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

SCALING LAWS AND WALL-ATTACHED STRUCTURES IN A
SUPERSONIC TURBULENT CHANNEL FLOW
Hyeon Gyu Hwang
Department of Mechanical Engineering
UNIST
50 UNIST-gil, Eonyang-eup, Ulsan, 44919, Korea
hyunq92@unist.ac.kr

Jae Hwa Lee
Department of Mechanical Engineering
UNIST
50 UNIST-gil, Eonyang-eup, Ulsan, 44919, Korea
jhlee06@unist.ac.kr

Jinyul Hwang
School of Mechanical Engineering
PNU
2 Busandaehak-ro 63beon-gil, Geumjeong-gu,
Busan, 46241, Korea
jhwang@pnu.ac.kr

Ji-Hoon Kang
Division of National Supercomputing
KISTI
245 Daehak-ro, Yuseong-gu, Daejeon, 34141,
Korea
jhkang@kisti.re.kr

Hyung Jin Sung
Department of Mechanical Engineering
KAIST
291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Korea
hjsung@kaist.ac.kr

ABSTRACTTo establish integrating explanation for characteristics ofcompressible wall-bounded turbulence, scaling laws andrelevant turbulent coherent structures in compressible wallbounded turbulence are investigated in light of Townsend’s
attached-eddy hypothesis. Using data of direct numericalsimulation of supersonic turbulent channel flow between
isothermal walls, we verified the scaling laws for mean velocity
and Reynolds stresses, which can be predicted by attached-eddy
hypothesis in a unified way. The transformed mean velocity
profiles and the Reynolds stresses using semi-local scale
supports the predicted scaling laws at certain extent. However,complete logarithmic dependence is not achieved due torelatively low-Reynolds number or insufficient scale separation,
especially for streamwise normal stress. In search of coherent
structures responsible for the logarithmic dependence of the
turbulent statistics, wall-attached structures of velocity
fluctuations are extracted from instantaneous flow fields.
Reconstructed flow field from wall-attached self-similar
structures fits more closer to the logarithmic variation. With
these findings, we have provided the convincing evidence for thevalidity of the scaling laws and the existence of coherent
structures corresponding the attached-eddy hypothesis for thecompressible wall-bounded turbulence.

efforts has been devoted to investigation of such coherent
structures because it has been a foundation for modelling andcontrol strategies of the wall-bounded turbulence. Along withthese principal thoughts, the study of the scaling laws and thecoherent structures has started from the study of incompressiblewall-bounded turbulence, due to relatively simple condition of
constant fluid properties, while the study of compressible wallbounded turbulence is preceded by, and hence, takes its root in
the incompressible counterpart.
From the classical point of view, the presence of a wall
naturally enforces multiscale characteristics on turbulence above
the wall, with length scale ranging from viscous length scale δv
to flow thickness δ. The scale separation between these distinct
length scales is usually expressed as friction Reynolds number,Reτ=δ/δv. Emerging with sufficient scale separation (high Reτ),
logarithmic variation of mean velocity profiles is recognized asa cornerstone for the wall-bounded turbulence. This region, so
called as logarithmic region, occurs at δv<<y<<δ (y is distancefrom the wall), and it is equivalent to the condition of thesufficient scale separation. In essence, characteristic velocity
scale is friction velocity uτ=(τw/ρ)1/2 (τw is wall-shear stress andρ is density) and any geometric length scale is proportional todistance from the wall, y (Marusic et al., 2013).
In this context, Townsend (1976) conjectured that the lengthscales of energy-containing eddies in the logarithmic region are
scaled by their position (y), and thus, they extend to the wall,
which is called Townsend’s attached-eddy hypothesis (AEH).
Based on the AEH by Townsend (1976), Perry and Chong (1982)
proposed the statistical description of logarithmic region in wallbounded turbulent flows such that hierarchies of geometrically
self-similar eddies are superpositioned with their population
inversely proportional to their length scale (y). The proposed
representation and following deductions, which is called asattached-eddy model (AEM), lead to important scaling laws for

INTRODUCTION
Scaling laws for statistical features of wall-bounded
turbulence has been an important issue since they are consideredas building blocks for understanding the characteristics of thewall-bounded turbulence. Furthermore, given that the wallbounded turbulence consists of coherent structures (i.e., eddies)
which naturally occur across a wide range of length scales andare considered as responsible for the relevant scaling laws,

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
the wall-bounded turbulence, such as logarithmic variation ofmean velocity and Reynolds stresses for wall-parallel
components, as well as their spectral imprint of k-1 scaling (k iswavenumber). Further refinement of attached-eddy model has
been achieved through the following studies (Perry et al., 1986;
Perry and Marusic, 1995 and so on). Recently, evidence of the
AEH (or the AEM) has been substantiated by observations forco-existence of logarithmic variations of mean velocity andnormal stresses in high-Reynolds-number flow fields with highfidelity (Hultmark et al., 2012; Marusic et al., 2013; Lee andMoser 2015), and hence its significance is more increasing.
In last decades, the efforts have been devoted in order to
identify the coherent structures corresponding to the AEH.
Although several studies (del Á lamo et al., 2006; Lozano-Duránet al., 2012) provided the existence of self-similar motions inphysical space, whether they are responsible for the formation oflogarithmic variation have not been elucidated. Among therelevant studies, studies by Hwang and Sung (2018; 2019) foundthat the wall-attached clusters of streamwise velocity
fluctuations are not only geometrically self-similar but also have
population density following the hierarchical length scale
distribution. Furthermore, they successfully showed areconstructed flow field from the identified structures displays
clear logarithmic variations of the statistics even at relatively
low- to moderate-Reynolds number, suggesting the identified
structures as prime candidates for the coherent structures
corresponding the AEH.
On the other hand, the established scaling laws for mean
velocity and Reynolds stresses in the incompressible wallbounded turbulence is not directly affordable for thecompressible wall-bounded turbulence because properties suchas thermodynamic properties (e.g., temperature, density andpressure) are not constant. One might expect that the variationsin properties would significantly alter various aspect of wallbounded turbulence. According to Morkovin’s hypothesis (MH),at moderate Mach number, compressibility effect of density
fluctuation on turbulence is small, and essential dynamics
closely follow those of incompressible flow (Morkovin, 1962).
Hereafter, compressible wall-bounded turbulence only indicatesthe moderate Mach number flow (i.e., non-hypersonic flow) if
without any specification.
Encouraged by the principle of the MH, the scaling laws forthe compressibile wall-bounded turbulence have been
established via compressibility transformation, which casts thecompressible turbulent statistics to incompressible counterpart.
Accordingly, the logarithmic variations for mean velocity andwall-parallel components of the Reynlolds stresses (AEH) are
expected through the transformations (Smits and Dussage, 2006).
Although van-Driest transformation and Morkovin’s scaling, in
general, are agreed upon based on large volume of experimental
data (Fernholz and Finley, 1980; Smits and Dussage, 2006), theiraccuracy has been questioned due to uncertainty in measurment.
This issue was resolved by performing direct numericalsimulation of the compressible flow, and modified
transformations are also suggested (Trettel and Larsson, 2016;
Huang et al., 1995) for strong wall heat transfer conditions.
However, the DNSs of the compressible flow have been
performed at relatively low-to-moderate Reynolds number
limited by computational cost, and complete logarithmic
variations for the turbulence statistics, especially for the
streamwise Reynolds stresses, are not observed due toinsufficent scale separations.
On the other hand, in similar context with the incompressible

counterpart, several studies have devoted efforts to elucidate theexistence of self-similar coherent structures in the sense of theAEH. Pirozzoli (2012) and Modesti and Pirozzoli (2016; 2019)
proposed an appropriate length scales for spanwise width of
spatial correlation and spanwise wavelength at peak of energy
spectra of stremwise velocity fluctuation in compressible flowof canonical geometries. In their studies, the proposed lengthscales showed good scaling accuracy in overall outer layer withreasonable Mach number range, and it is indicates increasing
trends of the spanwise length scales for the turbulent motions
with the wall-normal locations. Although these studies
successfully took account of the compressibility effects on thelength scales of the turbulent motions, and found the presence ofself-similar motions in the compressible wall-bounded
turbulence, it has not revealed for how they contribute to thesuggested scaling laws.
The objective of present study is to establish integrating
explanation for characteristics of compressible wall-bounded
turbulence. To this end, scaling laws and relevant turbulent
coherent structures in compressible wall-bounded turbulence areinvestigated in terms of the AEH. We verify the predictedscaling behaviors using data of direct numerical simulation ofsupersonic turbulent channel flow between isothermal walls at
moderate Reynolds number. In search of coherent structures
responsible for the logarithmic variation of the turbulent
statistics, wall-attached structures of velocity fluctuations are
identified by implementing the method of Hwang and Sung(2018; 2019) and their characteristics are investigated.

NUMERICAL METHODS
Direct numerical simulation of supersonic turbulent channel
flow between isothermal wall at Reb=24000 and Mab=1.5 (wheresubscript b denotes bulk variables) is performed by solving
governing equations of compressible perfect gas with thenumerical schemes of Bernardini et al. (2020). Based on the wall
variables (uτ and δv) semi-local variables (uτ* and δv*), the flow’s
friction Reynolds number Reτ =h/δv (h is channel half-height) andsemi-local Reynolds number Reτ*=h/δv* are 1390 and 930,
respectively, and it is denoted as C930. Throughout the present
study, x, y and z indicate streamwise, wall-normal and spanwise
directions, respectively, and u, v and w are velocity components
for each direction. Computational domain is set as Lx × Ly × Lz =10πh × 2h × 3πh, which is expected to be sufficiently long toaccommodate large- and very-large scale motions (Kim andAdrian, 1999; Ganapathisubramani et al., 2006). Uniform grid
spacing is implemented in the streamwise and spanwise
direction, while grids are clustered toward the walls using
hyperbolic tangent mapping function. Grid resolutions in eachdirection based on wall variable (∆x+, ∆y+ and ∆z+) are set as14.31, 0.16~9.41 and 7.38, respectively. For the comparison,
DNS data of incompressible turbulent channel flow at Reτ=930(Lee et al. 2014) are employed, which are denoted as I930. The
numerical details for DNS of I930 is referred to Lee et al. (2014).
For averaging methods, it is noted that <f> and f′ refer toReynolds-averaged and fluctuating quantities, while{f}=<ρf>/<ρ> (ρ is density) and f″ denote Favre-averaged andfluctuating quantities. And superscript + and * indicate
quantities scaled by wall and semi-local variables, respectively.
It is noted that, for incompressible flow, both scalings are
identical by definition and transformation has no effects on
quantities due to constant properties.

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
SCALING LAWS OF COMPRESSIBLE WALLBOUNDED TURBULENCE
Scaling laws for mean velocity profile for the compressiblewall-bounded flow is expressed in form of gradient as below (1)

d u
dy

=

u*
y

(1)

, where κ is the von-Kármán constant. This scaling law wasderived from extension of mixing length theory (van-Driest 1951)or law of the wall (Bradshaw 1994) with assumption of theexistence of constant stress region close to the wall andconsideration of the mean density variation. In their studies, thescaling constant κ is assumed to be identical with that of theincompressible counterpart.
In investigating the scaling law for mean velocity (1) based on
the obtained DNS data, compressibility transformations are
implemented. As following Modesti and Pirozzoli (2016; 2019),
the compressibility transformations for the distribution of mean
velocity are casted to forms of integral. The transformations forthe wall-normal coordinate and the mean velocity are expressedas yT =

Figure 1. Mean velocity profiles for C930 (blue symbols) and
I930 (green symbols): (upper) untransformed velocity, {u}un+(middle) D-transformation {u}D+ and (lower) TL-transformation{u}TL+. Former two profiles are displaced by ∆{u}T+=5. Coloredstraight lines are added as guides for logarithmic variation.
(1951) and {u}TL(yTL) by Trettel and Larsson (2016), as well as
untransformed (actual) mean velocity {u}un.={u}. The sets of thetransformation kernels for the van-Driest transformation (5) andthe Trettel and Larsson transformation (6) are described below.

uy f dy and u =  g d u , respectively. Here,
0

T

T

0

T

subscript T denotes transformed quantities and fT and gT aretransformation kernels. By changing the variable of integration
with the chain rule, the formula for {u}T is rearranged as

fD = 1gD = ( uT = 0 gT uy dy ({u}y denotes d{u}/dy). Then, using theirygradients (i.e., dyT/dy and d{u}T/dy), d{u}T/dyT can be expressed
as below (2).
d uTdyT

gT d{u}f T dy

TL

(2)

dyT

=

u

c y

(3)

, where κc is scaling constant, but not to be determined yet, since
it is not known a priori whether scaling constant κc is same withthe von-Karman constant κ. Here, the velocity scale for {u}T in(3) is uτ since compressibility transformations in use will adopt
constant variable (i.e., wall variables). Now, Equating (2) and (3)yields the relation for the actual (or untransformed) mean
velocity gradient (4).
d u

w





w

−1

 g  u= T dy
 f T   c yT

w

1/ 2

1/ 2

1 1 d 2  dy1 1 d 2  dy

1 d 
dy



1 d 
dy

(6)



In figure 1, profiles of mean velocities for C930 and I930 are
illustrated. As previously reported, an excellent accuracy for TLtransformation is observed where {u}TL(yTL) perfectly collapse
onto the profile of I930, and hence, it shows logarithmic
variation with the slope of 1/κc=1/0.41. Since {u}TL(yTL) satisfies(3), it is possible to use (4) to determine the property of d{u}/dy.
Direct substitution of fTL, gTL, yTL and κ (also found in scaling
law of dissipation) into fT, gT, yT and κc in (4) exactly leads tod{u}/dy~ uτ*/κy. Next, the profile of {u}D(yD) does not collapse
onto the I930, and this failure is not expected to be recovered
even for a comparison of matching Reτ. As noted by previous
studies (Modesti and Pirozzoli, 2016; Yao and Hussain, 2020),
however, profile still follows logarithmic variation with theidentical slope of 1/κc=1/0.41 with a certain extent. Then, theidentical approach with {u}TL(yTL) can be applied for {u}D(yD)
and it deduces d{u}/dy~ uτ*/κy. Therefore, both transformation
results resolve to same conclusion leading to the scaling law (1)and the successful transformations are consistent with the MH.
On the other hand, a region for the logarithmic variation with theslope of 1/κc =1/0.41 for {u}un is substantially narrowed (aroundat y+=100) so that the presence of the logarithmic region is notcertain. According to (1) consistent with present findings, it ispredictable because the scaling for mean velocity will fail whenusing constant value of uτ, instead of uτ* which varies with y.
On the other hand, the scaling laws for the Reynolds stresses(7-9) are predicted by the AEH and the Morkovin’s scaling(Smits and Dussage, 2006), the latter alternating the velocity
scale for the turbulent motions from uτ to uτ*.

Whilst, if profile of {u}T shows logarithmic variation in thetransformed coordinate (yT), the relation for the d{u}T/dyT
follows asd uT

(5)


y−
y
( )( ) 1 +
g = ( ) 1 +
y−
y




f TL =



=


1/ 2
 w ) (4)

It is stressed that relation (4) is valid only when the transformed
mean velocity follows logarithmic variation (3), while scaling
property of d{u}/dy expressed by (4) depends on the constitutionof compressibility transformation in use (i.e., kernel f T and gT).
From this procedure, one can obtain straight information about
the actual mean velocity {u} from scaling properties of thetransformed velocity. For the present analysis, we investigatetransformed mean velocity profiles of {u}D(yD) by van-Driest

3

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

uu = B1 − A1 log 
*

y


 

(7)

ww = B2 − A2 log (8)vv = B3

(9)

y

 

*

*

Here, Ai (i=1, 2, 3) and Bj (j=1, 2) are constant. According to theprediction of the AEH, scaling by inner variables fails for wallparallel components of Reynolds stresses due to thesuperposition of the energy-containing eddies having finite slip
velocity near the wall. While the wall-normal component isconstant due to restriction by the impermeable boundary
condition (Townsend 1976; Perry and Chong 1982).
Figure 2 provides profiles of Reynolds stress for C930 and
I930 in inner and outer coordinates. The profiles in innercoordinates adopt scaling by semi-local variables (Huang et al.,1995) and the profiles collapse well throughout whole layer asmatching the semi-local Reynolds number (Reτ*) between C930
and I930, except increased streamwise intensities near the wall.
While the profiles in the outer coordinate, consistent with the
Morkovin’s scaling, well matched in the outer layer even though
the frictional Reynolds numbers (Reτ) differ.
For each component, the scaling law of (8) for the spanwise
components is observed. For the incompressible case, thisbehavior is reported from previous study (Lee and Moser, 2015)
and it starts to emerge at Reτ≈1000 and, for the compressible case,scaling law of (8) is also observed from similar simulation
results by Modesti and Pirozzoli (2016) and Yao and Hussain(2020). Besides, the scaling law of (9) for the wall-normalcomponents are confirmed, also. This behavior is easily
observable in numerous literatures. However, the logarithmic
variation for the streamwise components (7) are not found forboth cases. Considering that the logarithmic variation of
streamwise normal stress is observable at much higher Reynolds
number in the incompressible flows (Hultmark et al., 2012;
Marusic et al., 2013), the logarithmic dependence for the
streamwise normal stress is remain unclear. But hopefully, thescaling by semi-local frictional velocity uτ* is very successful at
low Reynolds number, we can only expect the logarithmic
variation of streamwise normal stress is observable at muchhigher Reynolds number.
Although the above results showed that, at some extent, theturbulent statistics follow the scaling laws of (1) and (8-9) forthe compressible wall-bounded turbulence, using thecompressibility transformations and the semi-local variables,
rigorous logarithmic dependence are not achieved based on
indicator function, especially for the streamwise Reynolds stress,
due to relatively low-Reynolds number. Furthermore, even if thescaling laws (1) and (7-9) are valid, coherent structures

Figure 3. Iso-surfaces of identified wall-attached structures of
streamwise velocity fluctuations in an instantaneous flow field:(a) I930 and (b) C930. Red and Blue colors indicate positive andnegative fluctuations, respectively, and colors fade out as y/hincreases.

contributing to these scaling laws are not elucidated for thecompressible wall-bounded turbulence, yet.

WALL-ATTACHED STRUCTURES OF VELOCITY
FLUCTUATIONS
The other goal of the present study is to identify actual
coherent structures within random instantaneous flow fields,
which correspond to the AEH and contribute to the scaling laws.
To this end, clusters of velocity fluctuations are identified
through the methodology suggested by Hwang and Sung (2018;2019). The clusters of velocity fluctuations are defined as thegroups of connected points satisfying u″>+α{u″u″}1/2 or u″<α{u″u″}1/2, where α=1.6 based on percolation theory. Owing to
the advantage of DNS, three-dimensional spatial information of
the identified structures is accessible, and assigned to each of
structures. Based on their minimum distance from the wall, theidentified structures are classified into wall-attached anddetached structures. Examples of the wall-attached structures forC930 and I930 are provided in figure 3, and qualitative similarity
is found supporting the validity of the MH.
The dimensions of individual structures are determined bydimensions of their bounding boxes, and length, height andwidth of structures are denoted as lx, ly and lz, respectively.
Figure 4 illustrates the self-similarity nature of the identified
structures as mean length <lx> and width <lz> follow <lx>~ly0.74
and <lz>~ly, respectively. In addition, the number density of thestructures per unit area (ns) also showed their population
following the hierarchical length scale distribution (inversely
proportional to ly), which is not shown here. These results
indicates that the characteristics of the identified structures is
consistent with the AEH.
According to the AEH, the superposition of geometrically
self-similar hierarchies leads to the logarithmic variation of the
turbulent statistics. To determine whether the identified
structures exhibit the logarithmic variation of the turbulent
statistics in the sense of the AEH, statistical flow field isreconstructed from motions carried by the extracted wallattached structures showing both the self-similarity and thehierarchical length scale distribution, and detailed definition forreconstructed flow field can be found in Hwang and Sung (2018;2019). Figure 5(a) shows the logarithmic dependence of thereconstructed {u″u″}as from the identified structures, normalized

Figure 2. Reynolds stresses profiles in (a) inner and (b) outer
coordinates. Lines in each panel are profiles for C930 while
symbols indicate those for I930. Colored straight lines are added
as guides for logarithmic variation for the spanwise components

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

0

15
30

-2

10
20

-4



{u' ' u' ' }as*(a) 40

10

0.1

5

0.2

0.3 0.4

0{u' ' u' ' }as* /4

0
-5

8

10-3

6
4

y/h

10-1

100

{u' ' u' ' }*as{u' ' u' ' }*

2
0 -3
10

10-2

10

-2

y/h

10

-1

10

0

V10=V6/4.5
Figure 5. (a)pn+:(upper)Streamwise Reynolds stress reconstructed
V10=V6/4.0
from the identified wall-attached structures {u″u″}as based on
Morkovin’s scaling and (lower) comparison between
distribution of the reconstructed stress {u″u″}as and the actual
stress {u″u″}. Blue straight lines are added as guides for the
logarithmic variations. (b) Indicator functions Ξ=yd{u″u″}*as/dy
for estimating logarithmic variation. In each panel, comparisonis done as {u″u″}*as being reduced by factor of 4.

Figure 4. Variations of ensemble-averaged length <lx> (upper)
and width <lz> (lower) of the wall-attached structures withrespect to its height. Lines are for C930 while symbols represent
the I930. Colored right-angled triangles are added as guides.

by uτ*. Here, it is noted that due to the definition of the identified
structures (intense u″ by α=1.6) and {u″u″}as, the magnitude of{u″u″}as are much greater than that of {u″u″}. Since theobjective of the present study is to identify coherent structurescontributing to the scaling laws in terms of the AEH, comparison
here is done by reducing magnitudes as factor of 4, as focusing
the logarithmic variation, rather than the actual magnitude. In
this way, lower panel of figure 5(a) again confirms thelogarithmic dependence of {u″u″}as. which is absent in the actual
streamwise stress {u″u″}. The logarithmic dependence can beanalysed using indicator function Ξ=yd{u″u″} *as/dy, where aplateau region indicates complete logarithmic dependence. In
accordance with the figure 5(a), indicator function for {u″u″}asin figure 5(b) has been much flattened than that of {u″u″}, andit deviates from constant value much more gradually.
Additionally, similar analysis for the mean velocity for the
motions carried by the identified structures ({u}as) also revealed
more clear logarithmic dependence for the reconstructed flow
field (not shown here). As a result, we have confirmed thepresence of the coherent structures contributing to the scalinglaws for the compressible flow. And it is conjectured that theobserved logarithmic dependences in the reconstructed flowfields at the moderate-Reynolds number seem to be masked in
the actual mean velocity {u} and/or stress distribution {u″u″} byinsufficient scale separation or contamination of other scales.

laws. Through the present study, the integrating explanation forthe characteristics of the compressible wall-bounded turbulenceis established as providing the supporting evidence of the AEH
for the compressible wall-bounded turbulence.

REFERENCE
del Á lamo, J. C., Jiménez, J., Zandonade, P., and Moser, R.
D., 2006, “Self-similar vortex clusters in the turbulent
logarithmic region”, Journal of Fluid Mechanics, Vol. 561, pp.
329–358.
Bernardini, M., Modesti, D., Salvadore, F., and Pirozzoli, S.,
2021, “STREAmS: A high-fidelity accelerated solver for direct
numerical simulation of compressible turbulent flows”,
Computer Physics Communications, Vol. 263, 107906.
Bradshaw, P., 1994, “Turbulence: The chief outstanding
difficulty of our subject”, Experiments in Fluids, Vol. 16, pp.
203–216.
van Driest, E., 1951, “Turbulent boundary layer incompressible fluids”, Journal of the Aeronautical Sciences, Vol.
18, pp. 145-160.
Fernholz, H. H., and Finley, P. J., 1980, “A criticalcommentary on mean flow data for two-dimensional
compressible turbulent boundary layers”, AGARDograph 253.
Ganapathisubramani, B., Clemens, N. T., and Dolling, D. S.,
2006, “Large-scale motions in a supersonic turbulent boundary
layer”, Journal of Fluid Mechanics, Vol. 556, pp. 271–282.
Huang, P., Coleman, G., and Bradshaw, P., 1995,“Compressible turbulent channel flows: DNS results andmodelling”, Journal of Fluid Mechanics, Vol. 305, pp. 185-218.
Hultmark, M., Vallikivi, M., Bailey, S. C. C., and Smits, A.
J., 2012, “Turbulent pipe flow at extreme Reynolds numbers”,
Physics Review Letters, Vol. 108(9), 094501.
Hwang J., and Sung, H. J., 2018, “Wall-attached structuresof velocity fluctuations in a turbulent boundary layer”, Journalof Fluid Mechanics, Vol. 856, pp. 958–983.
Hwang, J., and Sung, H. J., 2019, “Wall-attached clusters forthe logarithmic velocity law in turbulent pipe flow”, Physics of
Fluids, Vol. 31 (5), 055109.
Kim, K. C., and Adrian, R. J., 1999, “Very large-scale
motion in the outer layer”, Physics of Fluids, Vol. 11(2), pp.
417–422.

CONCLUSION
The turbulent statistics obtained from direct numericalsimulation of supersonic turbulent channel flow between
isothermal walls showed the existence of y-scaling region withthe velocity scale to be uτ*. In this region, the transformed mean
velocity profiles showed logarithmic variation, while the
spanwise and wall-normal components of the Reynolds stresses
followed the prediction by the AEH. However, due to relatively
low-Reynolds number or insufficient scale separation, thecomplete logarithmic dependence is not achieved. Especially,
the streamwise normal stress is far from the prediction of theAEH. In search of the coherent structures responsible for the
logarithmic dependence of the turbulent statistics, reconstructed
flow field from the wall-attached self-similar structures showed
clear logarithmic variation for the streamwise normal stresses, as
manifesting their primary role for the formation of the scaling

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Lee, J., Lee, J. H., Choi, J.-I., and Sung, H. J., 2014, “Spatialorganization of large-and very-large-scale motions in a turbulent
channel flow”, Journal of Fluid Mechanics, Vol. 749, pp. 818–
840.
Lee, M., and Moser, R. D., 2015, “Direct numericalsimulation of turbulent channel flow up to Reτ≈5200”, Journalof Fluid Mechanics, Vol. 774, pp. 395–415.
Lozano-Durán, A., Flores, O., and Jiménez, J., 2012, “Thethree-dimensional structure of momentum transfer in turbulent
channels”, Journal of Fluid Mechanics, Vol. 694, pp. 100–130
Marusic, I., Monty, J. P., Hultmark, M., and Smits, A. J.,
2013, “On the logarithmic region in wall turbulence”, Journal of
Fluid Mechanics, Vol. 716, R3
Modesti, D., and Pirozzoli, S., 2016, “Reynolds and Mach
number effects in compressible turbulent channel flow”,
International Journal of Heat and Fluid Flow, Vol. 59, pp. 33–
49.
Modesti, D., and Pirozzoli, S., 2019, “Direct numericalsimulation of supersonic pipe flow at moderate Reynolds
number”, International Journal of Heat and Fluid Flow, Vol. 76,
pp. 100–112.
Morkovin, M., 1962, “Effects of compressibility on
turbulent flows”, Mécanique de la Turbulence. A. Favre, pp.
367–380.
Perry, A. E., and Chong, M. S., 1982, “On the mechanism ofwall turbulence”, Journal of Fluid Mechanics, Vol. 119, pp.
173–217.
Perry, A. E., Henbest, S., and Chong, M. S., 1986, “A
theoretical and experimental study of wall turbulence”, Journalof Fluid Mechanics, Vol. 165, pp. 163–199.
Perry, A. E., and Marusic, I., 1995, “A wall-wake model forthe turbulence structure of boundary layers. Part 1. Extension of
the attached eddy hypothesis”, Journal of Fluid Mechanics, Vol.
298, pp. 361–388.
Pirozzoli, S., 2012, “On the size of the energy-containing
eddies in the outer turbulent wall layer”, Journal of Fluid
Mechanics, Vol. 702, pp. 521–532.
Smits, A. J., and Dussage, J. P., 2006, Turbulent Shear
Layers in Supersonic Flow. Springer.
Townsend, A. A., 1976, The structure of turbulent shear
flow. Cambridge University Press.
Trettel, A. and Larsson, J., 2016, “Mean velocity scaling forcompressible wall turbulence with heat transfer”, Physics of
Fluids, Vol. 28 (2), 026102.
Yao, J., and Hussain, F., 2020, “Turbulence statistics andcoherent structures in compressible channel flow”, PhysicalReview Fluids, Vol. 5, 084603.

6