3.1.1. Mean velocity profiles

Most of the previous works in OCFs focused on the inspection of the mean velocity profiles to assess whether they follow an universal logarithmic law of the wall (e.g. Finley, Phoe & Poh Reference Finley, Phoe and Poh1966; Nezu & Rodi Reference Nezu and Rodi1986; Bergstrom, Tachie & Balachandar Reference Bergstrom, Tachie and Balachandar2001; Bonakdari et al. Reference Bonakdari, Larrarte, Lassabatere and Joannis2008). Figure 2(a) shows the non-dimensionalised mean streamwise velocity profile $U^+(=U/u_\tau )$ for all the datasets listed in tables 1 and 2. Hereinafter, the superscript $+$ indicates non-dimensionalisation in wall units, i.e. with kinematic viscosity $\nu$ and friction velocity $u_\tau$. As expected, the profiles for the OCF match well with those for CCF, with only a small discrepancy in the vicinity of the free surface, which is due to a stronger wake for the latter (Finley et al. Reference Finley, Phoe and Poh1966).

Figure 2. (a) Mean streamwise velocity and the log-law indicator function $\beta$ in (b) wall and (c) outer units. Here, solid and dashed lines represent OCF and CCF, respectively (see tables 1 and 2 for details of the line types and symbols). In (a), each profile has been vertically shifted for better display, and the dash-dotted lines represent the law of the wall: linear law $U^+=y^+$ and log law $U^+=(1/\kappa )\ln y^++B$ with $\kappa =0.363$ and $B=3.45$. In (b,c), the dash-dotted lines denote $1/\kappa$ with $\kappa =0.363$ and $0.384$ top down, respectively.

Despite extensive studies, it is still unclear whether/how the mean velocity profiles in OCF differ from other types of wall turbulence in the overlap region. Early DNS of OCF at very low $Re$ regimes (e.g. $Re_\tau =134$ for Handler et al. (Reference Handler, Swean, Leighton and Swearingen1993) and $171$ for Lam & Banerjee (Reference Lam and Banerjee1992)) suggest that there is an open channel effect on the log law: either a smaller Kárman constant $\kappa$ or smaller constant $B$. Later, Bauer (Reference Bauer2015) found that both $\kappa$ and $B$ match well between OCF and CCF for $Re_\tau$ up to $600$. Figures 2(b) and 2(c) shows the log-law indicator function $\beta =y^+\partial U^+/\partial y^+$ in inner and outer scales, respectively. It is clear that there is notable differences between OCF and CCF, particularly at the low $Re_\tau$, and the discrepancy in the near-wall region decreases with increasing $Re_\tau$. For $Re_\tau =2000$, a good agreement can be observed between OCF and CCF up to $y^+\approx 300$. It is consistent with that observed by Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009) for CCF, pipe and boundary layers, suggesting a near-wall universality of the inner scaled mean velocity for all types of canonical wall turbulence.

However, unlike CCF, where a constant region of $\beta$ has not yet developed for $Re_\tau \le 2000$, there is a distinct plateau for OCF, starting from $y^+=500$ and extending up to $y^+\approx 1200$. The experimental data of Duan et al. (Reference Duan, Chen, Li and Zhong2020) appear too noisy to confirm the existence of constant $\beta$. Note that high-order corrections to the log-law were sometimes introduced to better fit the mean velocity profile of CCFs at low $Re_\tau$ in the overlap region (Afzal & Yajnik Reference Afzal and Yajnik1973; Jiménez & Moser Reference Jiménez and Moser2007). Recent higher-Reynolds-number DNS of CCF by Lee & Moser (Reference Lee and Moser2015) and Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) found that a constant $\beta$ region would occur for $Re_\tau \ge 5200$, indicating that the minimum $Re_\tau$ threshold for $U^+$ to develop a logarithmic region should be lower in OCFs than in CCFs.

By fitting our OCF2000 data in the region $500\le y^+\le 1200$, we obtain $\kappa =0.363$, which is slightly smaller than $0.384$ found by Lee & Moser (Reference Lee and Moser2015) and $0.394$ by Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) for CCF. However, according to figure 2(c), the value of plateau may decrease with increasing Reynolds numbers, indicating a larger $\kappa$ at very high $Re_\tau$ that may eventually match with $\kappa$ reported for CCFs. Therefore, the possibility of a universal $\kappa$ for OCFs and CCFs as $Re_\tau \to \infty$ cannot be excluded. If so, it will lay as strong evidence for an universal wall function (Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). On the other hand, there is no reason to ignore the influence of outer geometry on the value of $\kappa$, which may indeed differ among different flows. In fact, the value of $\kappa$ and its universality among different flow geometries are highly debated (Nagib & Chauhan Reference Nagib and Chauhan2008; She, Chen & Hussain Reference She, Chen and Hussain2017). Moreover, the current range of fitting for the value of $\kappa$ is from $y/h\approx 0.25$ to $0.6$ for OCF2000, which is beyond the normally considered logarithmic region $y/h \approx 0.1$–$0.2$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). Hence, whether this choice is the true logarithmic range for OCF needs further study. In any case, higher $Re_\tau$ data for OCFs are needed to clarify this significant issue.

3.1.2. Reynolds stresses

Figure 3 shows the normalised non-zero Reynolds stress tensor components (or velocity variances and covariances) $\tau ^+_{ij}=\left\langle {u'_iu'_j}\right\rangle /u^2_\tau$, with $i,j=1,2,3$ denoting streamwise, wall-normal and spanwise directions. Hereinafter, the velocity fluctuations are denoted using the prime symbol (e.g. $u'$), and the ensemble-averaged (both in time and space) quantities of the mean velocity and velocity fluctuations are expressed using bracket (e.g. $\left\langle {u' v'}\right\rangle$). Overall, all these quantities agree well with the CCF at the same $Re_\tau$ in the near-wall region but differ in the outer region due to the presence of a free surface. Our DNS results differ from the experimental data EXP600, EXP1000 and EXP1900, particularly in the region close to the wall, where the experimental data have large uncertainties.

Figure 3. Reynolds stress components as a function of $y^+$: (a) $\tau ^+_{11}$, (b) $\tau ^+_{22}$, (c) $\tau ^+_{33}$ and (d) $-\tau ^+_{12}$. The legend for line types and symbols is also given in tables 1 and 2.

The streamwise component exhibits the well-known inner peak in the buffer layer. Consistent with other types of wall turbulence (Schlatter & Örlü Reference Schlatter and Örlü2010; Lee & Moser Reference Lee and Moser2015), the location of the peak is universal with $y^+_p\approx 15$, and the magnitude increases with $Re_\tau$, which is believed to be due to the modulation effect of LSMs and VLSMs in the logarithmic layer (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009). Interestingly, at a given $Re_\tau$, the peak value is slightly higher in OCF than CCF, presumably because the LSMs and VLSMs are stronger in OCF (to be discussed in § 3.2). As suggested by Townsend's attached eddy hypothesis (Townsend Reference Townsend1976), $\tau ^+_{12}$ and $\tau ^+_{22}$ exhibit plateaus whose extent increases with increasing $Re_\tau$, whereas $\tau ^+_{11}$ and $\tau ^+_{33}$ develop logarithmic regions. Similar to the CCFs, the logarithmic region of $\tau ^+_{11}$ has not yet develop for the Reynolds number considered, and $\tau ^+_{33}$ is only marginally for $Re_\tau =2000$. With increasing $Re_\tau$, the Reynolds shear stress $-\tau ^+_{12}$ profiles also tend to become flattened and approach unity at higher $Re_\tau$. In addition, a good agreement on $\tau ^+_{12}$ is observed between OCFs and CCFs, even near the free surface. This is because the asymptotic behaviour of $\tau _{12}$ in the vicinity of the free surface should be similar to that near the centreline of CCF, both of which, in the limiting form, should follow $\tau ^+_{12}\sim c_1(1-y)+c_3(1-y)^3$ with $c_1$ and $c_3$ the fitting coefficients (Handler et al. Reference Handler, Saylor, Leighton and Rovelstad1999).

In Chen & Sreenivasan (Reference Chen and Sreenivasan2021, Reference Chen and Sreenivasan2022), a law of bounded dissipation is developed, which predicts a $Re^{-1/4}_\tau$ defect power law for the averaged near-wall measures, i.e.

(3.1)\begin{equation} \varPhi_\infty -\varPhi = C_\varPhi Re_\tau^{{-}1/4}, \end{equation}

where $\varPhi$ represents the peak of Reynolds normal stress (e.g. $\tau ^+_{11,p}$, $\tau ^+_{33,p}$), or vorticity fluctuation in the $x$-direction ($\omega ^{2+}_{x,w}$) or $z$-direction ($\omega ^{2+}_{z,w}$); $\varPhi _\infty$ denotes the bounded asymptotic value, and the coefficient $C_\varPhi$ depends on the $\varPhi$ but is Reynolds number invariant. In contrast to the logarithmic growth (e.g. in Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017; Diaz-Daniel, Laizet & Vassilicos Reference Diaz-Daniel, Laizet and Vassilicos2017; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), i.e.

(3.2)\begin{equation} \varPhi = A \ln Re_\tau + B, \end{equation}

where $A$ and $B$ are parameters independent of $Re_\tau$, the defect power law of (3.1) supports the classical wall units scaling at asymptotically large $Re_\tau$ because of the bounded constant $\varPhi _\infty$. The underlying physics behind (3.1) is that the maximum turbulent production (which is exactly 1/4 after normalisation by the viscous unit $u^4_\tau /\nu$) provides a bounding constraint on the dissipation rate, and, in turn, on all wall quantities and near-wall peaks in the limit of infinite Reynolds number. What matters at any finite Reynolds number is the departure of the dissipation rate from its limiting value that scales as $u^3_\tau /\eta _o$, where $\eta _o$ is the Kolmogorov length scale of the outer flow. Consequently, the $Re^{-1/4}_\tau$ scaling is obtained from the departure of the wall dissipation from its limiting value.

Figure 4(a) shows the $Re_\tau$-dependence of the maximum of Reynolds normal stresses $\tau ^+_{11,p}$, as well as the scalings of (3.1) and (3.2) with fitting parameters given in the figure legends. As a reference, DNS data of CCFs by Lee & Moser (Reference Lee and Moser2015) and Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) are also included. The magnitude of $\tau ^+_{11,p}$ grows evidently with increasing $Re_\tau$. If the growth continues indefinitely with $Re_\tau$ (as indicated by the logarithmic proposal in prior works, e.g. in Marusic et al. Reference Marusic, Baars and Hutchins2017), the $\tau ^+_{11,p}$ would be infinitely large and, hence, the failure of the classical wall units scaling. In contrast, (3.1) indicates a bounded $\tau ^+_{11,p}$ and, hence, the validity of the wall units scaling at asymptotically large $Re_\tau$ (Klewicki Reference Klewicki2022).

Figure 4. Variations in $Re_\tau$ of the peak (a) $\tau ^+_{11,p}$ and (b) $\tau ^+_{22,p}$.

As noted in Chen & Sreenivasan (Reference Chen and Sreenivasan2022), different from $\tau ^+_{11}$ and $\tau ^+_{33}$, the peak locations of ‘active’ motions (i.e. $\tau ^+_{12}$ and $\tau ^+_{22}$) extend infinitely far away from the wall (in wall units) as $Re_\tau$ increases. Due to the Prandtl–Kármán log-law mean velocity, it is believed that the peak location $y_{12,p}^+$ of $\tau ^+_{12}$ follows a $Re_{\tau }^{1/2}$. Chen, Hussain & She (Reference Chen, Hussain and She2019) recently showed that there is a scaling transition for the peak location $y^+_{12,p}$ of $\tau ^+_{12}$. For low $Re_\tau$, $y^+_{12,p}$ is in the viscous buffer layer rather than in the logarithmic layer; thus, the previous derivation of the $Re_{\tau }^{1/2}$ scaling is valid only for high $Re_{\tau }$. Instead, Chen et al. (Reference Chen, Hussain and She2019) presented an alternative $Re_{\tau }^{1/3}$ scaling for low $Re_{\tau }$ (less than 3000) based on the peak located in the buffer layer. For $\tau ^+_{22}$, its peak is believed to follow the scaling of $\tau ^+_{12}$ under the ‘active’ motion (Townsend Reference Townsend1976), which is indeed validated in figure 4(b). The peak magnitude of both OCF and CCF follows $\tau ^+_{22,p}\approx c_0-c_1Re_\tau ^{-2/3}$ with $c_0\approx 1.34$ and $c_1\approx 19.6$ fitted from our DNS data. One can also use the high $Re_\tau$ scaling to fit data of $\tau ^+_{22,p}$, i.e. $\tau ^+_{22,p}=1.37-7.22Re_\tau ^{-1/2}$ shown in figure 4(b), but this would have a larger departure at low $Re_\tau$.

In the outer region, $\tau ^+_{11}$ of OCFs is larger than that of CCFs, also observed by Tachie & Adane (Reference Tachie and Adane2007). As shown in figure 5(a), $\tau ^+_{22}$ and $\tau ^+_{33}$ for CCFs at the centreline are almost identical and both slightly increase with increasing $Re_\tau$, in contrast to $\tau ^+_{11}$ which appears to saturate at high $Re_\tau$ (whereas the centreline vorticities decrease with increasing $Re_\tau$ in CCFs; see figure 5b). The trend for CCFs suggests that as $Re_\tau$ increases, the flow near the channel centre becomes more isotropic. Due to impermeability of the free surface, the Reynolds normal stresses of OCFs are highly anisotropic when compared with the CCFs. Specifically on the free surface of OCFs, as the wall-normal Reynolds stress $\tau ^+_{22}$ drops to zero with its energy transferring to the wall-parallel components, $\tau ^+_{11}$ and $\tau ^+_{33}$ are higher than those of CCFs. Moreover, the energy of the wall-normal component is almost equally transferred to the streamwise $\tau ^+_{11}$ and the spanwise $\tau ^+_{33}$ components, in agreement with that observed in Swean et al. (Reference Swean, Leighton, Handler and Swearingen1991), Bauer (Reference Bauer2015) and Pinelli et al. (Reference Pinelli, Herlina, Wissink and Uhlmann2022). In addition, for the $Re_\tau$ considered here, both $\tau ^+_{11}$ and $\tau ^+_{33}$ on the free surface rise with increasing $Re_\tau$, suggesting more energy transfer at higher $Re_\tau$. In this regard, Komori et al. (Reference Komori, Nagaosa, Murakami, Chiba, Ishii and Kuwahara1993) and Handler et al. (Reference Handler, Swean, Leighton and Swearingen1993) showed that the pressure strain and anisotropic dissipation are the main reasons for the anisotropy of Reynolds stresses. On the other hand, it would be interesting to see whether the saturation of $\tau ^+_{11}$ in CCFs at high $Re_\tau$ similarly occurs for OCFs. If this happens, the wall-normal energy component would then be mostly transferred to the spanwise component in OCFs.

Figure 5. Variation in $Re_\tau$ of (a) Reynolds stress and (b) root-mean-square (r.m.s.) vorticity fluctuations on the free surface for OCF and channel centreline for CCF.

3.1.3. Vorticity fluctuation

The root-mean-square (r.m.s.) vorticity fluctuations (variances) $\left\langle {\omega ^{\prime 2}_i}\right\rangle$ are examined in figure 6. When compared with CCFs, the vorticity profiles match well in the vicinity of the wall. In addition, $\omega ^+_{y,rms}$ is remarkably $Re_\tau$-invariant for the inner flow region, whereas both $\omega ^+_{x,rms}$ and $\omega ^+_{z,rms}$ show notable $Re_\tau$ dependence, particularly at the wall where they reach their corresponding maximums. Noting that near the wall $\omega _{y}$ is dominated by the streamwise velocity streaks (i.e. $\omega _{y} \approx \partial _z u$), whose spanwise separation distance is approximately 100 wall units for all $Re_\tau$ (Jiménez Reference Jiménez2012), the profiles of $\omega _{y,rms}^+$, thus, are notably $Re$-invariant in viscous scalings (figure 6b). On the other hand, the outer flow large-scale structures are more developed along the streamwise and spanwise directions than along the wall-normal direction (due to the constraint by the outer flow boundary), their footprint effects thus result in notable $Re$-dependence on $\omega _{x, rms}^+$ and $\omega _{z;rms}^+$.

Figure 6. Root-mean-square (r.m.s.) vorticity fluctuations as a function of $y^+$: (a) $\omega ^+_{x,rms}$, (b) $\omega ^+_{y,rms}$ and (c) $\omega ^+_{z,rms}$. The insets in (a–c) are enlarged views near the free surface.

Figures 7(a) and 7(b) show the $Re_\tau$ scaling of vorticities variances at the wall. Again, both $\omega ^{2+}_{z,w}$ (also the spanwise wall dissipation rate) and $\omega ^{2+}_{x,w}$ (also the streamwise wall dissipation rate) are in good agreement with the defect power law (3.1). In contrast, previously published logarithmic scalings (Diaz-Daniel et al. Reference Diaz-Daniel, Laizet and Vassilicos2017; Tardu Reference Tardu2017) show notable deviations from the DNS data, particularly for $\omega ^{2+}_{x,w}$. Note that the data at $Re_\tau =10^4$ for $\omega ^{2+}_{x,w}$ in (b) deviate from both the logarithmic and defect power law, which may be due to the small domain size used by Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022). Notable differences in vorticity fluctuations are observed between OCFs and CCFs in a narrow region near the free surface. For the CCFs, the centreline r.m.s. vorticity fluctuations are almost isotropic, and all components decrease with increasing $Re_\tau$ (figure 5b). For the OCFs, due to impermeability and shear-free boundary condition on the surface, wall-parallel vorticities (i.e. $\omega _x$ and $\omega _z$) vanish, leading to only non-zero perpendicular vortices attached to the free surface (Pan & Banerjee Reference Pan and Banerjee1995). As the r.m.s. velocity fluctuations (e.g. $u^+_{rms}$ or $w^+_{rms}$) on the free surface increase but the wall-normal r.m.s. vorticity fluctuation $\omega ^+_{y,rms}$ decreases, the characteristic eddy length scale, the ratio of the two, increases in wall units with increasing $Re_\tau$. In fact, the thickness of the anisotropic vorticity layer where the surface parallel vorticity fluctuation components rapidly drop in OCF is very thin, only approximately $20$ wall units according to our data, similar to that observed by Calmet & Magnaudet (Reference Calmet and Magnaudet2003) based on the large eddy simulation of OCF at $Re_\tau =1280$.

Figure 7. Variations in $Re_\tau$ of the maximum vorticity fluctuations located at the wall (a) $\omega ^{2+}_{z,w}$ and (b) $\omega ^{2+}_{x,w}$. The dashed lines indicate logarithmic growth in (3.2), whereas the dash-dotted lines indicate the scaling of Chen & Sreenivasan (Reference Chen and Sreenivasan2022) in (3.1).

3.2.1. Premultiplied spectra of streamwise velocity

The separation of scales between the near-wall and outer-layer structures enlarges as $Re_\tau$ increases. Following previous works (Kim & Adrian Reference Kim and Adrian1999; Guala, Hommema & Adrian Reference Guala, Hommema and Adrian2006; Lee & Moser Reference Lee and Moser2015; Duan et al. Reference Duan, Chen, Li and Zhong2020), the one-dimensional velocity spectra are employed here to examine the distribution of energy/Reynolds shear stress at different scales. Figure 8 shows the premultiplied spectrum of the streamwise velocity fluctuations $k_x\varPhi_{uu}/u^2_\tau$ of OCFs for different Reynolds numbers as functions of $\lambda ^+_x$ and $y^+$. Consistent with prior findings for other types of wall turbulence (Morrison et al. Reference Morrison, Jiang, McKeon and Smits2002; Schlatter et al. Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010; Lee & Moser Reference Lee and Moser2015), $k_x\varPhi_{uu}/u^2_\tau$ exhibits a strong inner peak at $y^+\approx 13$ with a typical wavelength $\lambda ^+_x \approx 1000$, associated with the energetic near-wall streaks in the buffer layer (Waleffe Reference Waleffe1997; Schoppa & Hussain Reference Schoppa and Hussain2002). As $Re_\tau$ increases, the energy content in the outer region continuously increases. Recent experiments in OCFs (Cameron et al. Reference Cameron, Nikora and Stewart2017; Wang & Richter Reference Wang and Richter2019; Peruzzi et al. Reference Peruzzi, Poggi, Ridolfi and Manes2020) showed that there exist dual peaks in the outer region for $Re_\tau \ge 550$, and Duan et al. (Reference Duan, Chen, Li and Zhong2020) further found that because of sufficient inner- and outer-scale separation, a distinct outer peak develops for $Re_\tau \ge 1800$. The typical wavelengths of these two peaks are around ${O}(3h)$ and ${O}(20h)$, which correspond to the scale of LSMs and VLSMs, respectively. These scales are identical to those observed for CCFs and pipe flows (Balakumar & Adrian Reference Balakumar and Adrian2007; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009), but much higher $Re_\tau$ is required in these flows. As the domain size employed here is comparable to the scale of VLSMs, such dual peaks are not observed in figure 8. Note that in the current numerical set-up, all structures with wavelength $\lambda _x/h\ge 8{\rm \pi}$ are considered as streamwise uniform (i.e. $k_x=0$). As demonstrated in the Appendix, a second peak with streamwise wavelength $\lambda _x/h$ around $10$–$20$ indeed occurs when the streamwise domain size is doubled (i.e. $L_x/h=16{\rm \pi}$).

Figure 8. Premultiplied spectra of streamwise velocity $k_x\varPhi_{uu}/u^2_\tau$ as a function of $\lambda ^+_x$ and $y^+$ for: (a) $Re_\tau =180$, (b) $Re_\tau =550$, (c) $Re_\tau =1000$ and (d) $Re_\tau=2000$.

Figure 9 further shows the premultiplied spanwise spectra of the streamwise velocity fluctuations $k_z\varPhi_{uu}/u^2_\tau$ for different Reynolds numbers. Again, there exists a strong inner peak in the buffer layer $y^+\approx 13$ with a typical wavelength $\lambda ^+_z \approx 100$, which corresponds to the spanwise spacing of the near-wall streaks. The energy contents at large wavelengths manifest as $Re_\tau$ increases. For the OCF2000 case, a distinct outer peak occurs with $\lambda _z\sim 1h$, which is similar to that identified in CCFs (Jiménez Reference Jiménez2012; Lee & Moser Reference Lee and Moser2015). Note that to the best of our knowledge, this is the first time such an outer peak of $k_z\varPhi_{uu}(k_z)$ has been observed for the OCFs.

Figure 9. Premultiplied spectra of streamwise velocity $k_z\varPhi_{uu}/u^2_\tau$ as a function of $\lambda ^+_z$ and $y^+$ for: (a) $Re_\tau =180$, (b) $Re_\tau =550$, (c) $Re_\tau =1000$ and (d) $Re_\tau =2000$.

Figure 10 compares the one-dimensional pre-multiplied spectra $k_x\varPhi_{uu}(k_x)$ and $k_z\varPhi_{uu}(k_z)$ at different $y$ locations between OCF and CCFs for $Re_\tau =2000$. Note that the area under the curves is proportional to the scaled turbulence intensity of the streamwise velocity. A good agreement between OCF and CCF is observed at $y^+=15$, particularly at small wavelengths, suggesting that the near-wall coherent structures in OCFs have similar characteristics as other wall turbulence. In the overlap and outer region, both $k_x\varPhi_{uu}$ and $k_z\varPhi_{uu}$ are slightly smaller in OCFs than in CCFs at small wavelengths, but the trend reverses at large wavelengths. It indicates that the large-scale structures are more energetic in OCFs than CCFs. This can be further confirmed by examining the cumulative contributions to Reynolds stresses from all wavenumbers between $0$ and $k=2{\rm \pi} /\lambda ^c$, which is defined as (Balakumar & Adrian Reference Balakumar and Adrian2007; Duan et al. Reference Duan, Chen, Li and Zhong2020)

(3.3)\begin{equation} \gamma_{ij}(k^c)=\frac{\int^{k}_0 \varPhi_{ij}(\tilde{k})\,\mathrm{d}\tilde{k}}{\int^\infty_0 \varPhi_{ij}(\tilde{k})\,\mathrm{d}\tilde{k}}. \end{equation}

In the literature (e.g. Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011; Dogan et al. Reference Dogan, Örlü, Gatti, Vinuesa and Schlatter2019), $\lambda _x=3h$ or $\lambda _z=h$ is widely used to separate LSMs and VLSMs. Figure 11 shows the fraction of kinetic energy carried by VLSMs as a function of $y/h$ for $Re=550$, $1000$ and $2000$ cases. The results from the CCFs are also included for comparison. Consistent with the finding in Duan et al. (Reference Duan, Chen, Li and Zhong2020), $\gamma _{uu}(\lambda _x=3h)$ is $0.5$–$0.65$ in the outer region of OCFs (e.g. $y/h>0.2$). The contribution of VLSMs to the streamwise kinetic energy is even higher when calculated based on the spanwise wavelength (i.e. $\lambda _z=0.5h$). It suggests that VLSMs contribute most to the streamwise Reynolds stress in the outer region. In addition, at the same $Re_\tau$, the streamwise kinetic energy carried by VLSMs is notably larger in OCFs than in CCFs, confirming that VLSMs are stronger in the former.

Figure 10. Comparison of premultiplied spectra of streamwise velocity (a) $k_x\varPhi_{uu}$ and (b) $k_z\varPhi_{uu}$ at different $y$ locations for $Re_\tau =2000$. The solid and dashed lines represent OCF and CCF, respectively.

Figure 11. The fraction of kinetic energy carried by VLSMs defined based on (a) $\lambda _x>3h$ and (b) $\lambda _z>0.5h$. The legend for line types and symbols is also given tables 1 and 2.

Further insights into the distribution of scales can be gained from figure 12, where we show instantaneous streamwise velocity fluctuations $u'$ in wall-parallel ($x$–$z$) planes at different wall-normal locations for $Re_\tau =2000$. The $u'$ close to the wall exhibits the organisation of the streaks with a streamwise length of approximately $1000$ and spanwise spacing of $100$. Consistent with the previous findings (Hutchins & Marusic Reference Hutchins and Marusic2007; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014), the outer layer flow exhibits very long regions of positive/negative $u'$. Similar to the finding from energy spectra (figure 9), these streamwise meandering streaks have characteristic lengths $\lambda _z \approx h$ in the spanwise directions. These streaks can be interpreted in terms of large-scale structures that in the form of streamwise vortices of diameters comparable to the channel height (Tamburrino & Gulliver Reference Tamburrino and Gulliver1999; Duan et al. Reference Duan, Chen, Li and Zhong2020) or spatial clusters of hairpin vortices (Kim & Adrian Reference Kim and Adrian1999). In addition, these structures have their footprint both near the wall and the free surface.

Figure 12. Instantaneous streamwise velocity fluctuations ($u'/u_\tau$) for $Re_\tau =2000$ in $x$–$z$ planes at (a) $y^+= 15$, (b) $y^+=384$ ($y/h=0.19$), (c) $y/h=0.5$ and (d) $y/h=1$. The inset in (a) shows an enlarged view of a box with $2000 \times 1000$ in wall units.

Compared with the CCFs (Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014), more small scales are observed at $y/h=1$ in OCFs (figure 12d). This can be further confirmed by showing the instantaneous wall-normal vorticity $\omega _y$ on the free surface for $Re_\tau =2000$ (figure 13a). Numerous small-scale $\omega _y$ patches (with both positive and negative signs) are observed. Figures 13(b) and 13(c) further show the vortical structures (visualised using the $\lambda _2$ criterion) near the free surface in regions I and II of figure 13(a). Compared with CCFs, more small-scale $\lambda _2$ structures are observed, mainly due to a stronger boundary layer effect near the free surface (Wang et al. Reference Wang, Zhong, Wang and Li2017). As expected, among these structures, some are aligned along the wall-normal direction, which might be generated due to the Kelvin–Helmholtz instability of the free-surface vortex sheet. There also exists significant amounts of $\lambda _2$ vortices that are parallel to the wall. Furthermore, different from the near-wall vortices, which are predominantly aligned along the streamwise direction, the wall-parallel vortices near the free surface have relatively larger tilting angle with respect to the streamwise direction. Interestingly, some of these vortices are even oriented along the spanwise direction, which is consistent with the recent finding by Duan et al. (Reference Duan, Zhong, Wang, Chen, Wang and Li2021a) that higher relative population density of spanwise vortices occurs in OCFs than in CCFs. Based on the ‘bottom-up’ perspective for the origin of VLSMs, in which VLSMs are viewed as the result of coherence in the pattern of vortices produced near the boundary (Kim & Adrian Reference Kim and Adrian1999), these additional vortical structures near the free surface may be one possible explanation for the increased VLSMs in OCFs: a topic that merits further study.

Figure 13. (a) Instantaneous wall-normal vorticity $\omega _y$ (normalised by its r.m.s. value) on the free surface for $Re_\tau =2000$ and enlarged view of (b) box I and (c) box II along with the vortical structures visualised using the $\lambda _2$ criterion within $\Delta y^+\le 200$ from the free surface.

3.2.2. Premultiplied spectra of Reynolds shear stress

To investigate the contribution of VLSMs to the Reynolds shear stress and also to the mean wall shear stress, premultiplied cospectra of streamwise and wall-normal velocities are examined here. Figures 14 and 15, respectively, show the premultiplied cospectra $k_x\varPhi_{uv}/u^2_\tau$ and $k_z\varPhi_{uv}/u^2_\tau$ for different Reynolds numbers in wall units. Similar to the streamwise velocity spectra $k\varPhi_{uu}$ (figures 8 and 9), an inner peak is observed, which is associated with the near-wall coherent structures. The peak occurs at a higher $y^+$ when compared with $k\varPhi_{uu}$. As $Re_\tau$ increases, the energy content in the outer region continues to increase, and a second peak located at $\lambda _z=h$ starts to occur at $Re_\tau =2000$ for the $k_z\varPhi_{uv}/u^2_\tau$ (similar to $k_z\varPhi_{uu}/u^2_\tau$ in figure 10e), but not for $k_x\varPhi_{uv}/u^2_\tau$ (again due to limited streamwise domain size).

Figure 14. Premultiplied cospectra of streamwise and wall-normal velocities $k_x\varPhi_{uv}/u^2_\tau$ as a function of $\lambda ^+_x$ and $y^+$ for: (a) $Re_\tau =180$, (b) $Re_\tau =550$, (c) $Re_\tau =1000$ and (d) $Re_\tau =2000$.

Figure 15. Premultiplied cospectra of streamwise and wall-normal velocities $k_z\varPhi_{uv}/u^2_\tau$ as a function of $\lambda ^+_z$ and $y^+$ for: (a) $Re_\tau =180$, (b) $Re_\tau =550$, (c) $Re_\tau =1000$ and (d) $Re_\tau =2000$.

Figure 16 compares the one-dimensional premultiplied cospectra of streamwise and wall-normal velocities $k_x\varPhi_{uv}$ and $k_z\varPhi_{uu}$ at different $y$ locations for $Re_\tau =2000$. Again, good collapse can be observed between OCF and CCF in the near-wall region (e.g. at $y^+=15$). In the overlap region, both $k_x\varPhi_{uv}$ and $k_z\varPhi_{uv}$ are slightly smaller in OCF than in CCF at small wavelengths, but the trend reverses at large wavelengths, akin to those observed for the streamwise velocity spectra. Interestingly, very close to the free surface (e.g. at $y/h=0.9$), both $k_x\varPhi_{uv}$ and $k_z\varPhi_{uv}$ become larger again in OCFs at small $\lambda _x$ and $\lambda _z$, confirming the prominence of small-scale structures due to the boundary layer effect of the free surface (figure 13). This is also consistent with the finding by Wang et al. (Reference Wang, Zhong, Wang and Li2017), who showed that the free surface restrains the characteristic scales of the streaks and the associated vortical structures. Following (3.3), figures 17(a) and 17(b) show the fraction of Reynolds shear stress carried by VLSMs: $\gamma _{uv}(\lambda _x=3h)$ and $\gamma _{uv}(\lambda _z=0.5h)$. Both of them increase with $y$ and reach approximately $0.5$–$0.6$ at $y/h=0.5$ and then decrease near the free surface, particularly for the spanwise case. Again, at a given $Re_\tau$, the maximum values are slightly larger than that for CCF, highlighting the free surface effect in OCFs.

Figure 16. Premultiplied cospectra of streamwise and wall-normal velocities (a) $k_x\varPhi_{uv}$ and (b) $k_z\varPhi_{uv}$ at different $y$ locations for $Re_\tau =2000$. The solid and dashed lines represent OCF and CCF, respectively.

Figure 17. The fraction of Reynolds shear stress carried by VLSMs defined based on (a) $\lambda _x>3h$ and (b) $\lambda _z>0.5h$. The legend of line types and symbols is also given tables 1 and 2.