ABSTRACT

INTRODUCTION
Wall turbulence is characterized by a broad range of scales, from the viscous length scale (δν) to the outer length scale (δ).  The friction Reynolds number ( Re τ = δ/δν ) de- scribes such a multiscale phenomena.  At extremely high Re τ , there is a region that is sufficiently far from the wall and far from the outer region.  This region is the so-called logarith- mic region where the only relevant length scale is the distance from the wall y, and the mean velocity profile follows the loga- rithmic variation with respect to y (Millikan, 1938). Townsend (1976) hypothesized that coherent motions in the logarithmic region are self-similar and that their sizes are proportional to y.  Townsend's hypothesis was further extended by Perry & Chong (1982), who deduced the attached-eddy model. Based on the attached-eddy model, they expected that the logarithmic variations occur in the mean velocity and in the wall-parallel 1 components of turbulence intensity. The model also shows that self-similar motions contribute to a k − 1 scaling in the one- dimensional spectra of the streamwise velocity, where k x is the streamwise wavenumber.  The k − 1 region was predicted by Perry & Abell (1977), who assumed that there is a spectral overlap region where y and δ scalings hold simultaneously. In this sense, the k – 1 region can be a consequence of the AEH and is deemed the spectral signature of attached eddies (Perry & Chong, 1982; Perry et al. , 1986). However, although some studies have shown empirical evidence for the existence of the k − 1 region (Nickels et al. , 2005; Vallikivi et al. , 2015), it re- mains unclear whether such scaling exists in a high-Reynolds- number flow (Rosenberg et al. , 2013; Lee & Moser, 2015; Ahn et al. , 2015; Agostini & Leschziner, 2017; Chandran et al. , 2017; Baars & Marusic, 2020). The ambiguities in the spectral signatures of the attached eddies are related to insufficient scale separation because the AEH requires that the Reynolds number approaches infinity. This means no region exists in which both y and δ scalings are valid over the same wavenumber space, even in the high- Reynolds-number experiments Re = O(10 4 −5) of the afore- mentioned studies.  However, given the fact that the self- similar energy-containing motions follow a hierarchical distri- bution (Perry & Chong, 1982), we may expect that self-similar motions can exist even if scale separation is insufficient. Ow- ing to insufficient separation of scales, self-similar motions are less statistically dominant than other coexisting motions, which in turn leads to ambiguity of the asymptotic behaviours in turbulent statistics. Recently, Hwang & Sung (2018) extracted the wall- attached structures of the streamwise velocity fluctuations ( u ) in turbulent boundary layer (TBL) and found that the identi- fied structures were self-similar in terms of their heights ( l y ), directly contributing to the logarithmic variation in the stream- wise turbulence intensity.  In addition, they showed that the population density of the identified structures was inversely proportional to l , , , reminiscent of the hierarchies of self-similar eddies (Townsend, 1976; Perry & Chong, 1982). Moreover, in pipe flows, the profile of the streamwise velocity reconstructed by the superposition of the wall-attached u structures exhibits the logarithmic variation (Hwang & Sung, 2019). Despite ev- idence on the self-similarity of wall-attached u structures and their contribution to the logarithmic behaviors, their spectral contributions and the turbulence motions contained within the identified structures have not been revealed. Because the wall- attached u structures are identified in instantaneous fluctuating velocity fields, a wide range of scales is contained within the individual structure. Therefore, spectral analysis is required to reveal whether the large scales contained within the identified object can be attributed to the logarithmic variation or exhibit the spectral overlap argument proposed by Perry and cowork- ers (Perry & Chong, 1982). The objective of the present study is to explore the spec- tral contribution of turbulence motions that comprise wall- attached u clusters by computing the two-dimensional spec- tra of u , in which the velocity signals contained within self- similar structures are isolated. We examined the direct numer- ical simulation (DNS) data of a fully developed turbulent chan- nel and pipe flows, along with zero-pressure-gradient TBL at Re τ ≈ 1000, and identify the wall-attached self-similar struc- tures by applying universal filters in terms of height. The wall- attached u clusters were decomposed into the buffer-layer, self-similar, and non-self-similar structures in terms of their height. The wall-attached self-similar structures were further examined using the two-dimensional energy spectra to reveal the self-similar behaviours in the logarithmic region of all three flows.  We then explored the one-dimensional stream- wise and spanwise spectra by comparing the energy contained in the spectral range where the energetic ridges in the two- dimensional spectra follow a linear relationship between the wall-parallel wavelengths.
NUMERICAL DETAILS
In this study, we analysed the DNS data of the zero- pressure gradient TBL (Hwang & Sung, 2017; Yoon et al., 2018) as well as the fully developed turbulent channel (Lee et al. , 2014, 2015) and pipe flows (Ahn et al. , 2013, 2015). A detailed description of the DNS can be found in the afore- mentioned studies. The friction Reynolds numbers, defined as the ratio of the outer length scale to the viscous length scale, match at Reτ = uτδ/ν ≈ 1000.  Here, uτ is the friction ve- locity, v is the kinematic velocity, and δ is the flow thickness (i.e.  channel half-height, pipe radius, or 99% boundary layer thickness).  Throughout the present work, the superscript ++ represents viscous scaling ( v / u τ and u τ ). In the present study, x y, and z indicate the streamwise, wall-normal, and span- wise directions, respectively. In the pipe flow, the wall-normal direction is defined as y = δ − r, where r denotes the radial direction.  In addition, for an analogy with z in the TBL and channel flows, the spanwise dimension of the pipe is defined as the arclength rθ , where θ denotes the azimuthal direction. We define the streamwise velocity fluctuations u = U − U (y), where U is the streamwise velocity, and the overbar denotes the ensemble average. We identified the clusters of u in the instantaneous flow fields by extracting the contiguous points of the intense u re- gion (Hwang & Sung, 2018, 2019; Han et al. , 2019; Yoon et al., 2020). In the three-dimensional flow field, the irregu- lar shapes of the objects are defined as
where u rms is the standard deviation of the streamwise veloc- ity and α is the threshold.  Individual objects are extracted using the connectivity rule, in which nodes are labelled among the six orthogonal neighbours of each node satisfying (1) in Cartesian coordinates (Moisy & Jiménez, 2004; del Álamo et al. , 2006; Lozano-Durán et al. , 2012; Hwang & Sung, 2018). and cylindrical coordinates (Hwang & Sung, 2019; Han et al. , 2019). We chose the threshold α = 1.5 for all three flows; fur- ther discussion can be found in our previous works (Hwang & Sung, 2018; Yoon et al. , 2020).  The present identification method allows us to measure the physical length scales of indi- vidual structures in instantaneous flow fields. Each structure is bounded by a box of size l x × l y × l z , where the corresponding length, height, and width are denoted by l x , l y , and l z . In the pipe flow, l z is computed in terms of the maximum arc length in the plane obtained by projecting each object onto the cross- stream plane.
RESULTS AND DISCUSSION
To extract the wall-attached self-similar structures (WASS), we examined the mean streamwise length ⟨ l x ⟩ and the mean spanwise width 〈 l z 〉 at a given l , (figure 1). In fig- ure 1(a), we can see a power-law behavior (i.e. 〈l x 〉 ～ l 0 74 ) over 3Re 1/2 < l,+ < 0.6δ+ (figure 1a ) in all three flows. On the other hand, for the width of the identified structures (fig- ure 1 b ), the linear relationship is observed; i.e., { z } ～ l , over l † > 3Re 1/2 .  Although no linear relationship exists for the wall-attached u structures over 3Re t //  2 ≤ I + < 0.6δ + in fig- ure 1( a ), the sizes of the structures are scaled with I , and show a good agreement regardless of flow geometry. Accord- ing to figure 1, we classified the wall-attached u structures as three components in terms of l y : buffer-layer structures, self- similar structures, and non-self-similar structures, defined as l † < 3Re † /2 , 3Re † /2 ≤ l † < 0.6δ † , and 0.6δ ≤ l † , respec- tively. Here, the lower bound (= 3Re 1 /2 ) of self-similar struc- tures corresponds to that of the logarithmic region in Marusic et al. (2013) and Hwang & Sung (2019). The present Reynolds number is Re τ ≈ 1000; therefore, the lower bound of the loga- rithmic region is approximately 100 wall units (3Re 7 ≈ 100), that is, a classical scaling for the lower bound of the logarith- mic region (Perry & Chong, 1982). Next, we conditionally sampled u signals contained within the bounded volume of each WASS in instantaneous flow field ( u w 5 ) to examine the spectral contributions of various scales associated within the WASS. The u signals outside the WASS are artifically imposed to zero (Lee et al. , 2015; Hwang et al. , 2016b , 2020). The contribution of coherent motions in- grained within WASS to the streamwise variance is explored by computing the premultiplied two-dimensional spectrum of Uws (figure 2).
where k x (= 2π/λ x ) and k z (= 2π/λ ) are the streamwise and spanwise wavenumbers, and îtws indicates the Fourier coeffi- cient of u w s and the asterisk denotes a complex conjugate. In general, all three flows show that Φ 2 D scales reason- ably well with y in the range y < λ r < 10y (figure 2a,c,e) and with δ in the range λ x > δ (figure 2b,d ,f).  In particu- lar, the bounds of the constant energy distribution are aligned along a linear relationship between the streamwise and span- wise wavelengths (λ x and λ z , respectively) in the large-scale. range (12y < λ r < 3–4δ).  This spectral feature is found in all three flows indicative of the universality of the self-similar nature of the large scales contained within WASSs.  In addi- tion, λ x > 12y is consistent with the lower limit of the k − 1 region in channel flows (Hwang, 2015), and is close to the inner-scaling limit (λ x = 14y) of self-similar motions in TBLs over a wide range of Re τ (Baars et al. , 2017). Deshpande et al. (2020) also found the existence of linear behaviour in two- dimensional cross spectra of wall-coherent u motions over a similar range (i.e. λ x > 15y). This shows that the WASS iden- tified in the present study is directly related to the self-similar behaviour in the wavenumber space. However, the wall-attached u structures defined in the present study (Eq.(1)) depend on the threshold value α be- cause we identified the structures by extracting the physically connected volumes of intense fluctuations. Therefore, we ex- amined the influence of the threshold value over a certain range. in the vicinity of α ≈ 1.5. The threshold effect on the popu- lation density and the sizes of the bounding box was reported in Hwang & Sung (2018, 2019). Figure 3 shows the effect of α on (Φ 2 D . Here, we plot Φ 2 D for TBL only to avoid any repetition.  We can see that all the contours collapse reason- ably well regardless of α . In particular, the linear relationship ( λ τ λ ε ) appears at λ τ > 12y, and the lower bounds align along λ x = 10λ, regardless of the threshold. This result shows that the conditionally sampled flow field u w s can represent a contin- uous range of scales related to the energy-containing motions in the logarithmic region. Figure 4 shows the spectral ridge of Φ 2 D . We determined the spectral ridge by identifying λ ς , where the maximum value of Φ 2 D occurs at a given λ x . As seen, the spectral ridges are found to agree reasonably well over a wide range of scales. In particular, we observed two growth rates, the power-law behaviour λ̥λ ～ λ ₂ (dashed line), and the linear relationship λ x ～ λ  z (solid line), at relatively large scales.  The transition of the ridges from the power law to the linear law appears at λ x = 12y.  This result is consistent with the variation of the lower and upper bounds of Φ 2 D found in figure 2. The spec- tral ridges become flatten for λ x > 3 − 4 δ because at this very long λ x , the spanwise wavelength λ z reaches δ and the growth of λ z is restricted. In figure 4, the linear behaviour seems to follow λ x ≈ 4λ z over the range 12y < λ τ < 3 − 4δ . Here, the inset shows the ridge scale ratio λ x / λ z for the TBL and channel data.  A plateau of λ x /λ z ≈ 4 is clear, indicating the linear relationship. There- fore, the range of the self-similar energetic motions ingrained in WASS can be expressed as
(3) Here, the upper limit for the streamwise wavelength in Eq. (3) is 3 – 4δ, similar to that of the criteria that distinguish large-scale motions and very-large-scale motions (Guala et al. , 2006; Wu et al. , 2012; Hwang et al. , 2016a ).  Given that the contours of Φ 2 D are restricted to below λ x ≈ 3 − 4δ in figure 2, this result also shows that Φ 2 P comprises the contribution from the turbulence motions that include large-scale motions and relatively smaller motions. According to Deshpande et al. (2020), the two-dimensional spectra of wall-coherent motions at Ret ≈ 15000 are aligned along a linear ridge λ t = 7λ τ, which is slightly steeper than the proportionality λ x = 4λ z found in the present work.  However, the spectra reported by Deshpande et al. (2020) includes the contributions from wall- attached self-similar and non-self-similar motions.  Given that very large scales can contaminate self-similar behaviours of turbulent motions (Jiménez & Hoyas, 2008; Hwang & Sung, 2018; Han et al. , 2019), it would be instructive in future efforts to examine the energetic ridges of Φ 2 D over a wide range of Re -
CONCLUSIONS
We explored the wall-attached self-similar structures (WASS) of u from the viewpoint of Townsend's AEH, with special focus on the spectral contribution of turbulence mo- tions contained within the identified structures.  We extracted the wall-attached structures of u in the DNS data of a zero- pressure-gradient TBL, and turbulent channel and pipe flows at Re τ ≈ 1000 by extracting the intense u regions across the instantaneous flow fields.  Then, the wall-attached structures of u are classified as buffer-layer, self-similar, and non-self- similar structures with respect to the height ( l y ).  The varia- tions in the physical sizes of WASS show a good agreement in all three flows as well as scale with l y .  We also examined the two-dimensional spectra of u within WASS to explore the spectral signatures of self-similarity.  Across the logarithmic region, we found that the lower and upper bounds of the two- dimensional spectra follow a linear relationship λ x ～ λ z in the large-scale range ( λ x > 12y).  Moreover, the spectral ridges exhibit λ x ≈ 4 x  over the range 12y < λ r < 3 − 4δ, showing that only the large scales contained in WASS are self-similar. Our results indicate that we can capture the asymptotic be- haviours of turbulent statistics, predicted by the AEH, when we adequately filter out contributions from coexisting non- self-similar motions.
ACKNOWLEDGEMENTS
This work was supported by the National Research Foun- dation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1048537) and partially supported by the Supercomputing Center (KISTI).