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

WALL-ATTACHED STRUCTURES IN A TURBULENT CHANNEL FLOW WITH NAVIERSLIPMin Yoon
Division of Mechanical Engineering
Korea Maritime and Ocean University
Busan 49112, Republic of Korea
minyoon@kmou.ac.kr

Hyung Jin Sung
Department of Mechanical Engineering
KAIST
Daejeon 34141, Republic of Korea
hjsung@kaist.ac.kr

ABSTRACT
We explore wall-attached structures in a drag-reduced turbulent
channel flow with the Navier slip boundary condition. The
three-dimensional coherent structures of the streamwise
velocity fluctuations (u) are examined in an effort to assess theeffects of wall-attached u structures on drag reduction. We
extract the u clusters from the direct numerical simulation(DNS) data; for comparison, the DNS data for the no-slip
condition are included. The wall-attached structures, which arephysically adhere to the wall, in the logarithmic region are selfsimilar with their height and contribute to the presence oflogarithmic behavior. The formation and hierarchy of wallattached self-similar structures (WASS) are not affected by astreamwise slip, of which influences on them are limited up to
the lower bound of logarithmic region, supporting Townsend’souter-layer similarity hypothesis. Weakened mean shear by thestreamwise slip results in the decrease in the population densityof wall-attached structures within the buffer layer. This leads tolower population of WASS, whereas the space occupied bythem in the fluid domain increases. The conditional statistics
are obtained to analyze the WASS in the vicinity of the wall.
The streamwise slip induces long tails in the near-wall part of
WASS, reminiscent of the footprints of large-scale motions.
The present results reveal that the characteristics of WASS isirrespective of the streamwise slip and their near-wall part ismain energy-containing motions.

drag adversely influences the efficiency and performance ofengineering applications. Drag-reduced turbulent flows can bemodeled by using the Navier slip boundary condition (Navier,1823) in the streamwise direction (Min and Kim, 2004).
Although many studies of turbulence statistics and structures indrag-reduced flows have been performed, not much attention
has been paid to wall-attached structures, especially in thevicinity of the wall. Given that wall-attached structures are themain energy-containing motions in the logarithmic region andphysically anchored to the wall, it is essential to establish therole of the roots of wall-attached structures in the frictionaldrag to understand the drag reduction mechanism.

NUMERICAL DETAILSIn the present study, the DNS data for turbulent channel flows
with the Navier slip and no-slip boundary conditions of Yoonet al. (2016) were used. The Navier−Stokes equations andcontinuity equation for an incompressible flow were discretizedby using the fractional step method (Kim et al. 2002). The bulk
Reynolds number (Reb) is equivalent to 10,333. The Navier slip
boundary condition was applied in the streamwise direction, i.e.
uS  LS (du /dy )|wall , where ũS is the slip velocity, LS is the slip
length, ũ is the streamwise velocity, and y is the wall-normal
location. The slip length is fixed at 0.01δ to give a dragreduction rate of 35% for the channel flow with the no-slip
condition. Here, δ is the channel half-height. The friction
Reynolds numbers (Reτ) for the slip and no-slip cases are 469
and 577, respectively. The characteristics of the computational
domain are summarized in Table 1, where x and z are thestreamwise and spanwise directions, respectively. The
superscript “+” denotes quantities normalized by the wall unitsof each case.

INTRODUCTION
Townsend (1976) posited that the main energy-containing
motions in the logarithmic region are self-similar with respect
to their height (ly) and attached to the wall. The attached-eddy
hypothesis can be used to predict turbulence statistics throughthe construction of randomly superimposed attached eddies,and can be rigorously applied to an inviscid fluid near the wallfor high-Reynolds-number turbulent flows. Perry and Chong(1982) extended Townsend’s attached-eddy hypothesis toestablish the attached eddy model by using a hierarchy of selfsimilar vortex eddies with population densities that are
inversely proportional to their height. Recently, Hwang andSung (2018) extracted wall-attached structures from the direct
numerical simulation (DNS) data for a turbulent boundary
layer; they reported that wall-attached structures of thestreamwise velocity fluctuations (u) with an inverse power-law
distribution contribute to the presence of the logarithmic region
and are physically anchored to the wall. The near-wall
turbulence is dominantly influenced by the viscosity and isclosely related to the skin-friction reduction in drag-reduced
flows (Min and Kim, 2004).
A strong shear layer is formed near the wall, which resultsin high skin friction in wall-bounded turbulence. Frictional

Table 1. Parameters of the computational domain and friction
Reynolds number. Li and Ni are domain sizes and the numberof grids in each direction, respectively. Δymin+ and Δymax+
denote resolutions of the first grid from the wall and the grid atthe channel half-height, respectively.

Lx/δ
Lz/δ
Nx
Ny
Nz
Δx+
Δz+
Δymin+
Δymax+
Reτ

1

Slip

No-slip

10π
3π
2,497
401
1,249
5.92
3.55
0.08
5.96
469

10π
3π
2,497
401
1,249
7.27
4.36
0.10
7.32
577

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
RESULTS
Figure 1 represents the turbulence statistics for the slip and noslip cases. The magnitude of streamwise mean velocity (U  ) islarger than that of no-slip case due to the streamwise slip. The
dashed line in figure 1(a) exhibits the mean velocity relative tothe wall (U   US ) for the slip case, which coincides with thewell in the outer region, supporting Townsend’s outer-layer
similarity hypothesis. This observation is a good agreementwith results of Flores and Jiménez (2006) and Chung et al.
(2014). Here, Uc is the mean velocity at the channel center.
The 3-D u clusters in instantaneous flow fields can be
defined as the groups of connected points satisfying u ≥ αurms
and u ≤ –αurms, where α is the threshold. The threshold can bechosen from the percolation diagram in figure 2. The total
number (N) and total volume (V) of clusters at a given α arenormalized by the maximum N (Nmax) and maximum V (Vmax),
respectively. As α decreases, new clusters arise, or someadjacent clusters merge. The peak in N/Nmax at α = 1.6 resultsfrom the tradeoff between these two influences. In addition,
V/Vmax increases as the decrease in α with strong variation in
1.4 < α < 1.8, where the percolation transition occurs. Hence,
we adopted α = 1.6 in the present study.

profile of U  for the no-slip case. The mean shear (dU  /dy  )is insensitive to the streamwise slip. The streamwise slip
induces a virtual origin in U+ at y+ = –LS+, leading to the
upward shift of U+ as U   U S  LS in the entire region(García–Mayoral et al., 2019).

Figure 2. Variations with α in the total volume (V) (square) andtotal number (N) (circle) of clusters.

Each cluster can be identified based on above condition byusing the connectivity of six-orthogonal grids at a given nodein Cartesian coordinates (Moisy and Jiménez, 2004). As aresult, the spatial information about each cluster can beobtained. The clusters with volumes less than 303 wall units aredeleted to avoid grid resolution (del Álamo et al., 2006). Figure
3 shows the population density (n*) of all the identified clusters
with respect to ymax and ymin, which are the maximum andminimum distances from the wall, respectively. Here, n* isdefined as the number of u clusters (n) per unit wall-parallel
area (Axz = LxLz), i.e. n *  n ( y max , y min )/( mAxz ), where m is thenumber of snapshots. Two distinct groups are evident at ymin+ ≈0 and ymin+ > 0. The former are wall-attached structures, and thelatter are wall-detached structures (Hwang and Sung, 2018).
Figure 1. Profiles of (a) mean velocity and (b) defect form ofmean velocity. A dashed line in (a) represents mean velocity
relative to wall for slip case. Profiles of streamwise turbulence
intensity (urms+) and Reynolds shear stress (uv  ) for (c)inner scaling and (d) outer scaling.
The magnitude of streamwise turbulence intensity (urms+) is
amplified in y+ < 10, especially very close to the wall (figure1c), due to the streamwise slip. On the contrary, the profile of

Reynolds shear stress (uv ) is well matched with that for
the no-slip case in the inner region as a result of animpermeable condition at the wall. For outer scaling, theFigure 3. The number of u clusters per unit wall-parallel area(n*) with respect to ymin+ and ymax+.

profiles of Uc  U  , urms+ and uv  for both cases collapse

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
The population density (na*) of wall-attached structures as afunction of ly+ is shown in figure 6. Here, na is the number ofwall-attached structures at a given ly, and na* is defined as na
per unit wall-parallel area, i.e. na* = na/(mAxz). The magnitudeof na* is smaller than that for the no-slip case in the entire
region. The wall-attached structures with ly+ = Ο (10) are
related to near-wall streaks. As shown in figure 1, the mean
shear is preserved near the wall for the slip case, sustaining
turbulence via the formation process of streaky structures(Hamilton et al., 1995). The streamwise slip attenuates
streamwise vortices near the wall, creating streaks sparsely(Min and Kim, 2004). The lack of streamwise vortices leads toreduced populations of wall-attached structures near the wall.
The magnitude of na* is inversely proportional to ly+ in 250+< ly < 400 for both cases, reminiscent of the distribution ofhierarchy length scales of attached eddies (Perry and Chong,1982). The inverse power-law dependence arises within theregion of ly+ = 0.3δ+–0.6δ+ (Hwang and Sung, 2018; Hwang
and Sung, 2019). The streamwise slip induces a decrease in the
population density of wall-attached structures, but the inverse
power law is still valid. The no-slip boundary condition is notindispensable to form hierarchical distributions of wallattached structures. In addition, a peak in na* arises at ly+ =1.06δ+ for both cases, which attributes the large population ofwall-attached δ-height structures (Perry et al., 1986).

Figure 4 illustrates the 3-D iso-surfaces of u of wallattached structures for the slip case. Green and gold represent
negative u and positive u, respectively. The characteristic
lengths of each cluster are defined in terms of the dimensionsof the box circumscribing the object. The inset in figure 4
shows a sample and its length scales, i.e. lx, lz and ly, which are
the streamwise and spanwise sizes and the height, respectively.
The wall-attached structures are coherent in the streamwise
direction, and they are aligned side by side in the spanwise
direction.

Figure 4. 3-D iso-surfaces of the wall-attached u structures in
an instantaneous flow field for the slip case.

Figure 5 shows the mean lx+ (l x  ) and mean lz+ (l z  )with respect to ly+, which is the height of wall-attachedstructures, for the slip (blue) and no-slip (grey) cases. The
mean lx+ is proportional to ly+ with the power of 0.74 (Hwang
and Sung, 2018) in ly+ ≥ 70 ≈ 3Reτ0.5, whereas not proportional
to ly+ in ly+ < 70. Here, 3Reτ0.5 is the lower limit of logarithmic
region (Marusic et al., 2013; Hwang and Sung, 2019), whichaccords with approximately 70ν/uτ at the present Reynolds
number, Reτ ≈ 500. In addition, there is a linear relationshipbetween the mean lz+ and ly+ (ly+ ≥ 70), showing that the
spanwise size of wall-attached u structures is proportional tothe distance from the wall (Tomkins and Adrian, 2003; Hwang,2015). In particular, the profiles of  l x  and  l z  for bothFigure 6. Population density (na*) of wall-attached structures asa function of ly+. Red line denotes an inverse power-law
distribution of na*.

cases coincide in the entire ly+, especially ly+ ≥ 70, in agreementwith the results of Lozano–Durán and Bae (2019). Accordingly,
we divide wall-attached structures into non-self-similar (ly+ <70) and self-similar (ly+ ≥ 70) structures.

Figure 7 shows wall-normal profiles of streamwise
Reynolds stresses (uu  *a ) carried by the wall-attachedstructures with ly+ = 300,

uu*a  Sa ( y, l y )1 u(x)u(x)dxdz ,Sa(1)

where Sa is the wall-parallel area of wall-attached structures.
The magnitude of uu*a for ly+ = 300 is logarithmically
proportional to y+ (uu  *a ~ ln y  ) in 100 < y+ < 220 (Perry andChong, 1982). Although the present Reynolds number isrelatively low (Reτ ≈ 500), the logarithmic variation is evident
in uu  *a (Hwang and Sung, 2018; Hwang and Sung, 2020;

Figure 5. The mean lx+ and mean lz+ of wall-attached u
structures. The yellow lines represent l x  ~ l y0.74 and lz  ~ l y ,Yoon et al., 2020). The logarithmic behavior of wall-attachedstructures is sustained regardless of the streamwise slip at thewall.

respectively.

3

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

that the near-wall part of WASS is more sparsely distributed
than those for the no-slip case.

Figure 7. Streamwise Reynolds stresses (uu  a ) reconstructed

Figure 8. Two-point correlations of unws. Solid lines represent
5% of the maximum of positive correlations, and dashed lines
denote 50% of the minimum of negative correlations. Cross
symbols indicate spanwise centers of negative correlations.

by wall-attached structures with the height of ly+ = 300. Red
dashed lines indicate the best fits in 100 < y+ < 220.

The characteristics of wall-attached structures can be
decomposed into buffer-layer (ly+ < 70 ≈ 3Reτ0.5) and selfsimilar (ly+ ≥ 70) structures. The upper part (y+ ≥ 70) of wallattached self-similar structures (WASS) (ly+ ≥ 70) is
responsible for the logarithmic behavior (figure 7). The nearwall part (y+ < 70) of WASS (ly+ ≥ 70) contributes togeometrical self-similarity. The wall-attached structures can bedivided into unws, uuws and uwb, which are defined asu if |u|  1.6urms , ymin  0, ymax  70, y  70,
unws  
 0 otherwise,

(2)



u if |u|  1.6urms , ymin
 0, ymax
 70, y   70,
uuws  
 0 otherwise,

(3)



u if |u|  1.6urms , ymin  0, ymax  70,
uwb   0 otherwise.

(4)

We scrutinize the turbulence statistics of the near-wall partof WASS with respect to y. The turbulence statistics carried by
the near-wall part of WASS are conditionally averaged basedon conditional fields, where velocity fluctuations are obtainedusing equations similar to (2). Conditionally averaged
quantities are used to assess contributions of the near-wall partof WASS to the turbulence statistics in y+ < 70.

Here, the subscripts ‘nws’, ‘uws’ and ‘wb’ represent the nearwall part of WASS, upper part of WASS and wall-attachedbuffer-layer structures (WABS), respectively. We focus on thenear-wall part of WASS, in which the viscosity effect isdominant.
To investigate characteristic lengths of the near-wall part of
WASS statistically, conditional two-point correlations of unws
are examined at the reference wall-normal location yref+ = 14.5,
(figure 1c). The two-point
where the inner peak appears in u rms
correlation of unws is defined as,

R[unws , unws ] unws ( x, yref , z )unws ( x  rx , y, z  rz )
,unws, rms ( yref )unws, rms ( y)(5)where unws,rms is the root-mean-square quantity of unws.
Wall-parallel views of R[unws, unws] are displayed in figure
8. Solid and dashed lines correspond to 5% of the maximum of
positive R[unws, unws] and 50% of the minimum of negative
R[unws, unws], respectively. The positive R[unws, unws] areextended over 4δ in the streamwise direction, which is similar
to streamwise lengths of the footprints of large-scale motions at
y+ ≈ 14.5 (Hwang et al., 2016). In addition, the negative R[unws,
unws] indicates that the near-wall part of WASS is aligned side
by side along the spanwise direction. The distance to the center(cross symbol) of negative R[unws, unws] at rx/δ = 0 is 0.6δ,
which is 20% larger than that for the no-slip case. It represents

Figure 9. Conditionally averaged turbulence statistics of thenear-wall part of WASS: wall-normal profiles of (a) Anws/Axz,/dy  .
(b) uu  nws , (c) uv nws , and (d) dU nws

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

To explore the characteristics of the near-wall part of
WASS with respect to ly at the near-wall region, conditional
averages of wall-parallel area, streamwise velocity, streamwise
Reynolds stress and Reynolds shear stress are examined at y+ =14.5. They are defined as,

Figure 9 shows wall-normal profiles of conditional
averages for the near-wall part of WASS. Profiles of Anws/Axz
represent wall-parallel areas occupied by the near-wall part of
WASS, which account for 5 ~ 7% of the area of the x–z plane.
Below y+ = 30, the magnitude of Anws/Axz is larger than that for
the no-slip case, in particular, approximately 75% larger near
the wall. As shown in figure 9(b), the magnitude of uu  nws islarger than that for the no-slip case below y+ = 30. The near-

Sa,i  S a (y , l y )| y  14.5 /( mAxz ),(6)

U a,i  U ( y, l y )| y  14.5 /U (y )| y  14.5 ,(7)uu a,i  uu*a (y, l y )| y  14.5 / uu (y )| y  14.5 ,(8)uv a,i  uv ( y, l y )| y  14.5 / uv (y )| y  14.5 .

(9)

*
a

wall part of WASS is strengthened by the streamwise slip.
The profiles of  uv  nws for both cases collapse (figure 9c),similar to uv . The near-wall part of WASS carries
approximately 30% of total uv in spite of small Anws/Axz as
the main energy-containing motions near the wall. Figure 9(d)represents the mean shear (dU nws/dy  ) carried by the near-*aFigure 10(a) shows profiles of the areas in the x–z plane (Sa,i)
occupied by the near-wall part of WASS at y+ =14.5. The
magnitude of Sa,i gradually decreases as the increase in ly+ witha peak at ly+ = 1.1δ+. The quantities of Ua,i,  uu  a,i and uv a,iwall part of WASS. Below y+ = 20, the magnitude of
dU nws/dy  is larger than that for the no-slip case, especially bydescribe the dependence on ly in their contributions to theturbulence statistics (i.e. U,  uu  and uv ) at y+ = 14.5.
Figure 10(b) represents profiles of Ua,i with respect to ly+. The
region of ly+, where the magnitude of Ua,i is larger than 1,
indicates that positive u of the near-wall part of WASS at given
ly+ is dominant near y+ = 14.5, and vice versa. A concave peakof Ua,i is evident at ly+ = 1.12δ+. In particular, the profile of Ua,iis shifted downward from that for the no-slip case, especially in
ly+ = 300–δ+. It shows that the population of wall-attachednegative-u structures with ly+ ≈ δ+ are more dominant than thatfor the no-slip case (Yoon et al., 2016).
The streamwise Reynolds stress (  uu  a,i ) at y+ = 14.5 is

9% close to the wall. The magnitude of dUnws+/dy+ at the wall
can be arranged to the ratio τw,nws/τw. Here, τw,nws is the wall
shear stress carried by the near-wall part of WASS. Given thatthe wall shear stress is related to the skin friction coefficient(Fukagata et al., 2002), the high dUnws+/dy+ at the wall is
responsible for the frictional drag. The mean shear of the nearwall part of WASS for the slip case is restored over y+ = 30.
The near-wall part of WASS is varied by the viscosity and wallconditions, but their contributions to the streamwise Reynolds
stress and Reynolds shear stress are more significant.

shown at figure 10(c). The magnitude of uu  a,i gradually
increases as ly+ increases. The contributions of the near-wallpart of WASS to the streamwise Reynolds stress are enhanced
as increasing ly+. In particular, the magnitude of uu  a,i islogarithmically proportional to ly+ in 100 < ly+ < 350 (uu  a,i ~ln l y ) (a yellow line). The inner-peak magnitude of streamwise

Reynolds stress logarithmically increases with increasing Reτ
(Jiménez and Hoyas, 2008; Marusic et al., 2017). Since ly isrelated to hierarchical scales (figure 7), the logarithmic
variation in uu  a with respect to ly reflects hierarchicalfeatures of wall-attached structures (Hwang and Sung, 2018).
The logarithmic variation in uu  a is evident in 200 < ly+ < 300.
A hierarchy of wall-attached structures is well established overthe broader range of ly. Figure 10(d) shows profiles of  uv  a,i ,of which the value is larger than 3 in ly+ < δ+. It means that thenear-wall part of WASS carry three times more Reynolds shear
stresses than  uv  in the near-wall region (y+ = 14.5). The
magnitude of  uv  a,i is larger than that for the no-slip casebelow ly+ = δ+. These observations imply that the contributionsof the near-wall part of WASS to the near-wall turbulence aredominant, since the Reynolds shear stress is directly related toturbulent contributions to the skin friction coefficient (Fukagata
et al., 2002).
CONCLUSIONS
We have explored wall-attached u structures in a drag-reduced
turbulent channel flow and demonstrated their contributions tonear-wall turbulence. We extracted 3-D u clusters by using theconnectivity of six-orthogonal neighbors in Cartesian

Figure 10. Conditional averages of the near-wall part of WASS
at y+ = 14.5: (a) Sa,i (b) Ua,i, (c) uu a,i , and (d) uv a,i . A
green line in (c) denotes the logarithmic variation.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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coordinates from the DNS dataset for turbulent channel flows
with a Navier slip wall (Reτ = 470) and no-slip wall (Reτ = 577).
We showed that wall-attached structures are formed in their
entirety in instantaneous flow fields on the wall with thestreamwise slip. The wall-attached structures with ly+ ≥ 70 ≈3Reτ0.5 were self-similar with ly, and their geometrical features
remained unchanged regardless of the streamwise slip. The
hierarchical distribution with an inverse power law and thelogarithmic behavior of WASS (ly+ ≥ 70) were not influenced
by the streamwise slip. Influences of the streamwise slip on thewall-attached structures were limited up to the lower bound (y+= 3Reτ0.5) of logarithmic region. The streamwise slip induced
the decrease in the population density of WABS (ly+ < 70) withweakened mean shear. The decrease in the population densityof WASS is attributed to that of WABS, whereas the space
occupied by WASS in the fluid domain is enlarged. In addition,
we have focused on the near-wall part (y+ < 70) of WASS. The
conditional average and conditional fields were employed. The
near-wall part of WASS was extended in the streamwise
direction over 4δ, reminiscent of the footprints of large-scale
motions. The near-wall part of WASS contributed toapproximately 30% of the Reynolds shear stress. The Reynolds
shear stress carried by the near-wall part of WASS is enhancedby the streamwise slip. The streamwise slip curtails the
population density of WASS. The present study enhances ourunderstanding of the behavior of wall-attached structures andfurthers the development of attached-eddy models of dragreduced flows.
ACKNOWLEDGMENTS
This work was supported by the National Research Foundationof Korea (NRF) grant funded by the Korea government (MSIT)(No. 2020R1A2C2008106 and 2021R1F1A1053438).

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