3.1. Flow visualization

The $Re$ effect on the flow structure is illustrated qualitatively in figure 2, showing cross-sectional views of the instantaneous axial velocity $u_z$. Although large scales dominate in the central region of the pipe for all $Re_\tau$ cases, there is a general increase in the range of scales with increasing $Re_\tau$. The average spacing between near-wall low-speed streaks is around $(R\theta )^+=100$. For the lowest $Re_\tau$ studied here, approximately ten evenly distributed low-speed structures are seen in figure 2(a), identified by the plume-shaped black regions ejecting from the wall. For our highest $Re_\tau$, the streak spacing is reduced to approximately $0.02R$, and these fine-scale streaks can hardly be identified from the full cross-section in figure 2(e). A zoomed-in view of the near-wall region with domain size $(1000,200)$ in wall units in $(r,\theta )$ directions is provided to better visualize these structures, which share patterns quite similar to those in low $Re_\tau$ cases.

Figure 2. Visualization of the instantaneous axial velocity $u_z/U_b$ in the $(r,\theta )$ plane for $Re_\tau$ values (a) $180$, (b) $550$, (c) 1000, (d) 2000, and (e) 5200; and inner-scaled instantaneous axial velocity fluctuations $u'_z/u_\tau$ on a cylindrical surface $(z - r\theta )$ for $Re_\tau =5200$ at ( f) $y^+=15$ and (g) $y/R=0.5$.

Figures 2( f,g) show the inner-scaled instantaneous axial velocity fluctuations $u'_z/u_\tau$ on an unrolled cylindrical surface $(z-r\theta )$ for $Re_\tau \approx 5200$ at $y^+=15$ and $y/R=0.5$, respectively. At $y^+=15$, $u'_z/u_\tau$ shows the organization of the streaks, whose characteristics are better illustrated in a zoomed-in view with domain size $(2000,400)$ in wall units. Consistent with previous findings (Hellström, Sinha & Smits Reference Hellström, Sinha and Smits2011; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014), the flow in the outer region exhibits very long positive/negative $u'_z/u_\tau$ patches – indicating the presence of LSMs and VLSMs.

3.2. Friction factor

The mean friction (or wall shear stress), which is proportional to the pressure drop or the amount of energy required to sustain the flow, is an important parameter and has been studied extensively (Blasius Reference Blasius1913; McKeon et al. Reference McKeon, Zagarola and Smits2005; Furuichi et al. Reference Furuichi, Terao, Wada and Tsuji2015; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). A semi-empirical relation between the friction factor $\lambda =8\tau _{z,w}/(\rho U^2_b)$ and $Re$ is given as (known as the Prandtl friction law)

(3.1)\begin{equation} 1/\lambda^{1/2}=A\log_{10}(Re_b\,\lambda^{1/2})-B, \end{equation}

where the constant $A$ is related to the von Kármán constant as $A=1/(2\kappa \sqrt {2} \log _{10}(e))$. Curve-fitting the experimental data over $3.1\times 10^3< Re_b<3.2\times 10^6$ by Nikuradse (Reference Nikuradse1933) yields $A=2.0$ and $B=0.8$, which corresponds to $\kappa =0.407$. However, notable deviations were observed when comparing the Prandtl friction law with other experimental data. For example, McKeon et al. (Reference McKeon, Zagarola and Smits2005) showed that for the Princeton Superpipe data (in the range $3.1\times 10^4 \leq Re_b \leq 3.5 \times 10^7$), the constants of the Prandtl law work only over a limited range of $Re_b$. New constants (i.e. $A=1.920$ and $B=0.475$) and additional $Re$-dependent corrections are needed to better fit the data in the entire $Re_b$ range. For the ‘Hi-Reff’ data, Furuichi et al. (Reference Furuichi, Terao, Wada and Tsuji2015) found that $\lambda$ deviates from the Prandtl law by approximately $2.5\,\%$ in the lower $Re_b$ region, and $-3\,\%$ in the high $Re_b$ region. In addition, $\lambda$, although agreeing with the Superpipe data in the low $Re_b$ range, deviates for $Re_b>2 \times 10^5$.

Figure 3(a) shows the friction factor $\lambda$ as a function of $Re_b$, along with other DNS and experimental data as well as the theoretical prediction $\lambda _p$ based on (3.1). Note that these DNS results are conducted with various domain sizes, and prior studies demonstrated that its effect on one-point statistics appears limited when $L_z/R\geq 7$ (Feldmann et al. Reference Feldmann, Bauer and Wagner2018; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). All DNS and experimental data seem to follow $\lambda _p$. However, the scatter is better highlighted by examining the relative error with respect to the Prandtl law (i.e. $\lambda /\lambda _p-1$). As depicted in figure 3(b), all DNS data overshoot $\lambda _p$ at the low $Re_b$ (i.e. $\le 4\times 10^4$). Our data, which agree with Wu & Moin (Reference Wu and Moin2008), El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and Chin et al. (Reference Chin, Monty and Ooi2014) and the new simulation at $Re=2000$ using NEK5000, exceed $\lambda _p$ by approximately 2 %, while the results of Ahn et al. (Reference Ahn, Lee, Jang and Sung2013) and Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) are closer to $\lambda _p$ (within 1 % for $Re_b<5\times 10^4$). Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) attributed this discrepancy to the different grid resolutions employed in the $\theta$-direction, which is $(R\theta )^+=4\unicode{x2013}5$ in theirs and Ahn et al. (Reference Ahn, Lee, Jang and Sung2013), but $7\unicode{x2013}8$ for Wu & Moin (Reference Wu and Moin2008) and Chin et al. (Reference Chin, Monty and Ooi2014). However, the data from El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and our DNS have azimuthal resolutions comparable to those in Ahn et al. (Reference Ahn, Lee, Jang and Sung2013) and Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), but produce results similar to Wu & Moin (Reference Wu and Moin2008) and Chin et al. (Reference Chin, Monty and Ooi2014) – suggesting that the azimuthal resolution is not the main reason for such discrepancy. Interestingly, the data of Ahn et al. (Reference Ahn, Lee, Jang and Sung2013) and Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) are consistently lower than our and other results in the whole $Re_b$ range. In particular, at the highest $Re_b$, the Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) data undershoot by 2 % from $\lambda _p$, but our data are lower by only approximately 0.6 %. Table 3 asserts that such differences in $\lambda$ are beyond the uncertainty limit.

Figure 3. (a) Friction factor $\lambda$ as a function of $Re_b$, and (b) the relative deviations from the Prandtl friction law.

Table 3. Summary of values and standard deviations of some key parameters: the friction factor $\lambda =8\tau _{z,w}/(\rho U^2_b)$; the peak of axial velocity variance $\langle{u'^2_z}\rangle^{+}_p$; the peak of the Reynolds shear stress $-\langle{u'_ru'_z}\rangle^{+}_p$; the axial $\langle{\tau '^2_{z,w}}\rangle^+$ and azimuthal $\langle{\tau '^2_{\theta,w} }\rangle^+$ wall shear stress fluctuations; and the wall ($p'^+_{w,rms}$) and peak ($p'^+_{p,rms}$) values of root-mean-square (r.m.s.) pressure fluctuations.

Our DNS and the experimental data of Furuichi et al. (Reference Furuichi, Terao, Wada and Tsuji2015) agree well for the $Re_b$ range studied. Fitting our DNS data with (3.1) yields $A=2.039\pm 0.083$, $B=0.948\pm 0.364$ with uncertainty estimates based on 95 % confidence bounds, giving $\kappa =0.399\pm 0.015$. The value reported by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) (i.e. $A=2.102$, $B=1.148$) are slightly larger than ours but still within the uncertainty range. However, large uncertainty is present in the fitted values due to the limited data points in $Re$. In addition, as $Re_b$ is still relatively low, the reported value of $\kappa =0.399\pm 0.015$ should not be used outside of the given $Re$ range. As will be shown in § 3.4, even for the highest $Re_b$ case, a distinct logarithmic region does not manifest itself in the mean velocity profile $U(y)$. Higher $Re$ data (e.g. $Re_\tau \ge 10^4$) are required to better estimate the constants in (3.1) and the associated $\kappa$ values.

3.3. Wall shear stress fluctuations

The $Re$-dependence of axial wall shear stress fluctuation $\langle{\tau '^2_{z,w}}\rangle^+$ is one of the highly debated issues in wall turbulence. Note that $\langle{\tau '^2_{z,w}}\rangle^+$ is also equivalent to the wall dissipation of the axial Reynolds stress component $\epsilon ^+_{z,w}$, the azimuthal vorticity variance at the wall $\langle{\omega '^2_\theta }\rangle^+$ or the limiting value of $\langle{ u'^2_z}\rangle ^+/U^2$ at the wall (Örlü & Schlatter Reference Örlü and Schlatter2011). Previous DNS studies of channel, pipe and turbulent boundary layer observed an increase in $\langle{\tau '^2_{z,w}}\rangle^+$ with $Re_\tau$, which reflects the increased contribution of large-scale motions to wall shear stress at high $Re$ values (Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010a). However, the exact dependence of $\langle{\tau '^2_{z,w}}\rangle^+$ on $Re_\tau$ is not well established. For example, Örlü & Schlatter (Reference Örlü and Schlatter2011) suggested that the r.m.s. $\langle{\tau '^2_{z,w}}\rangle^+$ follows

(3.2)\begin{equation} \langle \tau'^2_{z,w} \rangle^{{+}1/2}=C+D\ln(Re_\tau), \end{equation}

where the two constants $C$ and $D$ are chosen as $0.298$ and $0.018$ based on the DNS of turbulent boundary layer data.

Some works (Yang & Lozano-Durán Reference Yang and Lozano-Durán2017; Smits et al. Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021) also suggested that

(3.3)\begin{equation} \langle \tau'^2_{z,w} \rangle^+=E+F\ln(Re_\tau). \end{equation}

By fitting turbulent channel flow data of Lee & Moser (Reference Lee and Moser2015) and pipe flow data of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) for $Re_\tau \ge 1000$, Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021) obtained $E=0.08$ and $F=0.0139$.

Recently, Chen & Sreenivasan (Reference Chen and Sreenivasan2021) proposed a defect-power law, given as

(3.4)\begin{align} \langle \tau'^2_{z,w} \rangle^+=\epsilon^+_{z,w}=G-H\,Re^{{-}1/4}_\tau, \end{align}

where $G$ is the asymptotic value at infinite $Re$, and $H$ is a coefficient. The assumption for (3.4) is that the dissipation of the axial Reynolds stress component ($\epsilon ^+_z=\nu \langle{(\partial u^+_z/\partial x^+_j)(\partial u^+_z/\partial x^+_j)}\rangle$) balances the turbulent kinetic energy production ($P_k=-\langle{u_ru_z}\rangle^+(\partial U^+/\partial y^+)$) near the location of peak production. The fact that $P_k$ is bounded by $1/4$ (Sreenivasan Reference Sreenivasan1989; Pope Reference Pope2000) implies that $\epsilon ^+_{z,w}$ may also stay bounded, which is further assumed by Chen & Sreenivasan (Reference Chen and Sreenivasan2021) to be the same bound of $P_k$, i.e. $G=1/4$ (Chen & Sreenivasan Reference Chen and Sreenivasan2021). This argument was later criticized by Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021) for the following two reasons. First, based on the DNS data, the location of peak production is actually the place where the largest imbalance of production and dissipation occurs. Second, as the balance between different terms in the Reynolds stress transport equation changes rapidly near the wall, it is unclear how the balance between $P_k$ and $\epsilon ^+_{z,w}$ can be extended up to the wall, where $P_k=0$, and $\epsilon ^+_{z,w}$ equals the viscous diffusion.

The axial wall shear stress fluctuation $\langle{\tau '^2_{z,w}}\rangle^+$ as a function of $Re_\tau$ is depicted in figure 4(a). Akin to the observation in El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013), the values for the pipe are slightly lower than those for the channel, but the difference decreases with increasing $Re_\tau$. Note that the data of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) are slightly lower than others, which is consistent with the lower $\lambda$ in figure 3. The logarithmic law (3.2) proposed by Örlü & Schlatter (Reference Örlü and Schlatter2011) is higher than the DNS data. This is somehow expected as the fitting coefficients were obtained based on the DNS of turbulent boundary layer data, which are higher than in pipe and channel. In the high $Re_\tau$ range, both the logarithmic (3.3) and defect-power-law scalings (3.4) agree well with the data. However, both scalings exhibit notable disagreements in the low $Re_\tau$ range. The discrepancy seems not particularly surprising, given that the parameters in these equations are obtained from different datasets and different $Re_\tau$ ranges. As a reference, the inset in figure 4(a) shows the fitting results of (3.2)–(3.4) using our DNS data only. The fitted values are $C=0.225$, $D=0.0264$ for (3.2), $E=0.016$, $F=0.0218$ for (3.3), and $G=0.255$, $H=0.477$ for (3.4). For the $Re_\tau$ range studied, the data seem to match better the defect-power law but with a slightly higher asymptotic value than suggested by Chen & Sreenivasan (Reference Chen and Sreenivasan2021). Additional data at higher $Re_\tau$ are needed to confirm this finding.

Figure 4. (a) Axial ($\langle \tau '^2_{z,w}\rangle ^+$) and (b) azimuthal ($\langle \tau '^2_{\theta,w}\rangle ^+$) wall shear stress fluctuations as a function of $Re_\tau$. The dotted, dashed and dashed-dotted lines in the inset of (a) denote $\langle \tau '^2_{z,w}\rangle ^+=(0.225+0.0264 \ln (Re_\tau ))^2$, $\langle \tau '^2_{z,w}\rangle ^+=0.016+0.0218 \ln (Re_\tau )$ and $\langle \tau '^2_{z,w}\rangle ^+=0.255-0.477\,Re^{-1/4}_\tau$, respectively.

Figure 4(b) further shows the azimuthal wall shear stress fluctuation $\langle \tau '^2_{\theta,w} \rangle ^+$ as a function of $Re_\tau$. Similar to findings for $\langle{\tau '^2_{z,w}}\rangle^+$, our data agree well with El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and NEK5000 2K cases, and all of them become closer to Lee & Moser (Reference Lee and Moser2015) with increasing $Re_\tau$. Fitting data with the logarithmic and defect-power law yields $\langle \tau '^2_{\theta,w} \rangle ^+=(0.058+0.023\ln (Re_\tau ))^2$, $\langle \tau '^2_{\theta,w} \rangle ^+=-0.040+0.016\ln (Re_\tau )$ and $\langle \tau '^2_{\theta,w} \rangle ^+=0.135-0.353\,Re^{-1/4}_\tau$. Again, the defect-power law seems to match better the DNS data, but the agreement is not as good as for $\langle{\tau '^2_{z,w}}\rangle^+$.

3.4. Mean velocity profile

In an overlap region between the inner and outer flows, there is a logarithmic variation of the mean axial velocity $U^+$ profile, which is given as

(3.5)\begin{equation} U^+=\frac{1}{\kappa} \ln y^+{+}B. \end{equation}

In a true log layer, the indicator function $\beta =y^+( \partial U^+/\partial y^+)$ is constant and equals $1/\kappa$.

The $U^+$ profiles at different $Re_\tau$ are compared to previous DNS data in figure 5(a). First, as expected, the wake in $U^+$ for the pipe is stronger than in the channel by Lee & Moser (Reference Lee and Moser2015). Second, our data in the outer region agree well with those of El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013), but not with Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). This discrepancy was also noted by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), who found that their data, along with those of Wu & Moin (Reference Wu and Moin2008) and Ahn et al. (Reference Ahn, Lee, Jang and Sung2013), differ from those of El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and Chin et al. (Reference Chin, Monty and Ooi2014). This disparity is probably due to the numerical methods, where all of the former used low-order finite difference methods, while the latter two used high-order spectral-element methods.

Figure 5. (a) Mean velocity profiles $U^+$ and (b) log-law diagnostic function $\beta$. Profiles in (a) are offset vertically by two wall units, and the shaded area in (b) represents the standard deviation of our data at $Re_\tau =5200$.

The log-law diagnostics function $\beta$ is shown in figure 5(b) for all high DNS data (i.e. $Re_\tau >5000$). Interestingly, $\beta$ for our $Re_\tau \approx 5200$ agrees with the channel data of Lee & Moser (Reference Lee and Moser2015) and Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) up to $y^+\approx 250$. Note that a relatively small domain size was used by Hoyas et al. (Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022), but it is assumed large enough to yield accurate flow statistics (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). This suggests a near-wall universality of the inner scaled mean velocity – similar to that observed by Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009). For all three cases, the trough is $\beta \approx 2.30$, located at $y^+\approx 70$. However, the data of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) deviate from others for $y^+>40$ and have a larger magnitude of the trough, which is somewhat consistent with the slight upward shift of the $U^+$ observed in figure 5(a). This discrepancy is significantly larger than the statistical uncertainty (see Appendix B). Unlike channel flow, where a plateau starts to develop for $Re_\tau \ge 5200$, there is no plateau for pipe flow – suggesting that the minimum $Re_\tau$ for $U^+$ to develop a logarithmic region should be higher in the pipe than in the channel. The $\beta$ in the wake region of the pipe is distinctly larger than the channel, implying a notable difference in flow structures in the core region between these two flows (Chin et al. Reference Chin, Monty and Ooi2014).

High-order corrections to the log-law relation (3.5) were sometimes introduced to better describe the mean velocity profile in the overlap region (Buschmann & Gad-el Hak Reference Buschmann and Gad-el Hak2003; Luchini Reference Luchini2017; Cantwell Reference Cantwell2019). For example, based on refined overlap arguments expressed by Afzal & Yajnik (Reference Afzal and Yajnik1973), Jiménez & Moser (Reference Jiménez and Moser2007) proposed the indicator function

(3.6)\begin{align} \beta=\left(\frac{1}{\kappa_\infty}+\frac{\alpha_1}{Re_\tau}\right)+\alpha_2\,\frac{y}{R}, \end{align}

where $\alpha _1$ and $\alpha _2$ are adjustable constants, and $\kappa _\infty$ is the asymptotic von Kármán constant. Equation (3.6) allows for a $Re$-dependence of $\kappa =\kappa _\infty +(\alpha _1/Re_\tau )^{-1}$ and introduces a linear dependence on $y$. By fitting our $Re_\tau \approx 5200$ data in the region between $y^+=300$ and $y=0.16$, we obtain $\kappa =0.401$ and $\alpha _2=2.0$. This $\kappa$ is very close to $0.399$ estimated from the friction factor relation (3.1) and $0.402$ reported by Jiménez & Moser (Reference Jiménez and Moser2007) using channel data of $Re_\tau =1000$ from Del Alamo et al. (Reference Del Alamo, Jiménez, Zandonade and Moser2004), and $Re_\tau =2000$ from Hoyas & Jiménez (Reference Hoyas and Jiménez2006). It is slightly larger than $0.387$ by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) and $0.384$ by Lee & Moser (Reference Lee and Moser2015). In addition, $\alpha _2$ is generally much larger in the pipe than in the channel – suggesting a strong geometry effect on $\beta$. The value of $\alpha _2=2$ is consistent with the finding by Luchini (Reference Luchini2017), who suggested that the logarithmic law of the velocity profile is universal across different geometries of wall turbulence, provided that the perturbative effect of the pressure gradient is taken into consideration. Furthermore, a good collapse between our data and the analytical prediction by Luchini (Reference Luchini2017) is seen in figure 5(b).

3.5. Reynolds stresses

The non-zero components of the Reynolds stress tensor (or the velocity variances and covariance) are examined in this subsection (figures 6–10). For all datasets, the inner-scaled velocity variances and covariance increase with $Re_\tau$ in the whole wall-normal range. In terms of the axial velocity variance $\langle{u'^2_z}\rangle^{+}$ (figure 6a), our data agree well with El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) but differ from Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), which is notably smaller in the near-wall region, particularly at low $Re_\tau$. For the highest $Re_\tau$ cases, the agreement is reasonably good near the wall. However, note that the Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) simulation is at a slightly higher $Re_\tau$ (i.e. $\sim 6000$). The differences between our case and that of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) can be better highlighted in the diagnostic plot (figure 6b), where the r.m.s. axial velocity fluctuation $u'_{z,rms}$ is plotted against the mean velocity $U^+$. The diagnostic plot was introduced by Alfredsson & Örlü (Reference Alfredsson and Örlü2010) to assess if the mean velocity and velocity fluctuation profiles behave correctly without the need to determine the friction velocity or the wall position. Consistent with the observation in Alfredsson & Örlü (Reference Alfredsson and Örlü2010), the diagnostic plot collapses in the outer region for $Re_\tau >180$, and has a clear $Re$ trend around the peak value. Most importantly, the data by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) are consistently lower than ours, particularly in the near-wall region. Such inconsistency is also observed for $\langle{u'^2_\theta}\rangle^{+}$ (figure 10). The agreement for $\langle{u'^2_r}\rangle^{+}$ (figure 7a) is reasonably good among different pipe flow datasets, which is slightly larger than the channel, particularly in the outer region. The Reynolds shear stress $\langle{u'_r u'_z}\rangle^+$ shows excellent agreement among different datasets, even including the channel.

Figure 6. (a) Axial velocity variance $\langle{ u'^2_z}\rangle ^+$ as a function of $y^+$; (b) the diagnostic plot depicting $u'_{z,rms}/U_c$ as a function of $U^+/U_c$; and (c) the inner peak of axial velocity variance $\langle{ u'^2_z}\rangle ^+_p$ as a function of $y^+$. The dashed and dashed-dotted lines in the inset of (c) denote $\langle{u'^2_z}\rangle^{+}_p=3.251+0.687\ln (Re_\tau )$ and $\langle{u'^2_z}\rangle^{+}_p=11.132-17.402\,Re^{-1/4}_\tau$, respectively.

Figure 7. (a) Radial velocity variance $\langle u'^2_r \rangle ^+$, and (b) Reynolds shear stress $\langle u'_r u'_z \rangle ^+$, as functions of $y^+$.

Figure 8. Reynolds number dependence of (a) the peak of the Reynolds shear stress $\langle u'_r u'_z \rangle ^+$, and (b) the corresponding peak location in wall units.

Figure 9. Indicator function for Townsend's prediction among different high-$Re$ simulations: (a) axial velocity variance $y\,\partial _y\langle{ u'^2_z}\rangle ^+$, and (b) azimuthal velocity variance $y\,\partial _y\langle u'^2_\theta \rangle ^+$.

Figure 10. Azimuthal velocity variance $\langle u'^2_\theta \rangle ^+$ as a function of (a) $y^+$ and (b) $y/R$. Only the two highest $Re_\tau$ cases are included in (b).

Let us focus now on the inner peak of the axial velocity variance $\langle{u'^2_z}\rangle^{+}$. The inner peak is assumed to increase logarithmically with $Re_\tau$ – similar to the wall shear stress fluctuations due to the increased modulation effect of the large-scale structures in the logarithmic layer (Marusic & Monty Reference Marusic and Monty2019). Chen & Sreenivasan (Reference Chen and Sreenivasan2021) suggested recently that the growth of $\langle{u'^2_z}\rangle^{+}$ would eventually saturate. The argument is based on the balance between the viscous diffusion and dissipation at the wall, and the Taylor series expansion of the axial velocity variance near the wall, given as

(3.7)\begin{equation} \langle{u'^2_z}\rangle^+\sim D^+_{z,w}y^{{+}2}= \epsilon^+_{z,w} y^{{+}2}. \end{equation}

Note that a similar expression is obtained in Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021), where the axial wall dissipation is used instead, i.e. $\langle{u'^2_z}\rangle^{+}\sim \langle{\tau '^2_{z,w}}\rangle^+y^{+2}$. If the assumption of the boundedness of wall dissipation (3.4) is valid and the inner peak location of $\langle{u'^2_z}\rangle^{+}$ (denoted as $y^+_{z,p}$) is independent of $Re_\tau$, then (3.7) suggests that the peak of axial velocity variance should also be bounded in a defect-power form similar to the wall shear stress fluctuation:

(3.8)\begin{align} \langle{u'^2_z}\rangle^+_p=M- N\,Re_\tau^{{-}1/4}, \end{align}

where $M$ is the asymptotic value and $N$ is the coefficient.

This validity of (3.8) was challenged recently by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), who, based on their data, observed a slight increase of $y^+_{z,p}$ with $Re_\tau$, from $y^+_{z,p}=14.28$ for $Re_\tau \approx 500$ to $15.14$ for $Re_\tau \approx 6000$. We emphasize that such variation of $y^+_{z,p}$ with $Re_\tau$ is not observed in our case, where much finer near-wall resolutions than in Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) are used. The value of $y^+_{z,p}$ is approximately $15$ for all $Re_\tau$ (e.g. $y^+_{z,p}=15.07$, $15.03$, $15.50$ for $Re_\tau =180$, $2000$, $5000$, respectively) – akin to the findings by many others (Moser et al. Reference Moser, Kim and Mansour1999; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Chin et al. Reference Chin, Monty and Ooi2014; Smits et al. Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021).

Figure 6(c) shows $\langle{u'^2_z}\rangle^{+}_p$ for all the DNS data listed in table 2, along with the logarithmic law $\langle{u'^2_z}\rangle^{+}_p=3.8+0.64\ln (Re_\tau )$ by Marusic et al. (Reference Marusic, Baars and Hutchins2017) and the power law $\langle{u'^2_z}\rangle^{+}_p=11.5-19.32Re_\tau ^{-1/4}$ by Chen & Sreenivasan (Reference Chen and Sreenivasan2021). The difference between different DNS datasets is relatively small, except for those from Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), which are consistently lower than others for all $Re_\tau$. Note that this discrepancy is much larger than the uncertainty (standard deviation), which is less than $0.5\,\%$ (see table 3). Both the logarithmic and defect-power laws fit well with the data in the high $Re_\tau$ range but have certain discrepancies at low $Re_\tau$. This suggests that there might exist a transitional scaling – similar to that found for the Reynolds shear stress (Chen, Hussain & She Reference Chen, Hussain and She2019). The parameters in these two scaling laws can be adjusted to better fit our dataset. The inset shows the fitting results for our data without the one at $Re_\tau =180$: $\langle{u'^2_z}\rangle^{+}_p=3.251+0.687\ln (Re_\tau )$ and $\langle{u'^2_z}\rangle^{+}_p=11.132-17.402\,Re^{-1/4}_\tau$. With these new constants, the agreement is improved for both scaling laws. In summary, for the $Re_\tau$ range studied, both scaling laws provide a good match with $\langle{u'^2_z}\rangle^{+}_p$ data when the fitting parameters are properly adjusted. Data at even higher $Re_\tau$ are required to determine which law is more consistent with the data.

According to Townsend's attached eddy hypothesis, at sufficiently high $Re$, the Reynolds stress components in a certain $y$ range satisfy

(3.9)$$\begin{gather} \langle{u'^2_z}\rangle^+=A_1-B_1 \ln(y/R), \end{gather}$$

(3.10)$$\begin{gather}\langle{u'^2_z}\rangle^+=A_2, \end{gather}$$

(3.11)$$\begin{gather}\langle{u'^2_\theta}\rangle^+=A_3-B_3 \ln(y/R), \end{gather}$$

(3.12)$$\begin{gather}\langle{u'_r u'_z}\rangle^+=-1, \end{gather}$$

where $A_i$ and $B_i$ are universal constants.

Consistent with these relations, the radial velocity variance $\langle{u'^2_r}\rangle^{+}$ slowly develops a flat region as $Re_\tau$ increases. In addition, the Reynolds shear stress $\langle{-u'_ru'_z}\rangle^+$ profiles also tend to become flattened at higher $Re_\tau$. As noted by Afzal (Reference Afzal1982), the peak Reynolds shear stress at high $Re_\tau$ follows $\langle{-u'_ru'_z}\rangle^+_p\approx 1-2/\sqrt {\kappa \,Re_\tau }$, and the corresponding position $y^+_m$ shifts away from the wall following $y^+_m\sim \sqrt {Re_\tau /\kappa }$. Chen et al. (Reference Chen, Hussain and She2019) suggested that there is a non-universal scaling transition, where the peaks at low $Re_\tau$ scales as $\langle{u'_ru'_z}\rangle^{+}_p\approx Re_\tau ^{-2/3}$ and their locations scales as $y^+_m \sim Re_\tau ^{1/3}$. Figure 8 shows $\langle{-u'_ru'_z}\rangle^+_p$ and the corresponding $y^+_m$ as a function of $Re_\tau$. For $Re_\tau >1000$, $Re^{-1/2}$ for $\langle{-u'_ru'_z}\rangle^+_p$ and $Re^{-1/2}$ for $y^+_m$ are satisfied with good accuracy, and at the low $Re_\tau$ range, $Re^{-2/3}_\tau$ for $\langle{-u'_ru'_z}\rangle^+_p$ and $Re^{-1/3}_\tau$ for $y^+_m$ scalings proposed by Chen et al. (Reference Chen, Hussain and She2019) also yield good agreement.

Regarding the axial velocity variance $\langle{ u'^2_z}\rangle ^+$, the indicator function is not flat anywhere in the domain (figure 9a) – suggesting that no clear logarithmic region develops for the $Re_\tau$ range considered. As discussed in Lee & Moser (Reference Lee and Moser2015), $Re_\tau =5200$ is not quite high enough to exhibit such a region. Based on the Superpipe data, Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) suggested that a sensible logarithmic layer emerges only for $Re_\tau >10^4$. Consistent with the findings in Lee & Moser (Reference Lee and Moser2015) and Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), the azimuthal velocity variance $\langle u'^2_\theta \rangle ^+$ develops the logarithmic layer at lower $Re_\tau$. The indicator function of $\langle u'^2_\theta \rangle ^+$ shows a distinct plateau (figure 9b). Interestingly, when compared with other cases, the range of the logarithmic layer is wider in our $Re_\tau \approx 5200$ case. Fitting the data in the range $120\le y^+ \le 800$ yields $A_3=0.921$, $B_3=0.420$, which are close to $A_3=1.0$, $B_3=0.40$ by Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), and between $A_3=1.08$, $B_3=0.387$ obtained by Lee & Moser (Reference Lee and Moser2015) for the turbulent channel and $A_3=0.8$, $B_3=-0.45$ by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) in the turbulent boundary layer.

3.6. Production and dissipation of the turbulent kinetic energy

Figure 11(a) shows the production $P^+_k$ and dissipation $\epsilon ^+_k\ (=\nu \langle{(\partial u'^+_i/\partial x^+_j)(\partial u'^+_i/\partial x^+_j)}\rangle)$ of the turbulent kinetic energy (i.e. $k^+=(\langle u'^2_r \rangle ^++\langle u'^2_\theta \rangle ^++\langle{ u'^2_z}\rangle ^+)/2$). Other terms in the transport equations of the turbulent kinetic energy and individual Reynolds stress components are available at https://dataverse.tdl.org/dataverse/turbpipe. The production $P^+_k$ has a peak at around $y^+\approx 11$, and the magnitude approaches the asymptotic value $1/4$ as $Re_\tau$ increases. Despite notable differences in the mean axial/streamwise velocity profile observed in the outer region, $P^+_k$ is quite similar between pipes and channels. This explains why the higher velocity gradient of the pipe does not contribute an effect to the turbulence intensities. The magnitude of dissipation $\epsilon ^+_k$ increases continuously with $Re_\tau$, and the difference between the pipe and channel is mainly in the near-wall region and decreases with increasing $Re_\tau$.

Figure 11. (a) Production $P^+_k$ and dissipation $\epsilon ^+_k$ of the turbulent kinetic energy, and (b) balance of production and dissipation $P^+_k/\epsilon ^+_k-1$, as functions of $y^+$.

At sufficiently high $Re_\tau$, there is an intermediate region where production balances dissipation. Recent numerical results (Lee & Moser Reference Lee and Moser2015; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) suggest that such equilibrium between production and dissipation is violated due to the presence of LSMs and VLSMs. Figure 11(b) shows the relative excess of production over dissipation ($P^+_k/\epsilon ^+_k-1$). First, there is a near-wall region ($8\le y^+\le 35$) where $P^+_k$ distinctly exceeds $\epsilon ^+_k$. At high $Re_\tau$, another region of $P^+_k/\epsilon ^+_k>1$ develops, and the magnitude increases with $Re_\tau$. For the $Re_\tau \approx 5200$ case, the peak imbalance is approximately $11\,\%$ (located at $y^+\approx 330$), which is slightly larger than in the channel (i.e. $8\,\%)$.

3.7. Mean pressure and r.m.s. pressure fluctuation

The mean pressure and r.m.s. pressure fluctuations are displayed in figure 12. First, the mean pressure $P^+$ has different behaviour in the outer region between pipe and channel flows, with $P^+$ being substantially lower in the wake of the pipe. As discussed in El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013), this difference is related to the mean radial momentum equation, which, in pipe flow, is given as (Hinze Reference Hinze1975)

(3.13)\begin{equation} \frac{1}{\rho}\,\frac{\partial P}{\partial r}+\frac{\mathrm{d}}{\mathrm{d} r}\langle{u'^2_r }\rangle+\frac{\langle{u'^2_r }\rangle-\langle{u'^2_\theta }\rangle}{r}=0. \end{equation}

By changing variable (i.e. $r=R-y$) and then integrating the above equation, the mean pressure for pipe flow with the wall value set to zero can be expressed as

(3.14)\begin{equation} P^+(y)={-}\langle{u'^2_r}\rangle^+{+}\int^y_0\frac{\langle{u'^2_r }\rangle^+{-}\langle{u'^2_\theta}\rangle}{R-y'}^+\, \mathrm{d} y'. \end{equation}

In channel flow, the last term on the left-hand side of (3.13) is absent, and the mean pressure is solely balanced by the wall-normal velocity fluctuation, i.e. $P^+(y)=-\langle{v'^2}\rangle^+$. From figure 7(a), it is clear that the wall-normal velocity fluctuation is comparable between pipe and channel flows. However, as $\langle{u'^2_r}\rangle^+<\langle{u'^2_\theta}\rangle^{+}$ in pipe flow, the extra term in (3.14) is zero at the wall and decreases with increasing $y$ – resulting in a lower pressure in pipes than in channels.

Figure 12. (a) Mean pressure ($P^+$) and (b) r.m.s. pressure fluctuation ($P^{\prime+}_{rms}$), as functions of $y^+$.

Similar to that observed by El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013), the r.m.s. pressure fluctuation $p'^{+}_{rms}$ exhibits similar behaviour between the pipe and channel, except for slightly lower values for the latter. Minor differences are observed between our data and those of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), particularly near the peak value. The difference between our data and the channel data of Lee & Moser (Reference Lee and Moser2015) in the near-wall region decreases with increasing $Re_\tau$. Figure 13 further shows the peak and wall values of $p'^+_{rms}$, which has similar $Re$-dependence as for other measures, such as wall shear stress fluctuations and axial velocity variances. Again, for the $Re_\tau$ studied, both the logarithmic and defect-power laws fit the data well.

Figure 13. Reynolds number dependence of (a) peak ($P^{\prime+}_{p,rms}$) and (b) wall ($P'^+_{w,rms}$) values of r.m.s. pressure fluctuations.

3.8. Energy spectra

As $Re$ increases, the separation of scales between the near-wall and outer-layer structures enlarges. In this section, the scale-separation is examined with one-dimensional velocity spectra for $Re_\tau \approx 5200$ (figure 14). The energy spectrum of axial velocity $u_z$ in the axial direction has two distinct peaks – the inner one located at $k_zR=40$ ($\lambda ^+_z=816$) and $y^+=13$, and the outer one at $k_zR=1$ ($\lambda _z=2{\rm \pi} R$), $y^+=400$. The dual-peak nature is more discernible in the azimuthal spectra with peaks at $k_\theta =250$ ($\lambda ^+_\theta =2{\rm \pi} r^+/k_\theta =131$), $y^+=13$ and $k_\theta =6$ ($\lambda _\theta =2{\rm \pi} r/k_\theta =0.846R$), $y^+=1000$. The $k_\theta$ values of all these peaks coincide with those found by Lee & Moser (Reference Lee and Moser2015) in the channel at the same $Re_\tau$, but the physical scales are smaller than in the channel. It is well known that the inner peak at $y^+=13$ is associated with the streaks that are generated through the near-wall self-sustaining cycle (Waleffe Reference Waleffe1997; Schoppa & Hussain Reference Schoppa and Hussain2002). As seen frequently in experiments (Hutchins & Marusic Reference Hutchins and Marusic2007; Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013), the outer peak results from VLSMs. The outer peak in the $\theta$-direction (at $y=0.192$) is located farther away than the streamwise one (at $y=0.077$), which according to Wu, Baltzer & Adrian (Reference Wu, Baltzer and Adrian2012) suggests that the VLSMs in the outer region maintain their energy in the $\theta$-direction more strongly than in the $z$-direction. The pre-multiplied energy spectra of the Reynolds shear stress in axial ($k_zE_{rz}/u^2_\tau$) and azimuthal ($k_\theta E_{rz}/u^2_\tau$) directions as functions of $y^+$ are shown in figures 14(c,d), respectively. The inner peak is located at $y^+=30$ with $k_z R=49.2$ ($\lambda ^+_\theta =664$) for $k_zE_{rz}/u^2_\tau$, and $k_\theta =268$ ($\lambda ^+_\theta =120$) for $k_\theta E_{rz}/u^2_\tau$. Compared with the axial velocity spectra, although the wavelength of the outer peak remains identical, the magnitude is much weaker and farther away from the wall.

Figure 14. Wavenumber pre-multiplied energy spectra for $Re_\tau =5200$: (a) $k_zE_{zz}/u^2_\tau$, (b) $k_\theta E_{zz}/u^2_\tau$, (c) $k_zE_{rz}/u^2_\tau$, and (d) $k_\theta E_{rz}/u^2_\tau$.

Figure 15(a) shows the one-dimensional pre-multiplied energy spectra $k_zE_{zz}/u^2_\tau$ at different $y$ locations. For comparison, the channel data of Lee & Moser (Reference Lee and Moser2015) at the same $Re_\tau$ are also included. First, good agreement is observed at $y^+=15$ between channel and pipe, particularly at higher wavenumbers – suggesting insignificance of pipe curvature on fine-scale near-wall structures. The scaling analysis of Perry, Henbest & Chong (Reference Perry, Henbest and Chong1986) suggests that the energy spectral density of the axial velocity fluctuations $k_zE_{zz}/u^2_\tau$ should vary as $k^{-1}_z$ in the overlap region. The $k^{-1}_z$ region has previously been observed in the high-$Re$ experiments (Nickels et al. Reference Nickels, Marusic, Hafez and Chong2005; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). Recently, such $k^{-1}_z$ has also been discovered in DNS of turbulent channel flow at $Re_\tau =5200$ (Lee & Moser Reference Lee and Moser2015). Similarly, a plateau in the region $6\le k_zR\le 10$ is observed for $90\le y^+\le 170$, and the magnitude $0.8$ agrees with experiments (Nickels et al. Reference Nickels, Marusic, Hafez and Chong2005; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). A bimodal is observed for $y^+=90$, with the peak magnitude at low wavenumbers ($k_zR=1$) being smaller than at high wavenumbers ($k_zR=30$). Interestingly, $k_zE_{zz}/u^2_\tau$ values at low wavenumbers are slightly smaller in the pipe than in the channel. Figure 15(b) further shows the one-dimensional pre-multiplied energy spectra $k_\theta E_{zz}/u^2_\tau$ at different $y$ locations. Again, $k_\theta E_{zz}/u^2_\tau$ agrees well between the pipe and the channel at high wavenumbers. Consistent with those in the channel, a plateau appears for $5\le k_\theta \le 30$ in the overlap region, with the magnitude increasing with $y^+$. Such a plateau is present even in the viscous sublayer, which is the footprint of LSMs and VLSMs near the wall (Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016).

Figure 15. Comparison of the pre-multiplied energy spectra between pipe (solid) and channel (dashed) for $Re_\tau =5200$: (a) $k_zE_{zz}/u^2_\tau$, and (b) $k_\theta E_{zz}/u^2_\tau$.