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

ON THE STREAMWISE WALL-SHEAR FLUCTUATIONS GENERATED BY
ATTACHED EDDIES IN A TURBULENT CHANNEL FLOW

Cheng Cheng
Department of Mechanical and Aerospace Engineering
The Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
ccust@ust.hk

Lin FuDepartment of Mechanical and Aerospace Engineering
Department of Mathematics
Center for Ocean Research in Hong Kong and Macau (CORE)
The Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
Shenzhen Research Institute, The Hong Kong University of Science and Technology, Shenzhen, China
linfu@ust.hk

ABSTRACT

large-scale motions (VLSM) exert influences on the near-wall
turbulence. The former refers to the footprints of LSMs and
VLSMs on the near-wall turbulence, while the latter indicatesthe intensity amplification or attenuation of near-wall smallscale turbulence by the outer motions. Mathis et al. (2013)
extended the model to interpret the generation of wall-shear
stress fluctuations in high-Reynolds number flows. They emphasized that superposition and modulation are still two essential factors. This inner-outer interaction model (IOIM) has also
been successfully developed to predict the near-wall velocity
fluctuations with data inputs from the log layer (Marusic et al.,2010; Baars et al., 2016; Wang et al., 2021).

A growing number of studies suggest that the generationof wall-shear fluctuations is actively linked with the attachededdies populating the logarithmic region. In the present study,
we investigate the statistical properties of the streamwise wallshear fluctuations (τx′ ) generated by attached eddies in a turbulent channel flow at Reτ ≈ 2000. To this end, the superposition components of τx′ , which are correlated with logarithmic
motions are identified by adopting an inner-outer predictivemodel. The momentum generation functions carried by them
are calculated and compared with the scaling law predicted by
the attached-eddy model. Our results further verify that thegeneration of τx′ by attached eddies approximately follows anadditive process.

On the other hand, the most elegant conceptual model describing these energy-containing motions is the attached-eddy
model (Townsend, 1976). It hypothesizes that the logarithmic region is occupied by an array of randomly-distributedand self-similar energy-containing motions (or eddies) with
their roots attached to the near-wall region (see Fig. 1). During the recent decades, a growing body of evidence that supports the attached-eddy hypothesis has emerged rapidly, e.g.,
Hwang (2015), Hwang & Sung (2018), Hwang et al. (2020),
to name a few. The reader is referred to a recent review workby Marusic & Monty (2019) for more details. Throughout
the paper, the terms ‘eddy’ and ‘motion’ are exchangeable.
It should be noted that the terms of ‘wall-attached motions’and ‘wall-attached eddies’ used in the present study do not
only refer to the self-similar eddies in the logarithmic region,
but also the very-large-scale motions (VLSMs) or superstructures, as some recent studies have shown that VLSMs are also
wall-attached, despite that their physical characteristics do notmatch the attached-eddy model (Hwang & Sung, 2018; Yoonet al., 2020).

INTRODUCTION
Wall-shear stress fluctuation is a crucial physical quantity
in wall-bounded turbulence, as it is of importance for noiseradiation, structural vibration, drag generation, and wall heattransfer, among others (Diaz-Daniel et al., 2017; Cheng et al.,2020). In the past two decades, ample evidence has shown thatthe root mean squared value of streamwise wall-shear stress′fluctuations (τx,rms ) is sensitive to the flow Reynolds number(Abe et al., 2004; Schlatter & Örlü, 2010; Yang & LozanoDurán, 2017; Guerrero et al., 2020). It indicates that largescale energy-containing eddies populating the logarithmic andouter regions in high-Reynolds-number wall turbulence have
non-negligible influences on the near-wall turbulence dynamics, and thus the wall friction (de Giovanetti et al., 2016; Liet al., 2019).
Till now, several models have been proposed on the organization of motions in logarithmic and outer regions and theirinteractions with the near-wall dynamics. Marusic et al. (2010)
have established that superposition and modulation are the twobasic mechanisms that large-scale motions (LSM) and very-

Previous study (Yang & Lozano-Durán, 2017) verified
that the generation of wall-shear stress fluctuations can be interpreted as the outcomes of the momentum cascade acrossmomentum-carried eddies of different scales, and modeled by1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
y+s , the lower bound of logarithmic region) to 0.2Reτ (denoted
as y+e , the upper boundary of logarithmic region), i.e. the spanning of the logarithmic region (Jiménez, 2018). Once uL is obtained, the streamwsie wall-shear fluctuation deduced by thesuperposition can be calculated by definition (i.e. ν ∂∂uyL at thean additive process. Here, we first aim to couple the additive
description with the AEM to portray the generation processof streamwise wall-shear fluctuations, resulting from wallattached eddies. Then, we intend to isolate the streamwise
wall-shear stress fluctuations generated by attached eddies ina turbulent channel flow at Reτ = 2003 (Reτ = huτ /ν, h denotes the channel half-height, uτ the wall friction velocity andν the kinematic viscosity) by resorting to the IOIM (Marusic
et al., 2010; Baars et al., 2016). Direct comparison between
the statistics from these two models will be conducted to verify the consistency.

′wall). We denote superposition component of τx+ estimated′+ +from y+o as τx,L (yo ). According to the attached-eddy model,
′

′

+ +
+ +
τx,L
(ys ) − τx,L(yo ) can be interpreted as the streamwsie wallshear fluctuations generated by the wall-attached motions with+wall-normal heights within y+s and yo , see Fig. 1.

Scaling law predicted by attached-eddy model
DNS databasemethodandscale

decomposition

According to Yang & Lozano-Durán (2017), the wallstress fluctuation can be modeled as an additive process. Similarly, the streamwise wall-shear fluctuations generated by attached eddies within yo and ys can be viewed as an discreterandom contributions:

The DNS database adopted in the present study has been
extensively validated by Jiménez and co-workers (Hoyas &Jiménez, 2006; Lozano-Durán & Jiménez, 2014a). The caseat Reτ =2003 is used and named as Re2000. The data are provided by the Polytechnic University of Madrid. Details of theparameter settings are listed in Table 1.

′ReτLx (h)Ly (h)Lz (h)

NF

Re2000

2003

8π

2

3π

94

where ai are random additives, assumed to be identically and′independently distributed (i.i.d), representing τx+ associatedwith attached eddies at
a given wall-normal height. The numR
ber of addends is n ∝ 1y dy ∝ ln(yo /ys ). The momentum genDE′+ ) can be evaluated as
eration function exp(qτx,a
D
E
′
yo
+
exp(qτx,a) = ⟨exp(qa)⟩Nz ∼ ( )τq ,ys(4)

where q is a real number, τ(q) = C1 ln ⟨exp(qa)⟩ is called
anomalous exponent, C1 is an constant. (4) is called strong
self similarity (SSS). If a is a Gaussian variable, the anomalous exponent can be recast as

According to the inner-outer interaction model (Marusic
et al., 2010), the large-scale motions would exert the footprints on the near-wall region, i.e., the superposition effects.
Baars et al. (2016) demonstrated that this component (denoted′
as uL+ (x+ , y+ , z+ )) can be obtained by the spectral stochastic
estimation of the streamwise velocity fluctuation at the logarithmic region y+o , namely byτ(q) = C2 q2 ,(5)

where C2 is an another constant. On the other hand, an extended self-similarityis
D (ESS)
E defined
D to describe
E the rela′+ ) and exp(q τ ′ + ) (fixed q )tionship between exp(qτx,o0 x,o0(Benzi et al., 1993), i.e.,

n

 h ′
io
′
+
uL+ x+ , y+ , z+ = Fx−1 HL λx+ , y+ Fx uo+ x+ , y+
,
o ,z
(1)
′
where uo+ is the streamwise velocity fluctuation at y+o in thelogarithmic region, and, Fx and Fx−1 denote the FFT and theinverse FFT in the streamwise direction, respectively. HL isthe transfer kernel, which evaluates the correlation between′′+u + (y+ ) and uo+ (y+o ) at a given length scale λx , and can becalculated as
D
E

, z+
û′ λx+ , y+ , z+ û′ λx+ , y+o
HL λx+ , y+ = DE ,+ û′ λ + , y+ , z+û′ λx+ , y+
o ,z
x
o

(3)

i=1

Table 1. Parameter settings of the DNS database. Here,Lx , Ly and Lz are the sizes of the computational domain in
the streamwise, wall-normal and spanwise directions, respectively. NF indicates the number of instantaneous flow fields
used to accumulate statistics.

Case

n

+
τx,a
= ∑ ai ,D
E DEξ (q,q0 )′′++exp(qτx,o) = exp(q0 τx,o
)
,

(6)

where ξ (q, q0 ) is a function of q (fixed q0 ). Note that ESS
does not strictly rely on i.i.d of the addends, but the additive
process Eq. (3).
Considering that y+s is the lower bound of the logarithmic region, the increase of y+o in Eqs. (1) corresponds to theenlargement of the addends in the additive description (seeEq. (3)). In this way, the connection between AEM and IOIM
are established, and the AEM predictions (see Eqs. (4)-(6)) canbe verified directly.

(2)

Results and Discussion

where û′ is the Fourier coefficient of u′ , and û′ is the complex
conjugate of û′ .
In this paper, the main concern is the τx′ generated by attached eddies populating logarithmic region. Thus, only thesuperposition term is taken into account. The predicted position y+ is fixed at y+ = 0.3, and y+
0 varies from 100 (denoted as

To shed light on the generation mechanism of τx′ associated with attached eddy, we define a moment generating function based on the above-mentioned decomposition, i.e.
D
E
′
′
+ +
+ +
G(q, y+o ) = exp(q(τx,L (ys ) − τx,L (yo ))) .

2(7)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022+
Figure 1. A schematic of the attached-eddy model (Hwang, 2015). Each circle represents an individual attached eddy. y+s and ye are+++
the lower and upper bound of the logarithmic region, respectively. y0 is the outer reference height, and varies from ys to ye .

Figure 2. G as functions of yo /ys for q = ±5 and q = ±3.

Fig. 2 shows the variations of G as a function of yo /ys
for q = ±5 and q = ±3. Power-law behaviours can be foundin the interval between 1.7 ≤ yo /ys ≤ 2.9 for positive q and1.7 ≤ yo /ys ≤ 4 for negative q, justifying the validity of SSS,
i.e., Eq. (4). This observation highlights that the superpositionsof wall-attached log-region motions on wall surface follow theadditive process, characterized by Eq. (3). It is also worth
mentioning that the power-law behaviour can be observed forlarger wall-normal intervals for negative q. As G(q, y+o ) quan′′+ ++ +tifies τx,L(ys ) − τx,L(yo ), which features the same sign as q,
this observation is consistent with the work of Cheng et al.
(2020), which showed that the footprints of the inactive part of
attached eddies populating the logarithmic region are actively
connected with large-scale negative τx′ . Other q values yield
similar results and are not shown here for brevity.
The anomalous exponent s(q) can be obtained by fitting
the range 2 ≤ yo /ys ≤ 2.9, where both positive and negative q
display good power-law scalings. Fig. 3(a) displays the variation of the anomalous exponent s(q) as a function of q. The
solid line denotes the quadratic fit within −0.5 ≤ q ≤ 0.5.
It can be seen that the variation of s(q) is very close to themodel prediction, i.e., the quadratic function as Eq. (5) with
C2 = 0.00629. Only minor discrepancies between DNS data
and model predictions can be observed. As such, it is reasonable to hypothesize that the streamwise wall-shear stress fluctuation τx′ generated by attached eddies of a given size follows

Figure 3. (a) Anomalous exponent s(q) as a function of q.
The black line is a quadratic fit; (b) second- to sixth- order′moments of τx+ as functions of Reτ . The dashed lines are thelog-normal predictions from Eq. (8)-(10).

the Gaussian distribution. Moreover, we can also estimate the+ by taking the derivative of G(q, y+ )statistical moments of τx,oewith respect to q around q = 0 (Yang et al., 2016), i.e.,
D ′ E ∂ 2 G(q; y+ )
o
2+
τx,o
=
∼ 2C2 ln Reτ ,
∂ q2
q=0

3

(8)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
ESS scalings. Based on these observations, we conclude thatthe streamwise wall-shear stress fluctuations generated by attached eddies populating the logarithmic region can be treatedas Gaussian variables. Considering the fact that the intensity of wall-shear stress fluctuations is typically underpredictedby the state-of-the-art wall-modelled large-eddy simulation(WMLES) approaches (Park & Moin, 2016), the results proposed in the present study may be constructive for the development of the LES methodology.

ACKNOWLEDGMENT
We are grateful to the authors cited in Fig. 3(b) for making their invaluable data available. Lin Fu has been founded by
the fund from Guangdong Basic and Applied Basic Research
Foundation (No. 2022A1515011779), the fund from CORE asa joint research center for ocean research between QNLM andHKUST, and the fund from Shenzhen Municipal Central Government Guides Local Science and Technology Development
Special Funds Funded Projects (NO. 2021Szvup138).

Figure 4. G(q) as functions of G(−2) for q =−1, −3, −5, −7. Both vertical and horizontal axes in (a)are plotted in logarithmic form.
D ′ E1/2√∂ 4 G(q; y+o)4+)1/2 ∼ 2 3C2 ln Reτ ,
τx,o
=(
4
∂q
q=0

(9)

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the attached-eddy model.

CONCLUSIONSIn summary, the present study reveals that IOIM and
AEM are consistent to each other quantitatively. The statistical characteristics of the superpositions of log-region eddies follow the predictions of AEM, namely, the SSS and4

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