Streamwise inclination angle of wall-attached eddies in turbulent channel flows

It is generally recognized that the high-Reynolds number wall-bounded turbulence is filled with coherent motions of disparate scales, which are responsible for the energy transfer and the fluctuation generation of turbulence. Till now, the most elegant conceptual model describing these energy-containing motions is the attached-eddy model (Townsend, 1976; Perry & Chong, 1982). It hypothesizes that the logarithmic region is occupied by an array of randomly-distributed and self-similar energy-containing motions (or eddies) with their roots attached to the near-wall region (see Fig. 1). During the recent decades, a growing body of evidence that supports the attached-eddy hypothesis has emerged rapidly, e.g., Hwang (2015), Hwang & Sung (2018), Cheng et al. (2020b), Hwang et al. (2020), to name a few. The reader is referred to a recent review work by Marusic & Monty (2019) for more details. Throughout the paper, the terms ‘eddy’ and ‘motion’ are exchangeable. It should be noted that the terms of ‘wall-attached motions’ and ‘wall-attached eddies’ used in the present study do not only refer to the self-similar eddies in the logarithmic region, but also the very-large-scale motions (VLSMs) or superstructures, as some recent studies have shown that VLSMs are also wall-attached, despite that their physical characteristics do not match the attached-eddy model (Hwang & Sung, 2018; Yoon et al., 2020).

Previous studies have established that the energy-containing eddies populating the logarithmic and outer regions bear characteristic SIAs due to the mean shear (see Fig. 1($a$)). As early as the 1970s, Kovasznay et al. (1970) found that the large-scale structures in the outer intermittent region of a turbulent boundary layer have a moderate tilt in the streamwise direction. On the other hand, for the eddies in the logarithmic region of wall turbulence, the wall-attached $\Lambda-$vortex was used by Perry & Chong (1982) to illustrate them. According to Adrian et al. (2000), these $\Lambda-$vortexes are apt to cluster along the flow direction and form an integral whole (generally called as vortex packets). Further observations in channel flows (Christensen & Adrian, 2001) demonstrated that the heads of $\Lambda-$vortexes among the vortex packets tend to slope away from the wall in a statistical sense, with SIAs between $12^{\circ}$ and $13^{\circ}$. Most additional studies have shown a similar result, and it is widely accepted that the approximate SIAs of eddies are in the range of $10^{\circ}$ to $16^{\circ}$ (Boppe et al., 1999; Christensen & Adrian, 2001; Carper & Porté-Agel, 2004; Marusic & Heuer, 2007; Baars et al., 2016). Besides, the SIA is also found to be Reynolds-number independent (Marusic & Heuer, 2007).

However, the SIA estimated by experimentalists using the traditional statistical approach is the mean structure angle indeed (Marusic & Heuer, 2007; Deshpande et al., 2019). The common procedure to obtain the SIA is based on the calculation of the cross correlation between the streamwise wall-shear stress fluctuation ($\tau_{x}^{\prime}$) and the streamwise velocity fluctuation ($u^{\prime}$) at a wall-normal position in the log region ($y_{o}$). The cross correlation can be expressed as

where $<\cdot>$ represents the ensemble spatial average, and $\Delta x$ the streamwise delay. The SIA can be estimated by

where $\Delta x_{p}$ denotes the streamwise delay corresponding to the peak in $R_{\tau_{x}^{\prime}u^{\prime}}$. Considering that an array of wall-attached eddies with distinct wall-normal heights can simultaneously convect past the reference position $y_{o}$, $\alpha_{m}$ in Eq. (2) should be regarded as the mean angle of these eddies. Hence, the subscript ‘m’ in Eq. (2) refers to ‘mean’.

To estimate the SIAs of the largest wall-attached eddies, Deshpande et al. (2019) introduced a spanwise offset between the near-wall and logarithmic probes to isolate these wall-attached motions in the log region. They found that their SIAs are approximately $45^{\circ}$. This observation is consistent with several theoretical analyses. For example, Moin & Kim (1985) and Perry et al. (1992) proposed that for the flows with two-dimensional mean flows, the characteristic angles of the energy-containing eddies should follow the direction of the principal rate of mean strain. More specifically, their SIAs should be $45^{\circ}$ for a zero-pressure-gradient turbulent boundary layer (Perry et al., 1992). Marusic (2001) found that the mean SIA of the induced turbulence field by attached eddies is akin to the experimental measurements, if the hierarchical attached eddies tilt away from the wall with individual SIA being $45^{\circ}$ and organize like the vortex packets observed in numerical and laboratory experiments.

Reviewing the work of predecessors, it can be found the SIAs of attached eddies at a given length scale are ambiguous. Traditional measurements are only applicable for the assessment of the mean SIA (Brown & Thomas, 1977; Boppe et al., 1999; Marusic & Heuer, 2007). Moreover, the technique adopted by Deshpande et al. (2019) can only isolate the largest wall-attached motions in the logarithmic region. Considering the characteristic scale of an individual attached eddy being its wall-normal height as per the attached-eddy model (Townsend, 1976; Perry & Chong, 1982), it is self-evident that it is of great importance to assess the SIAs of attached eddies with any heights in the logarithmic region, not only for the completeness of attached-eddy hypothesis, but also the accuracy of turbulence simulations (Marusic, 2001; Carper & Porté-Agel, 2004). In the present study, we aim to achieve this goal by leaning upon the modified spectral stochastic estimation (SSE) proposed by Baars et al. (2016), and dissecting the direct numerical simulations (DNS) database spanning broad-band Reynolds numbers. We will also discuss the relationship between the mean SIA and the scale-based SIA.

Before investigating the SIAs of attached eddies, it is important to study the characteristic scales of $\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$ first. Figs. 4($a$) and 4($b$) show their streamwise premultiplied spectra at $y_{o}=0.1h$ and $y_{o}=0.2h$ for Re2000, respectively. The spectra of $\tau_{x}^{{}^{\prime}}$ and $u^{\prime}$ of the full-channel data are also included for comparison. Each spectrum is normalized with its maximum value. It can be seen that the spectra of $\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$ are roughly coincident, and peak at $\lambda_{x}=2.1h$ for $y=0.1h$, and $\lambda_{x}=4.2h$ for $y=0.2h$, respectively. By contrast, the spectra of $\tau_{x}^{{}^{\prime}}$ and $u^{\prime}$ do not share similar spectral characteristics. It is noted that $\Delta u_{W}^{{}^{\prime}2}$ and $\Delta\tau_{x,L}^{{}^{\prime}2}$ only account for very little energy of the full-channel signals at the same wall-normal positions. For example, $\Delta u_{W}^{{}^{\prime}2}$ at $y_{o}=0.1h$ and $y_{o}=0.2h$ occupies 0.0034$\%$ and 0.002$\%$ of $u^{{}^{\prime}2}$ at the corresponding positions, respectively, whereas $\Delta\tau_{x,L}^{{}^{\prime}2}$ for $y_{o}=0.1h$ and $y_{o}=0.2h$ occupies 0.012$\%$ and 0.0045$\%$ of $\tau_{x}^{{}^{\prime}2}$, respectively. Moreover, comparing with the spectra of the full-channel data, the spectra of $\Delta u_{W}^{{}^{\prime}}$ decay rapidly when $\lambda_{x}\geq 4h$ (see Figs. 4), which indicates that the effects of VLSMs on $\Delta u_{W}^{{}^{\prime}}$ are rather limited.

Fig. 5 shows the streamwise premultiplied spectra of $\Delta u_{W}^{{}^{\prime}}$ around $y_{o}=0.05h$ and $y_{o}=0.1h$. Each spectrum is normalized by the energy of $\Delta u_{W}^{{}^{\prime}}$ at a given $y_{m}$. Clear plateau regions can be observed around the spectral peaks. For $y_{o}=0.05h$, the region is between $18\leq\lambda_{x}/y_{m}\leq 30$, and for $y_{o}=0.1h$, it is between $17\leq\lambda_{x}/y_{m}\leq 31$ , which corresponds to the $k_{x}^{-1}$ region in the spectrum predicated by the attached-eddy model, and can be considered as the spectral signatures of the attached eddies (Perry & Chong, 1982; Perry et al., 1986; Hwang et al., 2020; Deshpande et al., 2021). Besides, the spectra shown here resemble the spectrum of the type A eddies hypothesized by Marusic & Perry (1995), i.e., the energy fraction captured by the attached-eddy model. These observations support the proposition that the $\Delta u_{W}^{{}^{\prime}}$ signals are the streamwise velocity fluctuations carried by the self-similar attached eddies predominantly. Moreover, they also indicate that the streamwise length scales of the dominant eddies increase with $y_{o}$, as the self-similar range is not altered significantly with increasing $y_{o}$.

To further investigate the scale characteristics of $\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$, the autocorrelation function of $\Delta u_{W,p}^{{}^{\prime}+}$ (the signals that are extracted from the spectral peaks shown in Fig. 5, i.e., filtered $\Delta u_{W}^{{}^{\prime}}$ with wavelength larger than $17y_{m}$, but smaller than $31y_{m}$) is considered, which takes the form of

and the counterpart of $\Delta\tau_{x,L}^{{}^{\prime}+}$ can also be defined similarly. Fig. 6($a$) shows the variations of $R_{\Delta u_{W,p}^{{}^{\prime}}\Delta u_{W,p}^{{}^{\prime}}}$ as functions of $\Delta x/h$ for two selected $y_{o}$. The larger $y_{o}$, the broader the $R_{\Delta u_{W,p}^{{}^{\prime}}\Delta u_{W,p}^{{}^{\prime}}}$. As a measure of the typical length scale, we employ $\Delta s/h$, which is the streamwise delay corresponding to $R_{\Delta u_{W,p}^{{}^{\prime}}\Delta u_{W,p}^{{}^{\prime}}}=0.05$ or $R_{\Delta\tau_{x,L,p}^{{}^{\prime}}\Delta\tau_{x,L,p}^{{}^{\prime}}}=0.05$ (here, 0.05 is an empirical small positive threshold). Fig. 6($b$) shows the variations of $2\Delta s/h$ as functions of $y_{m}/h$ for $\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$. For both $\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$, $2\Delta s/h$ increases linearly with $y_{m}/h$ throughout most of the logarithmic region. This observation is consistent with the attached-eddy hypothesis, which states that the length scales of the attached eddies grow linearly with their wall-normal heights (Hwang, 2015; Marusic & Monty, 2019). Moreover, both the streamwise length scales of $\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$ follow $2\Delta s=10.8y_{m}$ (consider the symmetry of the autocorrelation function with respect to $\Delta x=0$, $2\Delta s$ truly represents the streamwise length scale of the signals). This scale characteristic agrees well with some previous studies. For example, Baars et al. (2017) showed that the streamwise/wall-normal aspect ratio of the wall-attached eddy structure is $\lambda_{x}/y=14$ in turbulent boundary layers, which is close to the result here. Hwang et al. (2020) reported that the spectra of the self-similar wall-attached structures agree with the attached-eddy hypothesis at $\lambda_{x}=12y$, which is consistent with the estimation of the present study. All these observations indicate that $\Delta\tau_{x,L}^{{}^{\prime}+}$ and $\Delta u_{W}^{{}^{\prime}+}$ are representative of the attached eddies at a certain wall-normal height, though the minor influences of VLSMs still exist, and treating $y_{m}^{+}$ as their characteristic scales is reasonable.

In summary, all the observations mentioned above indicate that $\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$ are the outcomes of the energy-containing motions with the wall-normal heights approximately equal to $y_{m}$, and the cross correlation, i.e., Eq. (9), truly reflects the phase difference between the streamwise velocity fluctuations carried by these motions and their footprints in the near-wall region. Other wall-normal positions and DNS cases yield similar results and are not shown here for brevity.

Fig. 7($a$) shows the variations of $R_{LW}$ as functions of the streamwise delay for some selected wall-normal positions in the case Re2000. Since the streamwise length scales of the energy-containing motions are increased with their normal heights (see Fig. 4), $R_{LW}$ becomes wider about the peak with increasing $y_{o}$. $\Delta x_{p}$ can be identified obviously from the cross-correlation profiles, and the SIAs of the attached eddies at a given wall-normal height can be calculated according to Eq. (8). Fig. 7($b$) plots the variations of the normalized $R_{LW}$ as functions of $\Delta x/y_{m}$ for some selected $y_{o}$ in the case Re2000. The $R_{LW}$ distributions are normalized with their maximum values $R_{LW,max}$. It can be seen that the profiles of $R_{LW}/R_{LW,max}$ for different wall-normal heights coincide well with each other, which indicates the self-similar characteristics of the energy-containing motions in the logarithmic region. We have checked that the correlations calculated from the raw data, i.e., $R_{\tau_{x}^{\prime}u^{\prime}}$ in Eq. (1) can not coincide if normalized in this manner. Again, it demonstrates that the new methodology is capable of capturing the main properties of the attached eddies.

Fig. 8 plots the variations of $\alpha_{s}$ as functions of $y_{m}^{+}$ for all cases. $\alpha_{s}$ increases approximately from $27^{\circ}$ for Re550, to $40^{\circ}$ for Re4200. For a given case, $\alpha_{s}$ changes little spanning the logarithmic region except for the upper part of logarithmic region in Re4200. Deshpande et al. (2019) isolated the large wall-attached structures in a DNS of turbulent boundary layer at $Re_{\tau}\approx 2000$, and found the corresponding SIAs to be $32^{\circ}$ (see Fig. 4($a$) of their paper). Their observation is consistent with the results of the present study. However, Deshpande et al. (2019) only calculated the SIAs of the largest wall-attached motions in the logarithmic region due to the limitation of the methodology adopted in their study, whereas we make a thorough investigation on the SIAs of attached eddies with any wall-normal heights in the logarithmic region. Moreover, Deshpande et al. (2019) reported that the SIAs of the large wall-attached motions identified in a wind-tunnel boundary layer with $Re_{\tau}=14000$ are approximately $50^{\circ}$. They ascribed the result difference between DNS and experiment to the limited streamwise scale range owing to the DNS domain size selected for analysis. Our results reveal that the Reynolds number effects play a non-negligible role in the formation of SIAs of attached eddies. To the authors’ knowledge, this is the first time that the Reynolds-number dependence of SIAs of the wall-attached motions at a given length scale has been clearly shown. At last, it should be noted that $\alpha_{s}$ of Re4200 decreases rapidly for $y_{m}^{+}>500$ (not shown here). This diversity is due to the small computational domain size along the streamwise direction in this database. Thus, in the discussion below, the statistics of $\alpha_{s}$ in the range of $y_{m}^{+}>500$ in Re4200 will not be taken into account. The sensitivity of the presented results to the number of instantaneous flow fields employed for accumulating statistics is examined in the Appendix A.

Fig. 9($a$) shows the mean $\alpha_{s}$ ($\alpha_{s,m}$) distribution in the range of logarithmic region as a function of the friction Reynolds number. It can be seen that the SIA may reach the theoretical prediction angle $45^{\circ}$ (Perry et al., 1992) when $Re_{\tau}\sim O(10^{4})$. The results of DNS of a turbulent boundary layer and wind-tunnel experiment of Deshpande et al. (2019) are roughly agreed with the tendency. The minor differences may result from the distinct configurations of the wall-bounded turbulence.

In the present study, we develop a methodology to assess the streamwise inclination angles of the wall-attached eddies at a given wall-normal height in turbulent channel flows, by coupling the spectral stochastic estimation with the attached-eddy hypothesis. Our results show, for the first time, that the SIAs of the attached eddies are Reynolds-number dependent in low and medium Reynolds numbers and tend to be consistent with the theoretical prediction (i.e., $\alpha_{s}=45^{\circ}$) as Reynolds number increased. We further reveal that the mean SIA reported by vast previous studies are the outcomes of the additive effect contributed by multi-scale attached eddies.

The attached-eddy model has been the guidance for the reconstruction of the velocity field in wall turbulence (Perry & Marusic, 1995; Chandran et al., 2017; Baidya et al., 2017). Hierarchical vortex packets which consist of $\Lambda-$vortexes with $\alpha_{s}=45^{\circ}$ are distributed on the wall surface to mimic the attached eddies. The present results suggest that a lower SIA of representative structures might be helpful for a more accurate reconstruction when the Reynolds number is not high enough. Moreover, within the state-of-the-art wall-modelled large-eddy simulation (WMLES) framework, one may estimate the instantaneous $\tau_{x}$ based on the velocities carried by the log-region eddies (Fu et al., 2021, 2022). The Reynolds-number dependence of SIAs of these eddies should be accounted for by an advanced model in this sense.

L.F. acknowledges the fund from CORE as a joint research center for ocean research between QNLM and HKUST, and the fund from Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011779). We would like to thank Professor Jiménez for making the DNS data available. We also express our gratitude to the reviewers of this paper for their kind and constructive comments.

The authors report no conflict of interest.

The influences of the number of instantaneous flow fields for accumulating statistics are examined. Fig. 12 shows the effect of $N_{F}$ on the statistic $\alpha_{s}$ for the cases Re2000 and Re4200. Alteration of the statistical samples mainly affects the relative standard deviations (RSD) of the results. To be specific, when $N_{F}$ increases from $48$ to $94$, RSD decreases from $3.9\%$ to $3.3\%$ for Re2000; but for Re4200, RSD decreases from $6.5\%$ to $3.7\%$ when $N_{F}$ increases from $20$ to $40$. Given the fact that the case Re4200 has limited domain size, raising $N_{F}$ can effectively reduce the wiggles in the outputs. Nevertheless, the mean value of $\alpha_{s}$ in the logarithmic region seems to be insensitive to $N_{F}$.