened. Correspondingly, the generation of drag-inducing near-
wall streamwise vortices is also suppressed, with fewer vortices as expected.
Under SOJF, the basic near-wall streaks are destroyed [Figs. 4(c,d)], and only
a single wide meandering streak lies along the mid-span and extends the entire
flow length, as expected from the merger of the numerous background streaks
forced by the two wall-jets [38]. For Case III, most of the near-wall
streamwise vortices disappear, except near the central collision region, just
above the merged streak envelope. Consistent with the further suppression of
random turbulent fluctuations and associated dissipation, the vortices become
fewer for Case IV – clarifying the improved effectiveness of CDC. With OC
applied, the near-wall streaks become weak; hence less intensive large-scale
swirls are required to merge them together – the reason why the optimal
$A^{+}_{s}$ decreases from $0.015$ to $0.0067$ [Fig. 3(b)].

Figure 4:  $\lambda_{2}$ vortical structures associated with the streamwise
velocity fluctuations at $y/h=-0.92(y^{+}_{0}=15)$ on the bottom-half walls
for (a) Case I; (b) Case II; (c) Case III; and (d) Case IV.

###  IV.2 Flow Statistics

Figure 5(a) shows the mean streamwise velocity profile $U^{+}(=U/u_{\tau})$ in
outer units, with Figs. 5(b) and 5(c) showing the same quantity for the bottom
and top halves of the channel in the wall units. Compared to the uncontrolled
case (i.e., Case I), $U^{+}$ profile for Case II (corresponding to the optimal
OC case) in the near-wall region is almost unchanged, and $U^{+}$ in the log
and outer region shows a characteristic upward shift – similar to those
observed in Refs. [20, 50]. As the flow in the outer region is barely altered
by the OC, the upward shift is mainly attributed to a smaller $u_{\tau}$;
hence, its magnitude is directly linked to $\mathcal{R}$. For Case III
(corresponding to the optimal SOJF control case), $U^{+}$ profile is
significantly altered. In particular, due to the presence of large swirls
imposed by SOJF near the bottom wall, $U^{+}$ is no longer symmetric with
respect to the channel centerline, and the peak of $U^{+}$ is shifted towards
the bottom wall. For Case IV (corresponding to the optimal CDC case), $U^{+}$
resembles Case III, but with an upward shift.

(a)  (b)  (c)

Figure 5: Mean velocity $U(y)$ for (a) the whole channel, (b) bottom- and (c)
top-half channel.

Figures 6(a) and 6(b) show the cross-streamwise view ($y-z$ plane) of contours
of the streamwise velocity and the cross-plane velocity vectors of
$\overline{\mathbf{u}}$ for Case III and Case IV, respectively. As discussed
in Ref. [38], due to the lift-up effect of the counter-rotating control
swirls, a low-speed streak forms for Case III and extends up to the outer flow
region ($y/h>-0.5$). Near the wall, however, the streaks have dual peaks – due
to the secondary swirls caused by the no-slip wall. As for Case IV, the
topmost low-speed streak has a similar shape as Case III. However, due to the
presence of OC, the secondary swirls are suppressed; as a result, the dual
peaks of the near-wall streaks become less apparent. In addition, due to
smaller $A^{+}_{s}$, the primary counter-rotating swirls are also weaker than
that of Case III.

(a)  (b)

Figure 6: Cross-stream ($y-z$) plane view of the mean velocity field
($\overline{\mathbf{u}}$) with the contours indicating the mean streamwise
velocity (normalized by the bulk velocity $U_{b}$), and the vectors denote the
wall-normal and spanwise velocities for (a) Case III and (b) Case IV.

Figure 7 shows the root mean square (r.m.s) of total velocity fluctuations
($u^{\prime*}$, $v^{\prime*}$, and $w^{\prime*}$) and random velocity
fluctuations ($u^{\prime\prime*}$, $v^{\prime\prime*}$, and
$w^{\prime\prime*}$). [The superscript $*$ indicates non-dimensionalization by
the wall units of the controlled flow]. For Case II, OC establishes a virtual
wall as illustrated in Fig. 1b, which is typically located halfway between the
physical wall and the sensing plane $y^{+}_{d}$ [18, 20]. The location of the
virtual wall $y_{vw}$ can be reflected by a local minimum of
$v^{\prime*}_{rms}$. Turbulent intensities are significantly suppressed near
the virtual wall. In particular, $v^{\prime}_{rms}$ becomes almost zero at
$y=y_{vw}$, which inhabits the interaction between the near wall and the outer
core regions. As a result, $u^{\prime}_{rms}$ and $w^{\prime}_{rms}$ are
significantly suppressed below the virtual wall. Compared to the uncontrolled
case, the peak location moves away from the wall, consistent with the
observation in Refs. [18, 50]. For Case III, although all turbulence
intensities ($u^{\prime}_{rms}$, $v^{\prime}_{rms}$, and $w^{\prime}_{rms}$)
are suppressed in the near-wall region, they are significantly enhanced in the
outer region – mainly due to the increased momentum exchange caused by the
presence of large-scale control swirls. For random turbulence velocity
fluctuations ($u^{\prime\prime}_{rms}$, $v^{\prime\prime}_{rms}$, and
$w^{\prime\prime}_{rms}$), where the effects of large-scale swirls are
excluded, the peaks of them become smaller than those of the uncontrolled
case. However, they are still slightly larger than the uncontrolled case in
the core layer, mainly due to the advection of the large-scale control swirls,
which transport the near-wall turbulence and vortical structures to the outer
flow. For Case IV, the behaviors of all turbulence intensities are similar to
Case III, but with smaller magnitudes – suggesting a better suppression of
near-wall streamwise vortices.

(a)  (b)  (c)

Figure 7: Root mean square of velocity fluctuations: (a) $u^{\prime}_{rms}$;
(b) $v^{\prime}_{rms}$; (c) $w^{\prime}_{rms}$ . Note that the dashed lines
are $u^{\prime\prime}_{rms}$ and $v^{\prime\prime}_{rms}$ and
$w^{\prime\prime}_{rms}$ for Cases III and IV.

###  IV.3 Skin friction decomposition

The mean skin-friction $C_{f}$ can be decomposed into different physics-
informed contributions based on the mean and statistical turbulence quantities
across the wall layer. For example, the FIK (Fukagata, Iwamoto, Kasagi)
identity [52, 53] links $C_{f}$ with the total Reynolds shear stress
$\langle\overline{u^{\prime}v^{\prime}}\rangle$, which is given as

$\displaystyle C_{f}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{6}{Re_{b}}}_{C^{L}_{f,F}}+\underbrace{\frac{3}{U^{2}_{b}}\int^{1}_{-1}y\langle\overline{{u^{\prime}}{v^{\prime}}}\rangle\mathrm{d}y}_{C^{T}_{T,F}}.$
(8)

The drag coefficient $C_{f}$ thus has a laminar part $C^{L}_{f,F}=6/{Re_{b}}$,
which is the drag for laminar flow with the same flow rate, and a turbulent
part $C^{T}_{f,F}$, which is represented by the weighted integration of the
total Reynolds shear stress (RSS)
$\langle\overline{{u^{\prime}}{v^{\prime}}}\rangle$. As the latter is composed
of a coherent $\langle{\tilde{u}\tilde{v}}\rangle$ and a random part
$\langle{\overline{{u^{\prime\prime}}{v^{\prime\prime}}}}\rangle$, Eq. 8 can
be further decomposed as

$\displaystyle C_{f}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{6}{Re_{b}}}_{C^{L}_{f,F}}+\underbrace{\frac{3}{U^{2}_{b}}\int^{1}_{-1}y\langle{\tilde{u}\tilde{v}}\rangle\mathrm{d}y}_{C^{C}_{f,F}}+\underbrace{\frac{3}{U^{2}_{b}}\int^{1}_{-1}y\langle\overline{{u^{\prime\prime}}{v^{\prime\prime}}}\rangle\mathrm{d}y}_{C^{R}_{f,F}}.$
(9)

(a)  (b)

Figure 8:  (a) Total and (b) decomposed weighted Reynolds shear stress (WRSS)
distributions for different control cases. The symbols  and  in (b) represent
the coherent and random weighted Reynolds shear stress, respectively.

Since $Re_{b}$ is held constant for all cases, only changes in the coherent
$C^{C}_{f,F}$ and random $C^{R}_{f,F}$ contributions affect the drag. Figure
8(a) shows the profiles of the total weighted Reynolds shear stress (WRSS)
$-(1-y)\langle{\overline{{u^{\prime}}{v^{\prime}}}}\rangle$, and figure 8(b)
shows the corresponding coherent part
$-(1-y)\langle{\tilde{u}\tilde{v}}\rangle$ (denoted as C-WRSS) and the random
part $-(1-y)\langle{\overline{{u^{\prime\prime}}{v^{\prime\prime}}}}\rangle$
(denoted as R-WRSS). Following Eq. (8), for all controlled cases, the drag
reduction is mainly due to suppression of Reynolds shear stress in the near-
wall region. For Case II, the weighted Reynolds shear stress (WRSS) becomes
almost zero at the location of the virtual wall. For Case III, although SOJF
is applied only near the bottom wall, the WRSS is also notably decreased near
the top wall. Consistent with a larger $\mathcal{R}$, WRSS for Case IV is
significantly suppressed compared to the uncontrolled case, particularly in
the bottom half of the channel. Furthermore, from figure 8(b), C-WRSS
dominates R-WRSS to become the main contributor to WRSS in the bottom half
channel, but the trend reverses in the top half channel. Interestingly, the
main difference between Cases III and IV is on C-WRSS. It suggests that when
compared to the SOJF, the better performance of CDC is mainly due to smaller
Reynolds shear stress associated with the weakened large-scale control swirls.

Figure 9(a) shows the decomposed total skin friction (normalized by $C_{f,0}$)
based on the FIK identity. For the uncontrolled case (Case I), the laminar
($C^{L}_{f,F}$) and turbulent ($C^{T}_{f,F}$) parts contribute 25.8 % and 74.2
% to the total $C_{f,0}$, respectively. For Case II, $C^{T}_{f,F}$ decreases
to approximately 51.1% of $C_{f,0}$. For Case III, the random part
$C^{R}_{f,F}$ contributes only about 24.9% of $C_{f,0}$, while the coherent
part $C^{C}_{f,F}$ becomes the dominant contribution, about $32.1\%$. When
compared to Case III, both $C^{R}_{f,F}$ and $C^{C}_{f,F}$ are further
decreased for Case IV, contributing about $19.0\%$ and $22.7\%$, respectively.

Figure 9: The skin friction decomposition based on (a) FIK and (b) RD
identities for different cases.

The FIK identity are often criticized due to a lack of physical interpretation
for the linear weight of Reynolds shear stress. Renard and Deck [54] proposed
an alternative mean skin friction decomposition, referred to as the RD
identity. It is derived from the mean streamwise kinetic-energy equation in an
absolute reference frame and characterizes the power of skin-friction as an
energy transfer from the wall to the fluid by means of dissipation by
molecular viscosity and turbulent production. In the case of turbulent channel
flow, the RD identity can be written as

$\displaystyle C_{f}$ $\displaystyle=$
$\displaystyle\underbrace{\frac{1}{U^{3}_{b}}\int^{1}_{-1}\nu\left(\frac{\partial
U}{\partial
y}\right)^{2}\mathrm{d}y}_{C^{M}_{f,R}}+\underbrace{\frac{1}{U^{3}_{b}}\int^{1}_{-1}\langle\overline{-{u^{\prime}}{v^{\prime}}}\rangle\frac{\partial
U}{\partial y}\mathrm{d}y}_{C^{T}_{f,R}},$ (10) $\displaystyle=$
$\displaystyle\underbrace{\frac{1}{U^{3}_{b}}\int^{1}_{-1}\nu\left(\frac{\partial
U}{\partial
y}\right)^{2}\mathrm{d}y}_{C^{M}_{f,R}}+\underbrace{\frac{1}{U^{3}_{b}}\int^{1}_{-1}\langle{-\tilde{u}\tilde{v}}\rangle\frac{\partial
U}{\partial
y}\mathrm{d}y}_{C^{C}_{f,R}}+\underbrace{\frac{1}{U^{3}_{b}}\int^{1}_{-1}\langle\overline{-{u^{\prime\prime}}{v^{\prime\prime}}}\rangle\frac{\partial
U}{\partial y}\mathrm{d}y}_{C^{R}_{f,R}},$ (11)

where $C^{M}_{f,R}$ represents the contribution from direct molecular viscous
dissipation and $C^{T}_{f,R}$ characterizes the contribution associated with
the turbulent-kinematic-energy (TKE) production
$\langle-u^{\prime}v^{\prime}\rangle(\partial U/\partial y)$. Similarly, the
turbulent contribution can be further separated into coherent ($C^{C}_{f,R}$)
and random ($C^{R}_{f,R}$) parts, which are, respectively, associated with
coherent and random Reynolds shear stresses.

Figure 9(b) shows the decomposed total skin friction (normalized by $C_{f,0}$)
based on RD identity. For the uncontrolled case (Case I), $C^{M}_{f,R}$
overwhelm $C^{T}_{f,R}$ at this low $Re$, contributing 59.9 % of $C_{f,0}$.
Under OC, both $C^{M}_{f,R}$ and $C^{T}_{f,R}$ components are decreased to
51.8% and 25.2%, respectively. For case III, although the $C^{T}_{f,R}$ is
almost the same as Case II, the random contribution $C^{R}_{f,R}$ is only
16.5% – the rest is due to the coherent component $C^{C}_{f,R}$ that
associated with production of the large-scale control swirls. As discussed in
Ref. [38], effective drag reduction by SOJF results from a compensative effect
of suppressing random turbulence countered by enhancing the coherent part
induced by the forcing. For Case IV, $C^{M}_{f,R}$ is similar to Case II, but
$C^{R}_{f,R}$ is notably decreased. Compared to Case III, all three
contributions are decreased, especially for $C^{C}_{C,R}$ due to smaller
control amplitude. It suggests that, with the aid of OC, SOJF becomes more
effective due to the decreased counter effect caused by the large-scale
coherent swirls.

###  IV.4 Energy flux analysis

Figure 10: Energy flux box by (a) Reynolds decomposition and (b) triple
decomposition. Abbreviations MKE, TKE, CKE and RKE represent mean, turbulent,
coherent, and random kinetic energies, respectively. All the budget terms have
been normalized by the pumping power of the uncontrolled case, and their
numerical values for Cases I to IV are presented in order.

For a flow system at statistic equilibrium, the rate at which energy enters
the system (e.g., pumping power $P$ and control power input $W$) must equal
the rate at which energy is dissipated (i.e., the viscous dissipation
$\epsilon$). When employing the Reynolds decomposition, the kinetic energy of
the flow can be separated into the kinetic energy of the mean (MKE) and
turbulent fields (TKE). In this section, we examine how the energy transfer
between the mean flow and the turbulent fluctuations is affected by control.

As suggested by Ricco  _et al._ [55], the energy flux can be compactly
described using the so-called energy box, sketched in Figure 10a. The pumping
power is merely associated with the mean flow, while the energy dissipation is
split into parts corresponding to the mean ($\epsilon_{M}$) and turbulent
($\epsilon_{f}$) fields. In addition, the energy transfer from the mean to the
fluctuating field is embodied by the production of turbulent kinetic energy
$P_{M}$. $P_{M}$ acts as a sink for MKE, but as a source for TKE, which is
equal to the sum of $\epsilon_{f}$ and $W$. Hence, in contrast to RD identity,
Chen  _et al._ [45] provided an alternative expression for $C_{f}$ in terms of
dissipation and control power input, i.e.,
$C_{f}=(\epsilon_{M}+\epsilon_{f}-W)/U^{3}_{b}$. Similar to that shown in Fig.
9b, for the uncontrolled case, 59.9% and 40.1% of the energy that enters the
system (through pumping power) are dissipated by the mean and turbulent
fields, respectively. With control, the pumping power is decreased and the
amount of decrease corresponds to the rate of drag reduction $\mathcal{R}$
presented in Sec. III. Furthermore, the change of total dissipation
($\epsilon_{M}+\epsilon_{f}$) is the rate of net power saving $\mathcal{N}$
[45].

For all the control cases, the dissipation due to the mean flow fields
$\epsilon_{M}$ does not change much, which contributes slightly above 50% of
$P_{0}$. As a reference, the dissipation due to the laminar mean flow is about
one-fourth of $P_{0}$ [56]. Unlike $\epsilon_{M}$, the dissipation due to the
turbulent field $\epsilon_{f}$ vastly decreases under control – suggesting
significant suppression of near-wall random streamwise vortices. It is of
particular interest to look at Case IV, where the control power input
$W(=0.8\%)$ is in between Case II ($W=0.3\%$) and Case III($W=1.5\%$), but
$\epsilon_{f}$ is much smaller than these two cases.

To better understand how control flow affects the energy transfer, Gatti  _et
al._ [56] developed an elaborated energy flux analysis by separating the mean
flow into a laminar component and a deviation from it.  Although it provides
some further insight on the effect of control on energy flux, it lacks a
physical basis to separate the mean flow into laminar mean and the remaining
parts. Differently, Chen  _et al._ [45] introduced a triple-decomposition (eq.
3) into the energy budget analysis, which enables us to examine how controls
affect the energy transfer among the mean, coherent, and turbulent fields. The
corresponding energy box is shown in Fig. 10b. For flow under control, the
control power can be injected either through random velocity field
$W_{{}_{R}}$ (e.g., for OC) or through coherent velocity field $W_{{}_{C}}$
(e.g., for SOJF). Meanwhile, production converts mean flow energy to both
coherent ($P_{MC}$) and random ($P_{MR}$) velocity fields. There is an
additional production ($P_{CR}$) associated with the energy transfer from
coherent to the random velocity field. Note that similar energy budget
analysis has recently been employed by Alessio  _et al._ [57] to understand
drag reduction mechanisms of lubricated turbulent channel flows.

As the coherent part is absent for Cases I and II, the analysis is equivalent
to that based on Reynolds decomposition. For both Cases III and IV, the
dissipation due to coherent flow field
$\epsilon_{C}(=\langle\overline{\nu\partial_{i}\tilde{u}_{j}\partial_{i}\tilde{u}_{j}}\rangle)$
is rather small. For Case III, although the majority of the turbulent
production $P_{M}$ directly goes to the random turbulent field, around 40% of
$P_{M}$ is injected into the coherent field, most of which, together with the
coherent control power input $W_{C}$, is eventually transferred to the random
fields and is dissipated as
$\epsilon_{R}(=\langle\overline{\nu\partial_{i}u^{\prime\prime}_{j}\partial_{i}u^{\prime\prime}_{j}}\rangle)$.
For Case IV, due to a smaller forcing amplitude $A^{+}_{s}$, the large-scale
control swirls generated are weaker than Case III. As a consequence, the
production and dissipation associated with that are significantly decreased.
In addition, although the detecting plane $y^{+}_{d}$ is kept the same, the
random control power input $W_{R}$ is decreased compared to Case II. This is
not unexpected as the large-scale swirls weaken the low-speed streaks and
suppress the random turbulent fluctuations.

##  V Discussion and Conclusion

Composite drag control (CDC) based on the opposition control (OC) and spanwise
opposed wall-jet forcing (SOJF) methods is investigated via direct numerical
simulation of the incompressible Navier–Stokes equations in a turbulent
channel flow. A limited parameter search at $Re_{\tau}=180$ shows that a
maximum $\mathcal{R}$ of $32\%$ can be obtained with $y^{+}_{d}=15$ and
$A^{+}_{s}=0.0067$, which is much higher than that can be achieved by each
individual method. Furthermore, due to the small power input required,
significant net power saving can be achieved. Flow analysis, including skin
friction decomposition and flow visualization, reveals that CDC can take
advantage of both OC and SOJF methods. In particular, compared to OC, it is
more effective in suppressing the random turbulence due to the presence of
large-scale coherent swirls generated by SOJF. In addition, compared to SOJF,
it requires weaker large-scale coherent swirls due to the role of OC in
suppressing random turbulence – hence decreasing the drag contribution
associated with these swirls.

The imposition of SOJF improves the coherence between the measurable wall
quantities and the velocity fluctuations in the flow. This also indicates that
this control strategy could also yield better performance for more practical
wall-quantity-based drag control methods, such as the sub-optimal control and
those based on machine learning. For most of the drag control methods, the
performance decreases as $Re$ increases, mainly due to the increased
contributions of large-scale and very-large-scale structures [58]. As OC and
SOJF are designed to suppress small- and large-scale structures, respectively,
this CDC strategy might be employed as one possible way to overcome the
deterioration of drag reduction performance at high $Re$. All these questions
deserve further investigation.

###### Acknowledgements.

Computational resources provided by Texas Tech University HPCC, TACC Lonestar
are acknowledged. We appreciate Lee and Moser for providing us their DNS code,
which is used for this study. X.C. appreciates the funding support by the
National Natural Science Foundation of China No. 12072012, 11721202, 91952302.

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# New physics model constraints derived from SME coefficient limits using
IceCube astrophysical neutrino flavour data

Carlos. A. Argüelles1 Kareem Farrag2 and Teppei Katori3
1Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, Massachusetts 02138, USA  2Department of
Physics and Institute for Global Prominent Research,
Chiba University, Chiba 263-8522, Japan  3Department of Physics, King’s
College London, London WC2R 2LS, UK

###### Abstract

The IceCube collaboration has set stringent limits on neutrino sector
isotropic SME coefficients through the measurement of the astrophysical
neutrino flavor data. We investigate the consequences of these limits on
various new physics models.

## Introduction

Recently, the IceCube collaboration performed the first search of Lorentz
violation [1, 2] using the high-energy starting event astrophysical neutrino
sample [3, 4]. Astrophysical neutrinos mix, forming the largest neutrino
interferometers [5, 6] as they have the highest energy
($>10~{}$\mathrm{TeV}$$) and the longest uninterrupted trajectory ($\sim$1
Mpc) among all particles. Astrophysical neutrino flavor exhibit a strong
sensitivity to new physics [7] including Lorentz violation [8].

Although IceCube data have shown no sign of new physics, they set stringent
limits to the neutrino sector isotropic SME coefficients [9]. Limits on these
SME coefficients are defined where the Bayes Factor, which compares models
with versus without Lorentz violation, is $>10$, corresponding to a likelihood
with a one in ten chance assuming equal priors. In particular, limits of these
SME coefficients exceed naïve Planck scale physics expectations for non-
renormalizable dimension-five ($<E_{P}^{-1}$) and dimension-six
($<E_{P}^{-2}$) operators, where $E_{P}\sim 10^{19}$ GeV is the Planck energy.
At present, many IceCube limits depend on the assumed astrophysical neutrino
production model. Re-analysis with higher statistics data and an improved
flavor identification algorithm could remove such dependencies in the near
future.

Model-independent limits are obtained for the flavor-diagonal $\tau-\tau$
elements. For the dimension-three SME coefficient case, the effective
Hamiltonian can be explicitly written in the flavor basis as

$\displaystyle\begin{aligned}
H_{eff}\sim\frac{1}{2E}\cdot\left(\begin{array}[]{ccc}m^{2}_{ee}&m^{2}_{e\mu}&m^{2}_{\tau
e}\\\ m^{2*}_{e\mu}&m^{2}_{\mu\mu}&m^{2}_{\mu\tau}\\\ m^{2*}_{\tau
e}&m^{2*}_{\mu\tau}&m^{2}_{\tau\tau}\end{array}\right)+\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\
0&0&\accentset{\circ}{a}^{(3)}_{\tau\tau}\end{array}\right)=V^{\dagger}diag(\lambda_{1},\lambda_{2},\lambda_{3})V~{}.\end{aligned}$

Here, the first term is the standard neutrino mass matrix, and the second
matrix represents a dimension-three isotropic SME coefficient, where we set
only $\tau-\tau$ element to be nonzero. The sum of these two matrices is then
diagonalized, and the unitary matrix $V$ controls how astrophysical neutrino
flavor is modified through propagation. Since the mass matrix has off-diagonal
terms, flavor diagonal new physics operators also modify the astrophysical
neutrino flavors at detection. This is a type of Quantum Zeno effect [10].
Large nonzero $\tau-\tau$ element is equivalent to large $e-e$ and $\mu-\mu$
elements. This results in the electron neutrino and muon neutrino flavor
states to be locked throughout their propagation. Since the standard
astrophysical neutrino production models include only electron and muon
neutrinos, this means large nonzero $\accentset{\circ}{a}^{(d)}_{\tau\tau}$
and $\accentset{\circ}{c}^{(d)}_{\tau\tau}$ elements forbid flavor mixing of
astrophysical neutrinos, contrary to the current data whose best fit requires
flavor mixing.

A wide variety of models can produce non-standard flavor mixing. For example,
interactions between neutrinos and inter- or extra-galactic space can affect
astrophysical neutrino flavor, for both flavour diagonal and off-diagonal new
physics. Here, we investigate new physics models listed in the Snowmass21
white paper, “Beyond the Standard Model effects on Neutrino Flavor” [11].
Below, we describe how some of these new physics models can be tested with the
IceCube SME limits. For simplicity, we focus on the real positive part of the
dimension-three limit, $\Re(\accentset{\circ}{a}^{(3)}_{\tau\tau})<2\times
10^{-26}~{}$\mathrm{GeV}$$.

## New physics models

Neutrino magnetic moment — Neutrino magnetic moments, $\mu_{\nu}$, is order
$\sim 10^{-19}\mu_{B}$ in the Standard Model ($\mu_{B}$ is Bohr magneton).
Neutrino interactions with galactic magnetic field ($\sim 6\mu$G) make a
potential term $\mu_{\nu}\cdot B$ with order $\sim 10^{-42}$ GeV, which is far
below the limit the IceCube obtains from the astrophysical neutrino flavour.
On the other hand, current lab limit of the neutrino magnetic moment is order
$10^{-11}\mu_{B}$ from the GEMMA reactor neutrino experiment [12] and the
Borexino solar neutrino experiment [13]. If the order $10^{-12}\mu_{B}$
neutrino magnetic moment exists due to new physics processes, this can make a
potential term with $\sim 10^{-35}$ GeV. This is still many order smaller than
the current SME limit from the IceCube. Thus, any new electromagnetic
interactions on neutrinos are unlikely to contribute astrophysical neutrino
flavour mixings.

Neutrino self-interaction — The Universe is filled by the cosmic neutrino
background. These neutrinos produce a matter potential if there is a sizable
lepton asymmetry [14]. For the standard cosmological history, neutrino
temperature is 1.95 K and if we assume lepton asymmetry is $\sim 0.01$, the
matter potential derived from the cosmic neutrino background is $\sim
10^{-46}$ GeV, far below reachable value by astrophysical neutrino flavour
measurement. However, neutrinos may have new interactions between neutrinos,
called secret self-interactions by exchanging new scalar $\phi$ or vector
$\phi_{\mu}$ mediators [15]. If neutrinos have secret interactions like
$g_{\alpha\beta}\phi\nu_{\alpha}\nu_{\beta}$ or
$g_{\alpha\beta}\phi_{\mu}\nu_{\alpha}^{\dagger}\bar{\sigma}^{\mu}\nu_{\beta}$,
interactions between astrophysical neutrinos and cosmic neutrino background
may be enhanced. However, the matter potential or mass correction made by such
interactions seem negligible [16] because the number density of cosmic
neutrino is low. This means, astrophysical neutrino flavor is unlikely
affected by secret interactions.

Neutrino - Dark matter interaction — Dark matter is an optical invisible
substance which consists around 85% of all matter in the universe. There are
many candidates of dark matter if they are new particles. The axion-like
particles (ALPs) are a class of models with new light boson particles as dark
matter, but the mass is much lower than tradional weakly interacting massive
particles (WIMPs), lower than 0.01 eV to as low as $10^{-24}$ eV. In this low
mass region, ALPs behaves like a field $a$ [17], and it may couple with
neutrinos [18],
$g_{\alpha\beta}(\partial_{\mu}a)\bar{\nu}_{\alpha}\gamma^{\mu}\gamma_{5}\nu_{\beta}$.
By assuming local dark matter density $\rho_{DM}=0.3$ GeV/cm3, tau neutrino-
ALPs coupling $g_{\tau\tau}$ is limited order $10^{-5}$ GeV-1 from the IceCube
flavor data. This seems a strong limit compared with current and future
experimental limits. There are many different types of light dark matter [19]
and it may be possible to set limits on other light dark matter candidates
too.

Neutrino - Dark energy interaction — Dark energy is a missing energy density
which consists around 70% of all energy density in the universe. One candidate
uses a dynamic scalar field called quintessence $\phi$ which may couple with
neutrinos with coupling constant $\lambda_{\alpha\beta}$ [20]. Authors [21]
evaluate the effect of such coupling in terms of SME coefficients,
$\accentset{\circ}{a}^{(3)}_{\tau\tau}\sim\lambda_{\tau\tau}\dot{\phi}/M_{*}$,
where $M_{*}$ is the energy scale of the interaction. We are investigating the
implication of our SME limit
$\Re(\accentset{\circ}{a}^{(3)}_{\tau\tau})<2\times 10^{-26}~{}$\mathrm{GeV}$$
on quintessence.

Long range force — Neutrinos may feel a potential through unknown long-range
force. Such interaction can be speculated between neutrinos and any particles,
including electrons [22]. Assuming long-range Yukawa type coupling between tau
neutrinos and electrons, the potential can be written by the mediator mass
$m_{e\tau}^{\prime}$ coupling constant $g_{e\tau}^{\prime}$, electron number
$N_{e}$ and distance $d$, $V_{e\tau}={g_{e\tau}^{\prime}}^{2}N_{e}(4\pi
d)^{-1}e^{-d/m_{e\tau}^{\prime}}$. Very small coupling constant can be
explored through long range force between tau neutrinos and all electrons in
the visible universe. We will address the limits on long range force in
further study.

Acknowledgments — Authors thank to organizers of CPT’22 meeting to hold this
conference on online.

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