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

SPANWISE LARGE SCALE MOTIONS AND THEIR RELATION TO
ENERGY SPECTRUM IN TURBULENT SHEAR FLOWS

Yu Wang
Department of Energy Engineering and Science
Nagoya University
Furo-Cho, Chikusaku, Nagoya city, 464-8603, Japanwang.yu.k7@s.mail.nagoya-u.ac.jp

Yoshinobu Yamamoto
Department of Mechanical Engineering
University of Yamanashi
4-3-11, Takeda, Kofu 400-8511, Japan
yamamotoy@yamanashi.ac.jp

Yoshiyuki Tsuji
Department of Energy Engineering and Science
Nagoya University
Furo-Cho, Chikusaku, Nagoya city, 464-8603, Japan
c42406a@nucc.cc.nagoya-u.ac.jp

ABSTRACTThe velocity fluctuation is measured by the hot-wire operated
by constant temperature anemometer. It is attached to the traverse devise and the velocity distribution is obtained across thetest section. In the channel flow, the distance from the wall wasmeasured by 84 points from 0mm to 25mm. The hot-wire rake
made of 16 single I-probe was used in the measurement. I-typeprobe is made of tungsten wire with the sensor length, and its
diameter is φ = 5µm. The probes are arranged in the spanwise direction, and the spacing between each probe is 4mm.
The total span-wise extent of the probes is l = 60mm, and
it is approximately 2.4δ (δ is the channel half wall-height).
The hot-wire origin was placed slightly away from the wall
not to damage the wire due to the interaction with the wall.
The sampling frequency is 20 kHz, and the data are recordedfor 60 seconds at each measurement position. In this paper,
we compare the results of experimental and numerical simulations. The experimental data were obtained from measurements with a hot-wire anemometer, the details of which aresummarized in Table 1.
In this paper, the coordinates x, y, and z present thestream-wise, span-wise, and wall-normal direction, respectively. In channel, we defined that the boundary layer thickness is regarded as channel’s half height (that is, δ = h). The
upper superscript + indicates a non-dimensional values suchas wavelength λ + = λx uτ /ν and velocities u+ = u/uτ , whereν is the kinematic viscosity.

Studies on the structure of turbulent boundary layer haveattracted a significant amount of attention in the last few
decades, it has been recognized that wall turbulence is composed of certain recurring and quantifiable features collectively referred to as ‘coherent structures’. Kline et al. (1967)
observed the presence of streak structures in the viscous sublayer by flow field visualization. And recently, with the development of fluid measurement technology and fluid simulation
technology, there are many studies on the large-scale structure
in boundary layer (Kim & Adrian (1999); Ganapathisubramani
et al. (2003)). And there has been an increasing interest in thestatistical analysis of small-scale close to the wall and study oflarge-scale features, especially in boundary layers, Hutchins &Marusic (2007b) reported the presence of very long meandering structures in turbulent boundary layers, which they called‘super-structures’, whose length can reach to 20δ . Although agreat deal of research has been conducted on large scale structures, there are still many unknown points, However, there are
relatively few studies on the characteristics of span-wise direction of the turbulent structure. In this paper, the flow field isdecomposed using specific thresholds (along the span-wise direction, the thresholds used are described later), and the large
scale components defined in this paper are analytically discussed to have a significant contribution to the formation of
the outer peaks of the velocity pre-multiplied spectrum by calculating the flow field pre-multiplied spectrum.

THE NUMERICAL AND EXPERIMENTAL DATAMEAN VELOCITY AND TURBULENT INTENSITYPROFILEThe experiment was conducted in a channel wind tunnel and the results of the numerical simulations were obtained
from Yamamoto & Tsuji (2018), Most of the detailed discussions are based on the turbulence channel simulations of Yamamoto & Tsuji (2018).
And the experimental facility consist of wind tunnel andhot-wire anemometer. The channel wind tunnel is blowout
type wind tunnel, which consisting of a blower, a rectifier, anda test section. A dust collection filter is attached to the suction part of the blower, which removes the dust from the air.

Mean velocity and turbulent intensity profiles are plottedin Fig. 1 and Fig .2. In this study, the hot-wire array consistingof 16 I-probes arranged in the span-wise direction was used.
The distance between adjacent probes is 4mm. The velocity
profiles and turbulence intensity profiles of the random one
probe and numerical simulations are shown in Fig. 1 and Fig.2. The distance between probes is 4mm and the spatial resolution is not sufficient, which results in a discrepancy between
the experimental and numerical simulation values in the turbulence intensity profile. In comparison with DNS, the mea-

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 1: Parameter of data setup. Lx and Lz are the periodic stream-wise and span-wise dimensions of the numerical box,
and h is the channel half height. Nx , Ny and Nz are the grid points, where x indicates the stream-wise, y indicates thewall-normal, and z indicates the span-wise direction.

Name

line

Source

Reτ

Lx /h

Lz /h

∆x+

∆z+

Nx

Ny

Nz

case1

—

Experimental

930

–

–

–

–

–

–

–

case2

––

Simulation

1000

12.8

6.4

13.33

8.33

960

512

768

case3

–-–

Simulation

2000

12.8

6.4

16

8.33

2000

1024

1536

are usually represented as meandering elongated structures on
the contour maps. The low velocity region is flanked by thehigh velocity region, which is characteristic of the very large
scale motion (VLSM) from Hutchins & Marusic (2007b). But
only qualitative conclusions can be drawn from these figures.
Therefore a statistical analysis is required to determine the average structural characteristics. Eq .1 gives the definition of thetwo-point correlation along the span direction for quantifying
the span-wise scales of turbulent large-scale structures.

30
25
20
15
10
5
0
10 0

10 1

10 2

10 3

Ruu (∆z′ ) =< u′ (z,t)u′ (z + ∆z′ ,t) >< u′ (z,t)2 >(1)Figure 1: Mean velocity profiles. case1 — experimental
Here, u′ is the velocity fluctuating component and the angle brackets indicate time average, while z is the coordinate
along the probe or span-wise direction, ∆z′ indicates the distance along span-wise direction, and Ruu indicates the twopoint correlation along the span direction.
Figure 3 shows the result of two-point correlation of
stream-wise velocities at eight different wall heights. The data
are from case2 ,Reτ = 1000. And Fig .4 shows the result ofcase3, simulation data, Reτ = 2000. The span-wise typicalscale ls is defined by the two-point correlation function. It is
the distance between the points where Ruu has a small positive
value (but not zero). According to Monty et al. (2007), we
adopt a threshold of Ruu = 0.05, and this correlation thresholdis marked on Fig .3 and Fig .4 with a dashed line. The valueof ls is indicated by the double arrow in the figure. We can seethat in the channel flow (both experimental and simulations result) the value of ls becomes larger with the increase of the wallheight, and it is worth noting that the increase of the width of
the turbulent structure is also manifested in the boundary layer
flow (Hutchins et al. (2005); Adrian et al. (2000)). Fig .5 represents the variation of ls relative to the wall height at different Reynolds numbers, and the variation of ls at three cases isgiven on the way, corresponding to the results of experimental
and numerical simulations at different Reynolds numbers. The
horizontal coordinate in Fig .5 indicates the wall distance andthe vertical coordinate indicates the value of ls , which is normlized by the boundary layer thickness δ . We can observed thatthe variation of ls at different Reynolds numbers is consistent.
And it shows different growth rates near the wall and awayfrom the wall, and the wall height is about y/δ ∼ 0.12 when
the growth rate changes.
It is assumed that ls grows linearly, although it presents
different growth rates near and away from the wall (in agreement with Monty et al. (2007) results, although the wall height
where the growth raete changes does not agree with eachother). Outside of logarithmic region, the width of turbulentReτ = 930; case2 – – simulation Reτ = 1000; case3 – – simulation Reτ = 2000.

3
2.5
2
1.5
1
0.5
0
10 0

10 1

10 2

10 3

Figure 2: Turbulent intensity profiles. case1 — exper-

imental Reτ = 930; case2 – – simulation Reτ = 1000;
case3 – - – simulation Reτ = 2000.

surement underestimates the turbulent intensity (or RMS, root
mean square) near the wall region. This is not due to the probe
interaction among 16 ch probes, but attributes to the space resolution of sensor length of hot-wire. Since this paper focus
on the turbulent large-scale structure, the discrepancy can beneglected and does not influence the results of this study.

SPAN-WISE WIDTH SCALE
From previous studies, we know that in the contour figures of velocity fluctuations at different wall heights, there exist low and high velocity regions (Monty et al. (2007)), which212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 2: Definition of filter parameters for velocity de-1

composition in span-wise direction. Notation u′ denotes
the original component; u′L denotes the large scale; andu′s denotes the small scale component of velocity fluctuation.

0.5

0Notation

Designation

Cutoffu′original component

−−

u′L

large-scale component

λz /ls∗ > 1

u′s

small-scale component

λz /ls∗ < 1

-0.5
-1

-0.5

0

0.5

1

Figure 3: Two-point correlations for eight different wallheights. Data from case2, simulatiuon Reτ = 1000; Different symbols indicate different wall heights.

span-wise structures grows at a different rate up to the centerof the channel. In this paper, the velocity field is is decomposed into large and small scale components by the filtering of
span-wise wave length 2ls . This length scale is two times of ls
where the growth rate changes.

1

0.5

VELOCITY DECOMPOSITIONIn this paper the threshold is represented by ls∗ = 2 × ls (itis equal to the value of 2ls at y/δ = 0.12). The stream-wise velocity fluctuations whose span-wise wave length is larger thanls∗ are denoted as large scale components u′L . The stream-wisevelocity fluctuations whose span-wise length scale is smaller
than ls∗ are denoted as small scale components u′s . Scaled decomposition are summarized in Table 2. They are expressedby Eq .2.

0

-0.5
-1

-0.5

0

0.5

1

Figure 4: Two-point correlations for eight different wallu′ = u′L + u′sheights. Data from case3, simulatiuon Reτ = 2000; Different symbols indicate different wall heights.

(2)

Figure 6, 7 and 8 are examples of the decomposition ofvelocity fluctuations in the channel of experimental data and
DNS data, respectively. Each figure represents an contour surface map of velocity fluctuations for different velocity components, (a) original velocity component, (b) large-scale velocity
component, and (c) small-scale velocity component. In thiscontour plot, the blue color indicates the low velocity region
and the red color indicates the high velocity region. We can seethat the blue low velocity zone and the red high velocity zone
in the different velocity components meander alternately witheach other along the flow direction. In the following, we analyze the one-dimensional velocity pre-multiplied spectra forthe stream-wise direction of u′ , u′L and u′s .

1.2
1
0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

PRE-MULTIPLIED SPECTRUM

Figure 5: Variation of span-wise width scale, ls , with wall

By comparing the large-scale and the small-scale component, we defined the presence of a large-scale structure fluctuating in the span-wise direction in the large-scale component
and the absence of such a structure in the small-scale component. In this section, we will consider the pre-multipliedenergy spectra (PMS) of stream-wise velocity fluctuation ofdifferent velocity components of case1, case2 and case3. The
spectral density function of the stream-wise velocity fluctuation φuu is used and the stream-wise wavelength and wavenumber are denoted by λx and kx (kx = 2π/λx ), and kx φuu represent the pre-multiplied energy spectra of stream-wise velocity fluctuation. In this study all of the data are presented interms of stream-wise length-scale λx+ . Figure 9 shows the Premultiplied energy spectrum results of case1, case2 and case3.
In Fig. 9, the different cases are represented horizontally ascase1, case2 and case3. and the different velocity componentsu′ , u′L and u′s are represented vertically.
In previous study (Hutchins & Marusic (2007a,b)). The‘inner site’ and ’outer site’ are defined, where ’inner site’ isdistance. ⃝, data of channel from Monty et al. (2007),
Reτ = 3100 ; + denotes case1; □ denotes case2; ▽ denotes case3.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 20222

difficult to observe. Based on the decomposition method inthis study, in the large-scale velocity component, (u′L in Fig.9), for the experimental case the outer peak becomes obvious, but the inner layer peak still exists. For the numericalsimulation results, the large-scale velocity component has no
inner layer peak and only shows the characteristics of the outerlayer peak. Regarding the small-scale components, (u′s in Fig.9), for different cases (case1, case2 and case3), the velocity
pre-multiplied spectra corresponding to the small-scale components all have only the inner peaks. Taking into accountthe definition of the large-scale and the small-scale velocity
decomposition, we can draw an interesting conclusion: Thelarge-scale velocity components defined in this paper have an
important role in the generation of the outer peaks of the PMS,
and the velocity decomposition threshold, ls∗ , defined in thispaper, can be considered as the characteristic length for the
separation of the inner and outer peaks of the stream-wise velocity.

1

0
0

0

u'/u

2

2
0

-1

0

2

4

6

8

10

Figure 6: Example of the contour map of velocity fluc-tuations in channel of case1, Reτ = 930, normalized by
friction velocity uτ , and the wall-height is y/δ = 0.08.
(a)original velocity component; (b)large-scale velocity
component u′L ; (c)small-scale velocity component u′s .
2

1

In previous studies, meandering structures exist in the turbulent boundary layer up to 20δ in length, called Very large
scale motion (VLSM) (Balakumar & Adrian (2007); Kim &Adrian (1999)), and are associated with the generation of theouter peaks of PMS, which are similar to the u′L defined in thispaper, and it can be considered that the threshold (ls∗ ) definedin this paper separates out the VLSM turbulent structures.

0
0

0

u'/u

2

2
0

-1

0

4

8

Figure 7: Example of the contour map of velocity fluctu-

ations in channel of case2, Reτ = 1000, normalized by
friction velocity uτ , and the wall-height is y/δ = 0.08.
(Same as Fig. 6).

2

CONCLUSION
In this paper, the width scale ls in the span-wise direction
is defined by the data from experiments and numerical simulations in the channel flow, through the calculation of crosscorrelation coefficients, based on previous studies by Montyet al. (2007). ls increases with the distance from the wall andgrows linearly, but shows different growth rates near the wall
and away from the wall. In this paper, two times of ls chosen
as the threshold to decompose the flow field. Through a simple
Fourier filter and extraction of velocity components, velocity
components with different wavelengths in the span-wise direction are obtained. By comparing the relationship of large-scale
components and the small-scale components. The PMS of different scale is compared through statistical methods, and thefollowing conclusions are obtained:

1

0
0

0

u'/u

2

2
0

-1

0

4

8

Figure 8: Example of the contour map of velocity fluctu-

ations in channel of case3, Reτ = 2000, normalized by
friction velocity uτ , and the wall-height is y/δ = 0.08.
(Same as Fig. 6).

• The outer peaks of the PMS contour appear even at low
Reynolds number, and they are due to the structure of
the long wavelength velocity component that fluctuates
or meanders in the span-wise direction.
• The threshold ls∗ for the velocity field decomposition defined in this paper can be considered to decompose thevelocity components containing span wavelengths larger
than ls∗ and span wavelengths smaller than ls∗ , where u′L ,which can be considered as the VLSM defined in previous studies (as a conjecture), has a distinct outer peak in
its velocity pre-multiplied spectrum.
• It was clarified that the fluctuation of the large-scale structure in the span-wise direction and the effective usefulnessof the energy are deeply related.
• Regarding the distance from the wall when the growthrate of ls varies, the results of this paper differ from thoseof previous studies Monty et al. (2007), the results of thispaper are closer to the wall with y/δ = 0.1. However
this location indicates the position where the growth ratechanges.

a near-wall energy peak, and that this peak is approximately
fixed in viscous coordinates at y+ = 15 and λ + = 1000, and’outer site’ has a clear peak in the pre-multiplied spectra at
length-scales that are many times larger than the boundary+layer√ thickness (λx ≈ 6δ ). And the wall height is z =3.9 Reτ (consistent with the nominal area of log region,√ andexpressed in the coordinates of this study as y+ = 3.9 Reτ )(Mathis et al. (2009)). The position of the inner site (or innerpeak) and its corresponding wavelength are fixed, while theouter site (or outer peak) is related to the Reynolds number.
It is at high Reynolds number (Reτ ≥ 1700) that the near-walland logarithmic region can be clearly distinguished, seeing atleast a decade separation (one order of magnitude) between thetwo energy points.
In cases 1 and 2, where the Reynolds number is around
1000, it is difficult to see a clear separation between the inner and outer peaks, and even in case 3, the outer peaks are412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
case1
10

（a）u'

-3

10

-2

case2
10

-1

10

10-3

0

10-2

10-1

100

10

102

100

103

100

103

10-2

101

10-2

10

102

103

1

0
10

（b）u'
L

10

5

10

3

10

1

-1

0

10

1

10

2

10

10-3

10-2

10-1

100

10

100

10

1

-3

101

10

-2

102

103

-1

10

10

1

10

2

10

10-3

10-2

10-1

100

（c）u's

101

102

10-1

3

10-2

1

10-2

10

1

100

101

102

103

1

10-3

10-2

10-1

100

0

10

1

10-3

10

2

10-2

10

3

10-1
10

2

105

103

10

0

0

101

10

-2

-2

2

100

10

3

10

10-2

10

1

10

1

0
10

0

10

1

10

2

10

3

1

0
10

10

103

10-2

10

100

0
100

2

3

5

101

10

10

10

103

1

0

3

102

10

100

101

1

0
0

102

-2

105

102

10-2

10-3

0
10

-1

5

103

10

10

10

0

10

-2

1

0

102

2

10

100

10

105

10

-3

105

0

3

0

10

1

105

105

10

1

case3

1

0
100

101

102

103

Figure 9: Pre-multiplied one-dimensional spectra kx φuu /max(φuu ) as function of wavelength and wall distance, case1,
experimental; case2, simulatiuon Reτ = 1000; case3, simulatiuon Reτ = 2000. The vertical figure represents the different
velocity components u′ , u′L and u′s .

ACKNOWLEDGMENTS

cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. Journal of Fluid Mechanics 541, 21–54.
Hutchins, N & Marusic, Ivan 2007a Evidence of very longmeandering features in the logarithmic region of turbulent
boundary layers. Journal of Fluid Mechanics 579, 1–28.
Hutchins, Nicholas & Marusic, Ivan 2007b Large-scale influences in near-wall turbulence. Philosophical Transactionsof the Royal Society A: Mathematical, Physical and Engineering Sciences 365 (1852), 647–664.
Kim, Kyung Chun & Adrian, Ronald J 1999 Very large-scale
motion in the outer layer. Physics of Fluids 11 (2), 417–422.
Kline, Stephen J, Reynolds, William C, Schraub, FA & Runstadler, PW 1967 The structure of turbulent boundary layers. Journal of Fluid Mechanics 30 (4), 741–773.
Mathis, Romain, Hutchins, Nicholas & Marusic, Ivan 2009
Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. Journal of Fluid Mechanics 628, 311–337.
Monty, JP, Stewart, JA, Williams, RC & Chong, MS 2007
Large-scale features in turbulent pipe and channel flows.
Journal of Fluid Mechanics 589, 147.
Yamamoto, Yoshinobu & Tsuji, Yoshiyuki 2018 Numericalevidence of logarithmic regions in channel flow at r e τ=
8000. Physical Review Fluids 3 (1), 012602.

This work was financially supported by JST SPRING,
Grant Number JPMJSP2125. The author (Initial) would liketo take this opportunity to thank the “Interdisciplinary Frontier Next-Generation Researcher Program of the Tokai Higher
Education and Research System.”This work is partially supported by “Nagoya University High Performance Computing Research Project for Joint
Computational Science” in Japan. This work was supported
by JSPS KAKENHI Grant Number 19H00747.

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motions in channel and boundary-layer flows. Philosophical Transactions of the Royal Society A: Mathematical,
Physical and Engineering Sciences 365 (1852), 665–681.
Ganapathisubramani, Bharathram, Longmire, Ellen K &Marusic, Ivan 2003 Characteristics of vortex packets in turbulent boundary layers. Journal of Fluid Mechanics 478,
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Hutchins, N, Hambleton, WT & Marusic, Ivan 2005 Inclined5