3. Results

We begin by inspecting the instantaneous velocity fields in a cross-stream plane in figure 2. The figure highlights well that, as in all wall-bounded flows, the near-wall region with low-speed fluid becomes thinner as the Reynolds number increases, and finer-scale organization of the turbulent eddies is clearly visible. At the same time, large-sized towering eddies are well visible, which originate from the no-slip wall and extend up to the free surface. Despite differences in the details, the organization of the large-scale eddies seems to be similar, as will be confirmed quantitatively from the velocity spectra.

Figure 2. (a–f) Instantaneous streamwise velocity field (normalized by the mean free-surface velocity $U_{FS}$) in a cross-stream plane, for the flow cases A–F, respectively, listed in table 1. Here, $y=0$ corresponds to the no-slip wall, and $y/h=1$ corresponds to the free surface. Note that only a limited part of the domain span is shown.

The flow organization in wall-parallel planes is considered in figure 3, where we show streamwise velocity contours near the no-slip wall and at the free surface, at two Reynolds numbers. The flow near the bottom wall is dominated by streaks of alternating high- and low-speed fluid, which are the hallmark of wall-bounded turbulence (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). Organization of the streaks on two different scales is quite apparent, with large-scale streaks whose size is similar at the two Reynolds numbers, and smaller-scale streaks whose size scales in wall units, and which hence become smaller at higher ${{Re}}$. The large-scale streaks are the roots of the towering eddies noted previously. The imprinting of their tops is clear in the free-surface planes, which show the presence of streamwise-elongated eddies, whose spanwise size is similar at the two Reynolds numbers under scrutiny. Detailed information about the large-scale organization of the velocity field and the dynamical contributions of the large eddies can be found in the studies of Duan et al. (Reference Duan, Chen, Li and Zhong2020) and Yao et al. (Reference Yao, Chen and Hussain2022).

Figure 3. Instantaneous streamwise velocity fields in wall-parallel planes: (a,b) near the no-slip wall($y^+=15$), and (c,d) at the free surface ($y/h=1$), for (a,c) flow case C (${{Re}}_{\tau }=1000$), and (b,d) flow case F (${{Re}}_{\tau }=6000$).

Quantitative information about the flow organization can be gained from the spectral maps of ${u}$, which are depicted in figure 4, where $k_{z}=2 {\rm \pi}/ \lambda _{z}$ is the relevant wavenumber for the spanwise direction. At low Reynolds number, a single peak is present, associated with the wall regeneration cycle (Jiménez & Pinelli Reference Jiménez and Pinelli1999). At higher Reynolds number, a secondary energetic site emerges, which is associated with outer-layer, large-scale motions (Hutchins & Marusic Reference Hutchins and Marusic2007). In between the two sites, a spectral ridge is present, corresponding to eddies with typical spanwise length scale $\lambda _{z} \sim y$, here encompassing over one decade in flow case F, which can be interpreted as the footprint of a hierarchy of wall-attached eddies as in Townsend's model (Townsend Reference Townsend1976). Although the spanwise spectra may seem noisy in the outer part, their time convergence has been checked carefully. Hence the observed undulations are rather the result of difficulties for the largest structures to be accommodated in the relatively narrow computational box. These effects are commented on in Appendix A.

Figure 4. (a–f) Variation of pre-multiplied spanwise spectral densities of fluctuating streamwise velocity with wall distance, for cases A–F, respectively. Wall distances and wavelengths are reported both in inner units (bottom, left axes) and in outer units (top, right axes). The dashed line denotes the channel free surface. The crosses denote the positions of the inner and outer energetic sites. Contour levels from 0.2 to 2.0 are shown, in intervals of 0.2.

The spectral maps are compared with those found in closed channel flow (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2016), at matching Reynolds number (${{Re}}_{\tau }=2000$) in figure 5. Whereas the inner spectral peak is pretty much the same in the two flows (namely $\lambda _z^+ \approx 117$, at $y^+ \approx 13$), the outer peak is quite different. In fact, it occurs at a wavelength $\lambda _z = {\rm \pi}h / 2$, at a distance $y=0.28 h$ in the open channel, and it occurs at $\lambda _z = {\rm \pi}h / 3$, at a distance $y=0.18 h$ in the closed channel. This shows that in the open channel, the eddies that emerge from the no-slip wall can penetrate a greater distance, which also implies that the eddies are larger.

Figure 5. Comparison of spectral maps of fluctuating streamwise velocity in (a) open channel flow and(b) closed channel flow, at ${{Re}}_{\tau }=2000$ (Bernardini, Pirozzoli & Orlandi Reference Bernardini, Pirozzoli and Orlandi2014). Wall distances and wavelengths are reported both in inner units (bottom, left axes) and in outer units (top, right axes). The dashed line denotes either the channel free surface or the channel centreline. Contour levels from 0.2 to 2.0 are shown, in intervals of 0.2. The second group of contours that appears in (b), for $y > h$, is the symmetric equivalence of the $y < h$ pattern, condensed visually because of use of a logarithmic scale.

Reynolds number effects on the streamwise velocity spectra are scrutinized more closely in figure 6, where we show spectral densities at the inner energy site location ($y^+=15$), and at the outer site location ($y/h=0.28$). Figure 6(a) clearly brings out universality of the spectra in inner units at the small scales. Additional energy appears at large scales as the Reynolds number increases, as a result of influence from the outer energy site (Marusic & Monty Reference Marusic and Monty2019). At intermediate wavelengths, there is also mild evidence for an ${E}_{u} \sim \lambda _{z}$ spectral range, which is also predicted from Townsend's attached-eddy model (Nickels et al. Reference Nickels, Marusic, Hafez and Chong2005). The inset of figure 6(a) further compares the spectrum at ${{Re}}_{\tau }=2000$ with the corresponding case of a closed channel. The figure suggests that the distribution of energy across the flow scales is very much the same in the two flows, with deviations occurring only at the largest scales, where the open channel (solid line) is found to have more energy, which results from the different outer-layer organization noticed previously, and in agreement with the results of Yao et al. (Reference Yao, Chen and Hussain2022). The spectra corresponding to the outer energy site, shown in figure 6(b), are comparatively more scattered, with change of the peak value with ${{Re}}$ and occasional peak splitting, again on account of the limited spanwise size of the box. Nevertheless, we notice a tendency for the spectra to collapse at the highest Reynolds numbers under scrutiny. This seems to be evidence that besides the well recognized universality of the inner-layer dynamics in inner units, the outer-layer dynamics also becomes universal in outer units, however only at sufficiently high Reynolds number, as argued previously by Dennis & Sogaro (Reference Dennis and Sogaro2014) for pipe flow.

Figure 6. Pre-multiplied spectral densities of streamwise velocity at (a) $y^+=15$ and (b) $y/h=0.28$. The insets report a comparison with spectra in a closed channel at ${{Re}}_{\tau }=2000$ (solid lines indicate open channel, dashed lines indicate closed channel). The colour codes are as in table 1.

In figure 7, we show the inner-scaled mean velocity profiles obtained from the present DNS, which indicate agreement with a logarithmic distribution. Data fitting of the highest ${{Re}}$ data yields $U^+ = 1/k \log y^+ + 4.21$, with Kármán constant $k=0.375$, hence smaller than reported for channel flow and in pipe flow (Lee & Moser Reference Lee and Moser2015; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). It is important to note that no significant deviations from the logarithmic distribution are found, also far from the wall. Detailed comparison with other internal flows at ${{Re}}_{\tau } \approx 2000$ is reported in figure 7(b), where we also show experimental data of open channel flow from Duan et al. (Reference Duan, Chen, Li and Zhong2020). The figure shows quite good agreement with experiments, and that the open channel flow has a weaker wake than the closed channel flow, which in turn has a weaker wake than pipe flow. Hence one could argue that the open channel flow is closest to the ideal situation of a wake-free flow, which is the optimal testbed to quantitatively probe the properties of the logarithmic layer. Similar observations were reported in the context of shear-free Couette–Poiseuille flow by Coleman et al. (Reference Coleman, Pirozzoli, Quadrio and Spalart2017).

Figure 7. (a) Inner-scaled mean velocity profiles obtained from the present DNS. (b) Comparison with the case of closed channel and pipe flow at ${{Re}}_{\tau }=2000$. The dashed line in (a) refers to the logarithmic fit $U^+ = \log y^+ / 0.375 + 4.21$. In (b), the dashed line refers to the closed channel (Pirozzoli et al. Reference Pirozzoli, Bernardini and Orlandi2016), the dotted line to the case of pipe flow (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), and the blue circles to open channel experimental data at ${{Re}}_{\tau }=1903$ (Duan et al. Reference Duan, Chen, Li and Zhong2020). The colour codes are as in table 1.

More refined information on the behaviour of the mean velocity profile can be gained from inspection of the log-law diagnostic function, namely $y^+ \, \mathrm {d} {U}^+ / \mathrm {d} y^+$, which is shown in figure 8, and whose constancy would imply the presence of a genuine logarithmic layer. The figure supports universality of the inner-scaled streamwise velocity for ${{Re}}_{\tau } \gtrsim 10^3$, up to $y^+ \approx 60$, where the diagnostic function attains a minimum. The figure also shows clear flattening of this indicator as the Reynolds number increases. At ${{Re}}_{\tau }=2000$, an extended flat region is first observed, for which the extrapolated Kármán constant is in perfect agreement with the results of Yao et al. (Reference Yao, Chen and Hussain2022), and in very good agreement with the data of Coleman et al. (Reference Coleman, Pirozzoli, Quadrio and Spalart2017) for Couette–Poiseuille flow with a zero-shear wall, from which $k \approx 0.37$ can be inferred. At higher ${{Re}}_{\tau }$ the Kármán constant seems to decrease a little, and at ${{Re}}_{\tau }=6000$ there is a well developed logarithmic layer up to $y/h=0.5$, corresponding to $k=0.375$. These values of the Kármán constant seem to differ somewhat from the commonly quoted value $k \approx 0.39$ for closed channel and pipe flow (Lee & Moser Reference Lee and Moser2015; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022). However, in those cases, significant scatter and ambiguity persist on how the fitting is carried out, in the absence of extended regions with flat diagnostic functions. In fact, systematic deviations from a logarithmic distribution are observed in internal flows, which are typically modelled through additional linear terms (Afzal Reference Afzal1982; Jiménez & Moser Reference Jiménez and Moser2007; Luchini Reference Luchini2017). In particular, the latter study attributed the occurrence of additional linear terms in the overlap layer expansion of the mean velocity profile in internal flows to the effect of the imposed pressure gradient. The case of the open channel flow, in which a pressure gradient is present, but deviations from the log law are not evident, seems to rule out such explanation. On the other hand, the present findings seem to corroborate the interpretation given by Monkewitz (Reference Monkewitz2021), that the late start of the logarithmic layer in internals flows is due to intrusions from the opposite wall, which in fact are not present in the open channel flow. Differences with other canonical wall-bounded flows are made more explicit in figure 8(b), which makes it quite clear that the log-law diagnostic function is much flatter in the present case than in other flows, and even closed channel flow at substantially higher Reynolds number (Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) exhibits a much wider range of variation.

Figure 8. Log-law diagnostic function for mean streamwise velocity, as a function of (a) inner-scaled and(b) outer-scaled wall distance. The dashed horizontal line denotes the inverse Kármán constant, $1/0.375$. Lines denote present DNS data, with colour code as in table 1. Symbols denote data for closed channel flow at ${{Re}}_{\tau }=5200$ (triangles, Lee & Moser Reference Lee and Moser2015), boundary layer at ${{Re}}_{\tau }=6115$ (circles, Österlund et al. Reference Österlund, Johansson, Nagib and Hites2000), and pipe flow at ${{Re}}_{\tau }=6000$ (squares, Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021).

The structure of the wake region of the open channel is examined in detail in figure 9, where the mean velocity profiles are shown in defect form. Full outer-layer similarity is achieved even at the lowest Reynolds numbers under scrutiny, starting at $y/h \approx 0.2$. The defect logarithmic profile

(3.1)\begin{equation} U_{FS}^+{-} U^+{=} - \frac 1k \log (y/h) + B \end{equation}

fits the data very well with $k=0.375$, and with additive constant $B=0.128$. Deviations from the outer log law seem to be very minor, and concentrated in a narrow layer adjacent to the free surface.

Figure 9. Defect velocity profiles for DNS of open channel flow, in (a) linear and (b) semi-logarithmic scale. The dashed grey line marks the outer-layer logarithmic fit $U^+_{FS}-U^+ = -0.128 - 1/0.375 \log (y/h)$.

The previous observations regarding the structure of the mean velocity field can be leveraged to deduce that the dependence of the bulk and mean free-surface velocity on the friction Reynolds number should be logarithmic. This is checked in figure 10, where we show the mean velocity at the free surface and the bulk velocity normalized by the friction velocity, as functions of the friction Reynolds number. Consistently with the case of internal flows (e.g. Monkewitz Reference Monkewitz2021), the data in fact support logarithmic increase with ${{Re}}_{\tau }$, according to

(3.2a,b)\begin{equation} u_{b}^+{=} \frac 1{k} \log {{Re}}_{\tau} + B_b,\quad U_{FS}^+{=} \frac 1{k_{FS}} \log {{Re}}_{\tau} + B_{FS}, \end{equation}

with data fitting suggesting $k_{FS} \approx k=0.375$, and $B_b=1.57$, $B_{FS}=4.02$. As a result of the observed identity (or very close vicinity) of the Kármán constant for the centreline and for the bulk velocity, figure 10(b) highlights that their ratio approaches unity at large ${{Re}}$, supporting the inference that open channel flow asymptotes to plug flow in the infinite-Reynolds-number limit; see Pullin, Inoue & Saito (Reference Pullin, Inoue and Saito2013). Regarding that study, it is worthwhile noticing that one of the assumptions made in the analysis is that the wall-normal location of the onset of the logarithmic region is either finite or increases no faster than ${{Re}}_{\tau }$. Interpreting the near-wall minimum of the diagnostic function in figure 8 as the root of the (near) logarithmic layer, our data well support that assumption. However, as figure 10(b) suggests, this trend is extremely slow.

Figure 10. (a) Mean free surface velocity ($U_{FS}$) and bulk velocity, and (b) their ratio, as functions of the friction Reynolds number. DNS data are shown as circle symbols, and the corresponding logarithmic fits are shown as dashed lines. Black is used for $U_{FS}$, and purple for $u_b$.

Equations (3.2a,b) can be exploited in order to determine friction relationships for flow in smooth open channels, namely

(3.3a,b)\begin{equation} \sqrt{\frac2{C_f}}= \frac 1k \log \left( {{Re}}_b\,\sqrt{\frac{C_f}2} \right) + B_b - \frac 1k,\quad \sqrt{\frac2{c_f}}= \frac 1k \log \left( {{Re}}_{FS}\,\sqrt{\frac{c_f}2} \right) + B_{FS}, \end{equation}

whence the friction coefficient (either $C_f = 2 \tau _w / (\rho u_b^2)$ or $c_f = 2 \tau _w / ( \rho U_{FS}^2)$) can be evaluated as an implicit function of the bulk Reynolds number, or of the Reynolds number at the free surface (${{Re}}_{FS}=U_{FS} h / \nu$).

The distributions of the velocity variances along the coordinate directions are shown in figure 11, in inner scaling. As is well established (Marusic & Monty Reference Marusic and Monty2019), the streamwise ($u$) and spanwise ($w$) velocity fluctuations show slow increase with the Reynolds number, with commonly accepted logarithmic growth as in Townsend's attached-eddy model (Townsend Reference Townsend1976). On the other hand, the wall-normal velocity fluctuations seem to level off to maximum value approximately $1.30$ (Smits et al. Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021). It is remarkable that the general growth of the longitudinal and spanwise fluctuations is more evident in the outer layer, and in fact it has long been argued about the possible occurrence of a secondary peak of $\langle u^2\rangle$, besides the primary buffer-layer peak. Experiments carried out in the Superpipe (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012) and CICLoPE (Willert et al. Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017) facilities indeed support the occurrence of such a peak at ${{Re}}_{\tau } \gtrsim 10^4$. Whereas DNS data do not reach sufficiently high ${{Re}}_{\tau }$ to show this secondary peak, it appears that in flow case F, the streamwise velocity variance has attained a nearly horizontal inflectional point at $y^+ \approx 140$. Comparison with the data from closed channels shows generally higher values of all fluctuating velocity components, which is consistent with additional energy resulting from the more intense outer-layer eddies. This also yields higher probability that the outer-layer peaks of $\langle u^2\rangle$, if found, would occur earlier in the open channel than in the closed channel. Additional differences are related to the behaviour of the velocity variances near the free surface, where of course $\langle v^2\rangle$ must go to zero as a consequence of the rigid-lid assumption, and where the horizontal velocity variances show some pile-up and become very similar in magnitude (Nezu Reference Nezu2005). Comparison with the experimental data of Duan et al. (Reference Duan, Chen, Li and Zhong2020) in figures 11(a,b) shows generally good agreement, especially in the outer part of the flow. Differences observed towards the wall, and especially overestimation of the vertical velocity variance in the experiment, are the likely result of well-known difficulties in measuring in the vicinity of walls.

Figure 11. Distribution of velocity variances (a–c) and trend of peak streamwise velocity variance with Reynolds number (d). The blue circles in (a,b) denotes experimental data of Duan et al. (Reference Duan, Chen, Li and Zhong2020), at ${{Re}}_{\tau }=1903$. The circles in (d) denote the present open channel data, and the squares the closed channel data of Lee & Moser (Reference Lee and Moser2015), at ${{Re}}_{\tau } = 550, 1000, 2000, 5200$. The dashed and dot-dashed data denote the corresponding logarithmic fits, and the dotted line the trend predicted by Chen & Sreenivasan (Reference Chen and Sreenivasan2021).

Figure 11(d) displays the trend of the streamwise velocity variance with ${{Re}}_{\tau }$. This quantity is the subject of a lot of theoretical interest, and in particular its behaviour in the infinite-${{Re}}$ limit. In fact, the attached-eddy model (Townsend Reference Townsend1976; Marusic & Monty Reference Marusic and Monty2019) predicts that it should be increasing logarithmically with ${{Re}}_{\tau }$, on account of influence from distant, outer-layer eddies, which would imply a violation of the strict wall scaling. On the other hand, arguments have been made recently that this peak should stay finite, as a result of saturation of growth of the turbulence kinetic energy dissipation rate at the wall (Chen & Sreenivasan Reference Chen and Sreenivasan2021). Although difference between slow logarithmic growth and attainment of an asymptotic value is quite tenuous in practice, the theoretical interest is high as in the latter case universality of wall scaling would be restored eventually. The data shown in figure 11(d) generally seem to support logarithmic growth, with slope approximately $0.63$, very close to the case of closed channel (Lee & Moser Reference Lee and Moser2015), and as suggested from a collection of DNS and experiments on boundary layers (Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017). At the same time, the theoretical prediction of Chen & Sreenivasan (Reference Chen and Sreenivasan2021) (see the dotted purple line of figure 11d) seems to conform well with DNS data at the highest Reynolds number achieved. Discriminating between these two trends would probably require much higher Reynolds numbers than are currently feasible in DNS of wall-bounded flows.

Whereas our DNS data cannot be used to evaluate directly the theoretical predictions owing to limited achievable Reynolds number, they can be used to better scrutinize the foundations of the theoretical arguments. The main argument made by Chen & Sreenivasan (Reference Chen and Sreenivasan2021), although not strictly justified mathematically, was that since turbulence kinetic energy production is bounded, the wall dissipation must also stay bounded. Letting $P = - \left \langle u v \right \rangle \mathrm {d} U / \mathrm {d} y$ be the turbulence kinetic energy production rate, and $\epsilon _{11} = \nu \left \langle | \nabla u |^2 \right \rangle$ be the dissipation rate of the streamwise velocity variance, those authors argued that the wall limiting value of $\epsilon _{11}$ should scale as

(3.4)\begin{equation} {\epsilon_{11}}_w^+{=} 1/4 - \beta / {{Re}}_{\tau}^{1/4}, \end{equation}

with $\beta$ a suitable constant. In figure 12(a), we explore deviations of ${\epsilon _{11}}_w$ and of the peak turbulence kinetic energy production, say $P_{PK}$, from their asymptotic value, namely $1/4$. According to analytical constraints (Pope Reference Pope2000), we find that production tends to its asymptotic value quite rapidly, as approximately $1/{{Re}}_{\tau }$. Consistent with (3.4), the wall dissipation rate also tends to $1/4$, more or less at the predicted rate, thus validating the first assumption empirically. The next argument advocated by Chen & Sreenivasan (Reference Chen and Sreenivasan2021) is that wall balance between viscous diffusion and dissipation and Taylor series expansion of the streamwise velocity variance near the wall yields

(3.5)\begin{equation} \langle u^2 \rangle^+{\sim} \epsilon_{{11w}}^+ {y^+}^2, \end{equation}

whence, from assumed invariance of the peak location of $\langle u^2 \rangle$ (say, $y_{PK}^+$), saturation of growth of the peak velocity variance would follow. Figure 12(b) reports the position of the streamwise velocity variance peak position as a function of ${{Re}}_{\tau }$, as inferred from cubic spline interpolation of the DNS data. Also accounting for uncertainty associated with data interpolation, the figure suggests that this second assumption may not be valid, as the (inner-scaled) peak position shifts away from the wall at increasing ${{Re}}_{\tau }$, with non-negligible effect on the peak variance as it appears in squared form in (3.5). As a consequence, logarithmic growth of the peak velocity variance would still be possible even in the presence of saturation of the wall dissipation rate, at least based on DNS data in the currently accessible range of Reynolds numbers.

Figure 12. (a) Distributions of peak turbulence production ($P_{PK}$, squares), and wall dissipation of streamwise velocity variance ($\epsilon _{{11w}}$, circles). (b) Peak position of streamwise velocity variance. In (a), the dot-dashed and dotted lines denote fits of $P_{PK}$ and $\epsilon _{{11w}}$ in their tendency to the respective assumed asymptotic values.

It is well established that the intensity of wall pressure fluctuations in boundary layers and channels increases logarithmically with the Reynolds number, when scaled by the mean wall shear stress (Willmarth Reference Willmarth1975; Farabee & Casarella Reference Farabee and Casarella1991; Hu, Morfey & Sandham Reference Hu, Morfey and Sandham2006; Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007; Jiménez & Hoyas Reference Jiménez and Hoyas2008; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013). Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) found a similar trend for the peak pressure variance, which occurs at approximately $y^+=30$. Jiménez & Hoyas (Reference Jiménez and Hoyas2008) suggested that this logarithmic increment should be attributed to a growing hierarchy of self-similar wall-attached eddies (Townsend Reference Townsend1976; Hwang Reference Hwang2015; Marusic & Monty Reference Marusic and Monty2019). Pressure, like the wall-parallel velocity components, can indeed be regarded as a wall-attached quantity (Bradshaw Reference Bradshaw1967), which explains why profiles of the pressure variance also follow a logarithmic trend with respect to the wall distance. The same behaviour can, however, also be predicted based on inner/outer layer overlap arguments (Panton, Lee & Moser Reference Panton, Lee and Moser2017). Mehrez et al. (Reference Mehrez, Philip, Yamamoto and Tsuji2019) explored the DNS database of closed channel flow at friction Reynolds number up to 4000, and found that the logarithmic trend shows up only for ${{Re}}_\tau \gtrsim 500$.

The distributions of the pressure variances obtained from the present DNS are displayed in figure 13(a). These are found to remain nearly constant within the viscous sublayer (say, $y^+ \le 5$), then increase to attain a peak value at $y^+ \approx 30$, and decrease thereafter. A region with distinctly logarithmic decrease is found from $y^+ \approx 100$ basically all the way to the free surface, with slight deviations. As the Reynolds number increases, the pressure variances at the wall and their peak values increase systematically, whereas the value at the free surface is not much affected. Compared with the variances in closed channels (dashed lines), we note that pressure fluctuations in open channels tend to be a bit less in the buffer layer, whereas they are virtually identical towards the wall and in the outer layer. When plotted against outer wall coordinates, the logarithmic portions of the pressure variance distribution collapse for ${{Re}}_\tau \gtrsim 500$, consistent with the case of closed channel flow (Mehrez et al. Reference Mehrez, Philip, Yamamoto and Tsuji2019). Expressing the outer distribution as

(3.6)\begin{equation} \langle p^2 \rangle^+ (y/h)= A_p \log \left( y/h \right) + B_p, \end{equation}

curve fitting within the range $y = 0.02h\unicode{x2013}0.5h$ for flow case F yields the values of parameters $A_p=-2.40$ and $B_p=1.23$, with error $0.4\,\%$.

Figure 13. (a) Wall-normal distribution of pressure variance, and (b) wall pressure variance as a function of Reynolds number. In (a), the dashed lines correspond to closed channel flow DNS data (Lee & Moser Reference Lee and Moser2015) at ${{Re}}_{\tau } = 180, 550, 1000, 2000, 5200$, and the grey line denotes the logarithmic fit given in (3.6). In (b), circles are used for the open channel data, and squares for the closed channel data; the dashed lines denote the logarithmic data fit given in (3.7). Refer to table 1 for colour codes.

We further report the wall pressure variances as a function of ${{Re}}_{\tau }$ in figure 13(b). With exclusion of the lowest Reynolds number case at ${{Re}}_\tau \approx 180$, the wall pressure variances can be well characterized in terms of the log law

(3.7)\begin{equation} \langle p^2_w \rangle^+{=}2.22\log {{Re}}_\tau -8.99, \end{equation}

thus supporting theoretical inferences quite well. The logarithmic growth rate seems to be identical to that for closed channel flow.