3.1 TKE production and dissipation distribution

TKE production $\mathscr{P}_{k}$ and dissipation $\mathscr{E}_{k}$ profiles have been commonly presented in DNS studies of turbulent wall-bounded flows (see, for example, Eggels et al. Reference Eggels, Unger, Weiss, Westerweel, Adrian, Friedrich and Nieuwstadt1994; Abe et al. Reference Abe, Kawamura and Matsuo2001; Hoyas & Jiménez Reference Hoyas and Jiménez2008; Wu & Moin Reference Wu and Moin2009; Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010). For the convenience of readers, figure 1 presents $\mathscr{P}_{k}$ and $\mathscr{E}_{k}$ versus wall-normal distance from two independent DNS studies of turbulent channel flows by Iwamoto, Suzuki & Kasagi (Reference Iwamoto, Suzuki and Kasagi2002) and Lee & Moser (Reference Lee and Moser2015). The agreement between the DNS of Lee & Moser (Reference Lee and Moser2015) at $Re_{\unicode[STIX]{x1D70F}}=550$ and the DNS of Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) at $Re_{\unicode[STIX]{x1D70F}}=640$ indicates the high quality of the simulations.

Figure 1. TKE production and dissipation versus wall-normal distance. (a) Inner-scaled wall-normal distance $y^{+}$ . (b) Outer-scaled wall-normal distance $y/\unicode[STIX]{x1D6FF}$ . Data are from independent DNS studies by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) and Lee & Moser (Reference Lee and Moser2015).

In figure 1(a) the wall-normal distance is inner scaled as $y^{+}=y/(\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}})$ . To better show the near-wall region, the data are plotted on semilogarithmic axes. To show the trend in the outer region, figure 1(b) presents TKE production and dissipation versus the outer-scaled location $y/\unicode[STIX]{x1D6FF}$ .

Figure 2. Reynolds number dependence of the global integral of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ (magenta squares), $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ (green squares) and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}$ (red squares) in turbulent channel flows. Also shown are the sum of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ (black squares), the $U_{b}^{+}$ from DNS (black $\times$ ) and $U_{b}^{+}$ from experimental measurement (black star) by Wei & Willmarth (Reference Wei and Willmarth1989). (a) On linear–linear axes. (b) On semilogarithmic axes. DNS data are from four independent studies: Abe et al. (Reference Abe, Kawamura and Matsuo2001), Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002), Lee & Moser (Reference Lee and Moser2015), Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016).

Three notable features of figure 1(a) are as follows. (i) The inner-scaled TKE production $\mathscr{P}_{k}^{+}$ peaks at $y^{+}\approx 12$ with a maximum value of about $\mathscr{P}_{k,max}^{+}\approx 0.25$ , with no apparent Reynolds number dependence (the $Re_{\unicode[STIX]{x1D70F}}=180$ case clearly has a low Reynolds number effect). (ii) The inner-scaled TKE dissipation at the wall $\mathscr{E}_{k}^{+}(y=0)$ clearly increases with Reynolds number. (iii) The TKE production $\mathscr{P}_{k}^{+}$ and dissipation $\mathscr{E}_{k}^{+}$ are not always in equilibrium. For example, the balance of the TKE budget equation in the viscous sublayer is between the dissipation $\mathscr{E}_{k}^{+}$ and the viscous diffusion term. However, the viscous diffusion, turbulent transport and pressure velocity correlation terms in the TKE budget equation only redistribute TKE among different layers, and the global integral of these transport terms can be shown to be zero in turbulent channel flow and turbulent pipe flows. Therefore, the global integral of TKE production and dissipation must have the same magnitude, but with opposite signs. The global identity for the TKE production should also apply to the global integral of the TKE dissipation.

The balance of the TKE budget equation among different terms is commonly presented in DNS studies of turbulent wall-bounded flows, as in Bernardini et al. (Reference Bernardini, Pirozzoli and Orlandi2014) and Lee & Moser (Reference Lee and Moser2015), and is not discussed here. The focus of this work is on the integral properties of the TKE production and dissipation, that is, the area under the $\mathscr{P}_{k}^{+}$ curve and $\mathscr{E}_{k}^{+}$ curve in figure 1(b).

3.2 Global integral of TKE production and dissipation

Using $R_{12}$ and $U$ data from four independent DNS studies of turbulent channel flows by Abe et al. (Reference Abe, Kawamura and Matsuo2001), Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002), Lee & Moser (Reference Lee and Moser2015) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016), the global integrals $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ are calculated and presented in figure 2 as a function of Reynolds number. Also shown in the figures is the global integral $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}$ calculated from the DNS data of TKE dissipation term. To illustrate the global integral identity equation (2.11) for channel flows, the bulk mean velocity $U_{b}^{+}$ from DNS is also plotted in figure 2. As a test of accuracy, the experimental measurement of $U_{b}^{+}$ by Wei & Willmarth (Reference Wei and Willmarth1989) is plotted in the figure, and can be seen to agree well with the DNS data.

Figure 2 shows that the sum of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ (black open squares in the plot) agrees very well with $U_{b}^{+}$ (black $\times$ or star in the plot), and this supports the validity of the global integral identity equation (2.11).

Figure 2 shows that in low Reynolds number turbulent channel flow, $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\text{d}y^{+}>\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ (in a laminar channel flow, $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}=U_{b}^{+}$ ). However, as Reynolds number increases, $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ decreases to a constant value of approximately $9.13$ for $Re_{\unicode[STIX]{x1D70F}}>300$ . This constant value is consistent with the results of Laadhari (Reference Laadhari2007) and Abe & Antonia (Reference Abe and Antonia2016). Thus, at sufficiently high Reynolds number, the global integral of the TKE production and dissipation in turbulent channel flows can be approximated as

(3.1a,b ) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}=\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx U_{b}^{+}-9.13\quad \text{or}\quad \int _{0}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y=\int _{0}^{\unicode[STIX]{x1D6FF}}\mathscr{E}_{k}\,\text{d}y\approx U_{b}u_{\unicode[STIX]{x1D70F}}^{2}-9.13u_{\unicode[STIX]{x1D70F}}^{3}.\end{eqnarray}$$

To date, the highest $U_{b}^{+}$ obtained in superpipe experiments by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013) is around $U_{b}^{+}\sim 35$ , so the contribution of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}\approx 9.13$ is not negligible in the identity equation (3.1). However, at infinite Reynolds number, $U_{b}^{+}\gg 9.13$ and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}=\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx U_{b}^{+}$ .

To better show the Reynolds number dependence, figure 2(b) presents the data on semilogarithmic axes. In figure 2(b) a logarithmic function is used to approximate the Reynolds number dependence of the inner-scaled bulk mean velocity as $U_{b}^{+}\approx 2.5\ln (Re_{\unicode[STIX]{x1D70F}})+2.5$ . From (3.1), the approximate function for the global integral of the TKE production and dissipation becomes

(3.2) $$\begin{eqnarray}\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}=-\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx 2.5\ln (Re_{\unicode[STIX]{x1D70F}})-6.6.\end{eqnarray}$$

Given the complicated nature of turbulent wall-bounded flows, it is unlikely that the exact Reynolds number dependence of $U_{b}^{+}$ is a simple logarithmic function, but it is not the purpose of this work to determine the best fitting function or parameters for the Reynolds number dependence. Figure 2(b) shows that the increase of $U_{b}^{+}$ , $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}$ with Reynolds number is in fact nearly logarithmic. Similar logarithmic functions have been used by Laadhari (Reference Laadhari2007), Zanoun, Nagib & Durst (Reference Zanoun, Nagib and Durst2009) and Abe & Antonia (Reference Abe and Antonia2016) to approximate the bulk mean velocity in high Reynolds number turbulent channel flow.

Figure 3 presents the Reynolds number dependence of the global integral results in turbulent pipe flows. As shown in the figures, the sum of $(1/R^{+})\int _{0}^{R^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}r^{+})^{2}r^{+}\,\text{d}r^{+}$ and $(1/R^{+})\int _{0}^{R^{+}}\mathscr{P}_{k}^{+}r^{+}\,\text{d}r^{+}$ agrees excellently with the $U_{b}^{+}$ data, and this strongly supports the global integral identity equation (2.12). The deviation of the superpipe data from the trend is caused by the spatial resolution in the near-wall region. At such a high Reynolds number, the first data point is close to or beyond the peak Reynolds shear stress location. In the superpipe experiments by Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015), the Reynolds shear stress $R_{12}$ is not measured. For the calculation of $\mathscr{P}_{k}$ , $R_{12}$ is calculated indirectly using $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y$ and the once-integrated mean momentum balance equation (see appendix B).

Figure 3. Reynolds number dependence of the global integral of $(1/R^{+})\int _{0}^{R^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}r^{+})^{2}r^{+}\,\text{d}r^{+}$ (magenta circles), $(1/R^{+})\int _{0}^{R^{+}}\mathscr{P}_{k}^{+}r^{+}\,\text{d}r^{+}$ (green circles) and $(1/R^{+})\int _{0}^{R^{+}}\mathscr{E}_{k}^{+}r^{+}\,\text{d}r^{+}$ (red circles) in turbulent pipe flows. Also shown are the sum of $(1/R^{+})\int _{0}^{R^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}r^{+})^{2}r^{+}\,\text{d}r^{+}$ and $(1/R^{+})\int _{0}^{R^{+}}\mathscr{P}_{k}^{+}r^{+}\,\text{d}r^{+}$ (black circles), $U_{b}^{+}$ from DNS (black $\times$ ) by El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and $U_{b}^{+}$ from superpipe (black star) by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013). (a) On linear–linear axes. (b) On semilogarithmic axes.

Figure 4 shows the global integral results for ZPG TBL. Due to the advection term (the last two integrals) in (2.13), the sum of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ will be smaller than $U_{\infty }^{+}$ , as shown in the figure, but figure 4 shows that the difference is small, indicating that the mean advection has a minor role in the global balance of the TKE budget. The deviation of the experimental data at $Re_{\unicode[STIX]{x1D70F}}\approx 10\,000$ is caused by the spatial resolution of data points in the experiments of De Graaff & Eaton (Reference De Graaff and Eaton2000).

Figure 4. Reynolds number dependence of the global integral of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ (magenta triangles), $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ (green triangles) and $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}$ (red triangles) in ZPG TBL. Also shown are the sum of $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}+\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ (black triangles), $U_{\infty }^{+}$ from DNS (black $\times$ ) and $U_{\infty }^{+}$ from experimental measurement (black asterisk) by De Graaff & Eaton (Reference De Graaff and Eaton2000), Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015). (a) On linear–linear axes. (b) On semilogarithmic axes. DNS data are from two independent studies by Simens et al. (Reference Simens, Jiménez, Hoyas and Mizuno2009) and Schlatter & Örlü (Reference Schlatter and Örlü2010).

To sum up, the global integral identity of the TKE production (2.11)–(2.13) has been verified with high resolution experimental and DNS data from turbulent channel flows, pipe flows and ZPG TBL. Next, we will present the partition of the integral of the TKE production and dissipation in turbulent wall-bounded flows. We find that at sufficiently high Reynolds number, the integrals of the TKE production and dissipation are equally partitioned around the peak Reynolds shear stress location $y_{m}$ . Given the importance of $y_{m}$ and meso-scaling in the partition of the integral of TKE production and dissipation, we will briefly review the meso-layer and meso-scaling before presenting the partition of the integrals.

3.3 Meso-layer, meso-length scale and peak location of Reynolds shear stress

The concept of a meso-layer has been proposed in a number of studies, including Long & Chen (Reference Long and Chen1981), Afzal (Reference Afzal1982, Reference Afzal1984), Sreenivasan & Sahay (Reference Sreenivasan, Sahay and Panton1997), Wosnik, Castillo & George (Reference Wosnik, Castillo and George2000) and Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005). For example, in Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005) a four-layer structure is proposed for turbulent wall-bounded flows, based on the force balance of the mean momentum balance equation. Their Layer III (the meso-layer) centres around the peak Reynolds shear stress location $y_{m}$ .

A meso-length scale for turbulent channel/pipe and ZPG TBL flows has been proposed as the geometric mean of the inner length scale $l_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ and the outer length scale $l_{o}=\unicode[STIX]{x1D6FF}$ , as $l_{m}=\sqrt{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}}$ (Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005a ,Reference Fife, Wei, Klewicki and McMurtry b ; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). Meso-length scale $l_{m}$ has been advocated as a fundamental length scale in turbulent channel/pipe flow and ZPG TBL, like the $l_{\unicode[STIX]{x1D708}}$ and $l_{o}$ (Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005a ,Reference Fife, Wei, Klewicki and McMurtry b ; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). It has long been known that the peak Reynolds shear stress location $y_{m}$ scales as $y_{m}=O(\sqrt{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}})$ (see, for example, Long & Chen Reference Long and Chen1981; Sreenivasan Reference Sreenivasan1989; Sreenivasan & Sahay Reference Sreenivasan, Sahay and Panton1997; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). Thus, the peak Reynolds shear stress location $y_{m}$ scales as the meso-length scale.

Figure 5. Dependence of Reynolds shear stress peak location $y_{m}$ on Reynolds numbers. Squares are channel flow data from DNS of Abe et al. (Reference Abe, Kawamura and Matsuo2001), Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002), Lee & Moser (Reference Lee and Moser2015), circles are pipe flow data from DNS of El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013) and superpipe by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012, Reference Hultmark, Vallikivi, Bailey and Smits2013) and triangles are ZPG-TBL data from DNS by Schlatter & Örlü (Reference Schlatter and Örlü2010) (filled triangles) and Simens et al. (Reference Simens, Jiménez, Hoyas and Mizuno2009) (open triangles). Note Reynolds shear stress $R_{12}$ is not directly measured in the superpipe, but calculated indirectly as $R_{12}^{+}=1-y^{+}/R^{+}-\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+}$ (see appendix B for more details).

The meso-length scaling is defined as

(3.3) $$\begin{eqnarray}\frac{y}{l_{m}}=\frac{y}{\sqrt{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}}}=\frac{y^{+}}{\sqrt{\unicode[STIX]{x1D6FF}^{+}}}.\end{eqnarray}$$

Given the spatial resolution limitation inherent in physical experiments, it has been challenging to determine $y_{m}$ precisely. However, during the past thirty years, more DNS data at higher Reynolds numbers, and higher resolution experimental data, have made it possible to determine more precisely the Reynolds number dependence of $y_{m}$ . Figure 5 shows the Reynolds number dependence of the meso-scaled peak location $y_{m}/l_{m}$ . The experimental and numerical data in figure 5 offer strong evidence that at sufficiently high Reynolds number the peak Reynolds shear stress location $y_{m}$ scales as the meso-length scale: $y_{m}=O(l_{m})$ .

Figure 5 shows that the peak Reynolds shear stress location $y_{m}$ in turbulent channel flows is approximately $y_{m}\approx 1.5l_{m}$ or $y_{m}^{+}\approx 1.5\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ for $Re_{\unicode[STIX]{x1D70F}}>2000$ . The peak Reynolds shear stress location in ZPG TBL is slightly larger, at $y_{m}\approx 2.4l_{m}$ or $y_{m}^{+}\approx 2.4\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ . In the 1980s, Long & Chen (Reference Long and Chen1981) proposed a similar scaling but with a numerical factor of 1.87, and Sreenivasan (Reference Sreenivasan1989) suggested a numerical factor of 2.0. Given the spatial resolution of the instrumentation available then, and the Reynolds number range of the data, these predictions are remarkably close to the recent DNS and high resolution experimental data shown in figure 5.

One motivation for the present work is to assess the role of the meso-layer and meso-length scale in the partition of the integrals of TKE production and dissipation. In figure 6 we present the distribution of the TKE production and dissipation versus the meso-scaled distance from the wall, to complement the inner-scaled distance and the outer-scaled distance presented in figure 1. The peak location of TKE production in meso-scaling becomes smaller with increasing Reynolds number. This is not surprising, as the peak location of the TKE production scales with inner scaling.

Figure 6. TKE production and dissipation versus meso-scaled distance from the wall $y/l_{m}$ . Data from DNS of channel flow Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) and Lee & Moser (Reference Lee and Moser2015).

3.4 Pre-multiplied TKE production and dissipation

To better show the near-wall region, it is common to plot the wall-normal distance on a logarithmic scale. For the convenience of depicting the area under the curve (its integral) on semilogarithmic axes, the TKE production and dissipation have been pre-multiplied by the wall-normal distance, because the derivative of a logarithmic function has the following property:

(3.4) $$\begin{eqnarray}\text{d}\log _{10}(y)=\frac{1}{\ln (10)}\frac{1}{y}\,\text{d}y.\end{eqnarray}$$

Thus the area under the pre-multiplied TKE production and dissipation curves on semilogarithmic axes equals the integrals of TKE production and dissipation. For the inner-scaled $y^{+}$ and meso-scaled $y/l_{m}$ , the integrals of TKE production (or dissipation) can be written, respectively, as

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}=\int _{\log _{10}(y_{1}^{+})}^{\log _{10}(\unicode[STIX]{x1D6FF}^{+})}[\ln (10)]y^{+}\mathscr{P}_{k}^{+}\,\text{d}(\log _{10}y^{+}), & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}=\int _{\log _{10}(y_{1}^{+}/\sqrt{\unicode[STIX]{x1D6FF}^{+}})}^{\log _{10}(\sqrt{\unicode[STIX]{x1D6FF}^{+}})}[\ln (10)]y^{+}\mathscr{P}_{k}^{+}\,\text{d}\left(\log _{10}\left(\frac{y^{+}}{\sqrt{\unicode[STIX]{x1D6FF}^{+}}}\right)\right), & \displaystyle\end{eqnarray}$$

where $y_{1}$ is the location of the first data point above the surface. As $\log (0)$ is undefined, the first data point $y_{1}$ should not be at the wall. In practice, the first data point is at a small distance from the wall.

In figure 7(a) the pre-multiplied TKE production and dissipation are plotted versus the inner-scaled distance from the wall $y^{+}$ on semilogarithmic axes. For convenience, the xticks on the bottom show values of $y^{+}$ and the xticks on the top show values of $\log _{10}y^{+}$ . As in figure 1(a), the pre-multiplied TKE production also peaks within the viscous buffer layer at $y^{+}\approx 12$ .

Figure 7. Pre-multiplied TKE production and dissipation versus distance from the wall on semilogarithmic axes. (a) Distance from the wall is inner scaled $y^{+}$ . (b) Distance from the wall is meso-scaled $y/l_{m}$ . The xticks on bottom denote values of $y^{+}$ or $y/l_{m}$ and xticks on top denote values of $\log _{10}(y^{+})$ or $\log _{10}(y/l_{m})$ .

Figure 7 shows that as Reynolds number increases the pre-multiplied TKE production and dissipation approach a plateau away from the wall, before dropping sharply towards the channel centre. It has been observed that at sufficiently high Reynolds number, say $Re_{\unicode[STIX]{x1D70F}}>20\,000$ , a second bump appears away from the wall in the pre-multiplied TKE production profile. For example, using a composite mean velocity profile and the corresponding Reynolds shear stress profile proposed by Perry, Marusic & Jones (Reference Perry, Marusic and Jones2002), Marusic et al. (Reference Marusic, Mathis and Hutchins2010a ) sketched the pre-multiplied TKE production profiles $y^{+}\mathscr{P}_{k}^{+}$ for different Reynolds numbers. The value of the plateau is approximately $y^{+}\mathscr{P}_{k}^{+}\approx 2.5$ . Their sketch for $Re_{\unicode[STIX]{x1D70F}}=10^{6}$ shows a smaller second bump away from the wall (Marusic et al. Reference Marusic, Mathis and Hutchins2010a ). The second bump is also observed in the plot by Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016).

In figure 7(b), the pre-multiplied TKE production and dissipation are plotted versus the meso-scaled distance from the wall $y/l_{m}$ on semilogarithmic axes. The advantage of the meso-scaled length scale is shown in figure 8, in which the area under the curves of the pre-multiplied TKE production and dissipation is approximated by a rectangle.

Figure 8. Sketch approximating the integral of TKE production and dissipation at high Reynolds number. Dashed curves approximate the area under the pre-multiplied TKE production and dissipation curves of DNS data at $Re_{\unicode[STIX]{x1D70F}}=5300$ by Lee & Moser (Reference Lee and Moser2015). The dot-dashed curves are approximations for a higher Reynolds number at $Re_{\unicode[STIX]{x1D70F}}=10^{6}$ . The trend will be similar as Reynolds number increases further. The vertical dashed black line denotes the peak Reynolds shear stress location $y_{m}/l_{m}\approx 1.5$ , as shown in figure 5.

In figure 8 the dashed rectangle approximates the integral of the TKE production and dissipation at $Re_{\unicode[STIX]{x1D70F}}=5300$ of Lee & Moser (Reference Lee and Moser2015), and the dot-dashed rectangle is an extrapolation for a higher Reynolds number at $Re_{\unicode[STIX]{x1D70F}}=10^{6}$ . The height of the rectangle is approximately $\ln (10)y^{+}\mathscr{P}_{k}^{+}\approx 5.7$ , which is consistent with the value used in the sketch of Marusic et al. (Reference Marusic, Baars and Hutchins2017), differing by a factor of $\ln (10)$ .

The width of the rectangle in figure 8 can be approximated as $\log _{10}(\unicode[STIX]{x1D6FF}^{+})$ (see the xticks on the top of figure 8). Thus the area under the curves of the pre-multiplied TKE production and dissipation can be approximated as $5.7\log _{10}(\unicode[STIX]{x1D6FF}^{+})\approx 2.5\ln (Re_{\unicode[STIX]{x1D70F}})$ . This approximation is consistent with the Reynolds number dependence in the global integral identity equation $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}=\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx U_{b}^{+}-9.13$ and the data for $U_{b}^{+}\approx 2.5\ln (Re_{\unicode[STIX]{x1D70F}})+2.5$ , as shown in figure 2(b).

The vertical dashed line at $y/l_{m}=1.5$ in figure 8 indicates the peak Reynolds shear stress location, as shown in figure 5. Figure 8 shows that the width of the rectangle on both sides of $y_{m}$ is of the order of $0.5\log _{10}(\unicode[STIX]{x1D6FF}^{+})$ (see the xticks on the top).

The equal partition of the integral of the TKE production and dissipation around $y_{m}$ can also be explained using table 1. The ranges of boundary layer around $y_{m}$ , in meso-scaling, are $O((\log _{10}(\unicode[STIX]{x1D6FF}^{+}))/2)$ on both sides. Assuming $A_{k}$ for the height of the rectangle that approximates the pre-multiplied TKE production or dissipation, the area or the integral of TKE production and dissipation between the first data point $y_{1}$ and $y_{m}$ is $O((\log _{10}(\unicode[STIX]{x1D6FF}^{+})A_{k})/2)$ , and the area between $y_{m}$ and $\unicode[STIX]{x1D6FF}$ is also $O((\log _{10}(\unicode[STIX]{x1D6FF}^{+})A_{k})/2)$ .

Table 1. Partition of boundary layer around the meso-layer. For convenience, the first data point is taken as $y_{1}^{+}=1$ . (One can choose other values for $y_{1}$ . For example, if $y_{1}^{+}=0.1$ , then $\log _{10}(0.1)=-1$ , and $\log _{10}(0.1/\sqrt{\unicode[STIX]{x1D6FF}^{+}})\approx -0.5\log _{10}(\unicode[STIX]{x1D6FF}^{+})-1$ . As $\unicode[STIX]{x1D6FF}^{+}\rightarrow \infty$ , the constant $-1$ becomes negligible. Furthermore, as $y_{1}$ becomes smaller than 0.1, the pre-multiplied TKE production and dissipation quickly approach 0.)

Figure 8 and table 1 indicate that the integrals of the TKE production and dissipation are equally partitioned around $y_{m}$ with a value of $1.25\ln (\unicode[STIX]{x1D6FF}^{+})$ . The advantage and importance of meso-scaled length can be summarized simply: at sufficiently high Reynolds number, the integrals of TKE production and dissipation are equally partitioned around the peak Reynolds shear stress location $y_{m}$ .

The approximation of pre-multiplied TKE production or dissipation as a rectangle is crude, but it is interesting that meso-scaled distance from the wall reveals that the TKE production and dissipation seem to centre the integrals evenly across the meso-layer. Next we will use DNS data to assess quantitatively the partition of the integral of the TKE production and TKE dissipation around the peak Reynolds shear stress location $y_{m}$ .

3.5 Partition of the TKE production and dissipation around $y_{m}$

Using the TKE budget data from DNS by Abe et al. (Reference Abe, Kawamura and Matsuo2001), Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002), Lee & Moser (Reference Lee and Moser2015) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016), the integrals of TKE production from $0$ to $y_{m}$ and from $y_{m}$ to $\unicode[STIX]{x1D6FF}$ are calculated and presented in figure 9(a). Also plotted is the $(U_{b}^{+}-9.13)/2$ from (3.1). To better show the partition, the integrals are divided by $(U_{b}^{+}-9.13)$ in figure 9(b), which clearly shows that at low Reynolds number, the integral from $0$ to $y_{m}$ is larger than that from $y_{m}$ to $\unicode[STIX]{x1D6FF}$ . For example, at $Re_{\unicode[STIX]{x1D70F}}=100$ , the value from $0$ to $y_{m}$ (solid green square) is approximately $0.6$ , and the value from $y_{m}$ to $\unicode[STIX]{x1D6FF}$ (open green square) is approximately $0.25$ . Note that they do not add up to 1 because $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ is larger than $9.13$ at this low Reynolds number. The dominance of the near-wall region at low Reynolds number is consistent with the observation of Marusic et al. (Reference Marusic, Mathis and Hutchins2010a ).

At sufficiently high Reynolds number, figure 9(b) shows that the integral of TKE production indeed approaches an equal partition around $y_{m}$ , with $\int _{0}^{y_{m}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx \int _{y_{m}^{+}}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx 0.5(U_{b}^{+}-9.13)$ .

Figure 9. Partition of the integral of the TKE production about the peak Reynolds shear stress location $y_{m}$ . (a) Integral of the TKE production from $0$ to $y_{m}$ and from $y_{m}$ to $\unicode[STIX]{x1D6FF}$ versus Reynolds number. (b) Integrals divided by $(U_{b}^{+}-9.13)$ versus Reynolds number. The filled squares represent $\int _{0}^{y_{m}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ calculated from DNS of turbulent channel flow by Abe et al. (Reference Abe, Kawamura and Matsuo2001), Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002), Lee & Moser (Reference Lee and Moser2015). The open squares represent $\int _{y_{m}^{+}}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}$ . Circles represent turbulent pipe flow data from El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013).

Figure 10. Partition of TKE production and dissipation around the peak Reynolds shear stress location $y_{m}$ . Turbulent channel flows (squares), turbulent pipe flows (circles), ZPG TBL (triangles). A value of $a=9.13$ is used for turbulent channel and pipe flows and $a=12$ is used for ZPG TBL.

Partition of the integral of TKE dissipation is calculated from DNS data and presented in figure 10. It can only be proven that the global integral of the TKE production must equal the global integral of the TKE dissipation in turbulent channel flows and turbulent pipe flows. Given the complicated roles of the transport terms in the different layers, it is not known a priori that the partition of the integral of the TKE dissipation will be the same as that of the TKE production. However, the DNS data shown in figure 10 strongly suggest that the integral of the TKE dissipation data is also equally partitioned around the peak Reynolds shear stress location $y_{m}$ at sufficiently high Reynolds number, $\int _{0}^{y_{m}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx \int _{y_{m}^{+}}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx 0.5(U_{b}^{+}-9.13)$ .

3.6 ‘Log layer’ in the integral profiles of TKE production and dissipation

Finally, we investigate the integral profiles of the TKE production and dissipation. Denote the integral of TKE production and dissipation as

(3.7a,b ) $$\begin{eqnarray}{\mathcal{I}}_{\mathscr{P}_{k}}(y)=\int _{0}^{y}\mathscr{P}_{k}\,\text{d}y;\quad {\mathcal{I}}_{\mathscr{E}_{k}}(y)=\int _{0}^{y}\mathscr{E}_{k}\,\text{d}y.\end{eqnarray}$$

Using inner-scaled variables, the inner-scaled integral functions can be written as

(3.8a,b ) $$\begin{eqnarray}{\mathcal{I}}_{\mathscr{P}_{k}}^{+}(y)=\frac{\displaystyle \int _{0}^{y}\mathscr{P}_{k}\,\text{d}y}{u_{\unicode[STIX]{x1D70F}}^{3}};\quad {\mathcal{I}}_{\mathscr{E}_{k}}^{+}(y)=\frac{\displaystyle \int _{0}^{y}\mathscr{E}_{k}\,\text{d}y}{u_{\unicode[STIX]{x1D70F}}^{3}}.\end{eqnarray}$$

The integral profiles of TKE production and dissipation are calculated from the DNS data of turbulent channel flow by Abe et al. (Reference Abe, Kawamura and Matsuo2001) and Lee & Moser (Reference Lee and Moser2015) and presented in figure 11. Also plotted in the figure is the inner normalized mean streamwise velocity $U^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=5200$ shifted downwards by 9.3 and the log-law equation for $U^{+}$ , but with an additive constant of $-4.3$ .

Figure 11. Integral profiles of the TKE production and dissipation versus $y^{+}$ . Data are from DNS of turbulent channel flows by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) and Lee & Moser (Reference Lee and Moser2015).

In the viscous sublayer and buffer layer, the integral of the TKE dissipation is larger than the integral of the TKE production. For example, the integral of the TKE production at $y^{+}\approx 5$ (the end of the viscous sublayer) is $\int _{0}^{5}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx 0.06$ . The integral of the TKE dissipation at the end of viscous sublayer is $\int _{0}^{5}\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx 0.8$ . As shown in figure 1(a), in the viscous sublayer the TKE production term is small, and the TKE dissipation is balanced by the viscous diffusion term.

Figure 12(a) shows the integral profiles of TKE production and dissipation for turbulent pipe flows, and figure 12(b) shows the same profiles for ZPG TBL. To facilitate comparison, the same logarithmic equation $(1/0.4)\ln (y^{+})-4.3$ is plotted in all three figures. Notable features in figures 11 and 12 are as follows. (i) At the channel or pipe centreline, the magnitude of the global integral of the TKE production is the same as that of the dissipation. However, outside the boundary layer of ZPG TBL, the magnitude of the global integral of TKE production is larger than that of the dissipation, as shown in figure 12(b). (ii) Although all three types of turbulent wall-bounded flows exhibit a logarithmic-like layer at sufficiently high Reynolds number, noticeable differences exist among the three flows, especially in the wake region.

Figure 12. Integral profiles of the TKE production and dissipation versus $y^{+}$ . (a) Turbulent pipe flows at $Re_{\unicode[STIX]{x1D70F}}=181,\,550,\,1000$ by El Khoury et al. (Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013). (b) ZPG TBL at $Re_{\unicode[STIX]{x1D703}}=670,\,2000,\,3270$ ( $Re_{\unicode[STIX]{x1D70F}}=252,\,671,\,1043$ ) by Schlatter & Örlü (Reference Schlatter and Örlü2010).

The logarithmic-like layer shown in figures 11 and 12 can be represented as

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{I}}_{\mathscr{P}_{k}}^{+}(y^{+})\approx \frac{1}{\unicode[STIX]{x1D705}}\ln (y^{+})+C_{\mathscr{ P}_{k}}, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{I}}_{\mathscr{E}_{k}}^{+}(y^{+})\approx \frac{1}{\unicode[STIX]{x1D705}}\ln (y^{+})+C_{\mathscr{ E}_{k}}. & \displaystyle\end{eqnarray}$$

The von Kármán ‘constant’ in the log law is approximately the same as the one in the $U^{+}$ profile. Discussions on the existence of the log layer and the ‘universality’ of the von Kármán constant can be found in the work of Nagib & Chauhan (Reference Nagib and Chauhan2008). The downward shift in the integral profiles of TKE production and dissipation is directly related to the constant $-9.13$ in the global integral identity equation (3.1).

The logarithmic-like behaviour of the TKE production integral in turbulent channel flow can be explained by (2.10):

(3.11) $$\begin{eqnarray}{\mathcal{I}}_{\mathscr{P}_{k}}^{+}(y^{+})=U^{+}-\left\{\frac{y^{+}U^{+}-\int _{0}^{y^{+}}U^{+}\,\text{d}y^{+}}{\unicode[STIX]{x1D6FF}^{+}}+\int _{0}^{y^{+}}\left(\frac{\unicode[STIX]{x2202}U^{+}}{\unicode[STIX]{x2202}y^{+}}\right)^{2}\,\text{d}y^{+}\right\}.\end{eqnarray}$$

Thus the integral of TKE production can be approximated by $U^{+}$ with a downward shift. It has been shown above that, at sufficiently high Reynolds number, $\int _{0}^{y^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}\lesssim 9.13$ . Moreover, outside the viscous sublayer and buffer layer, it can be shown that $(y^{+}U^{+}-\int _{0}^{y^{+}}U^{+}\,\text{d}y^{+})/\unicode[STIX]{x1D6FF}^{+}$ is small. In other words, the logarithmic-like behaviour of the TKE production is directly related to the logarithmic-like behaviour of the mean velocity $U^{+}$ .

The logarithmic-like scaling for the global integrals of the TKE production and dissipation, equation (3.1), is also directly related to the logarithmic-like behaviour of the integral profiles of TKE production and dissipation, equations (3.9) and (3.10). At sufficiently high Reynolds number, equations (3.9) and (3.10) indicate that

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx \frac{1}{\unicode[STIX]{x1D705}}\ln (\unicode[STIX]{x1D6FF}^{+})+C_{\mathscr{ P}_{k}}=O(\ln (\unicode[STIX]{x1D6FF}^{+})), & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}^{+}}-\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx \frac{1}{\unicode[STIX]{x1D705}}\ln (\unicode[STIX]{x1D6FF}^{+})+C_{\mathscr{ E}_{k}}=O(\ln (\unicode[STIX]{x1D6FF}^{+})). & \displaystyle\end{eqnarray}$$

It is known that, at sufficiently high Reynolds number, $y_{m}=O(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}})$ or $y_{m}^{+}=O(\sqrt{\unicode[STIX]{x1D6FF}^{+}})$ . From (3.9) and (3.10), the scaling for the integral of TKE production and dissipation from wall to $y_{m}$ can be estimated as

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{y_{m}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx \frac{1}{\unicode[STIX]{x1D705}}\ln (\sqrt{\unicode[STIX]{x1D6FF}^{+}})+C_{\mathscr{P}_{k}}=O(0.5\ln (\unicode[STIX]{x1D6FF}^{+}))\approx 0.5\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}, & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{y_{m}^{+}}-\mathscr{E}_{k}^{+}\,\text{d}y^{+}\approx \frac{1}{\unicode[STIX]{x1D705}}\ln (\sqrt{\unicode[STIX]{x1D6FF}^{+}})+C_{\mathscr{E}_{k}}=O(0.5\ln (\unicode[STIX]{x1D6FF}^{+}))\approx 0.5\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}-\mathscr{E}_{k}^{+}\,\text{d}y^{+}. & \displaystyle\end{eqnarray}$$

Hence, the equal partition of TKE production around $y_{m}$ is also directly related to the logarithmic-like behaviour of the integral profiles of TKE production and dissipation, equation (3.9).

The noticeable difference among the integrals of the TKE production and dissipation in channel flows, pipe flows and ZPG TBL is similar to the differences observed in the low-order statistics on different types of turbulent wall-bounded flows, in particular within the wake region, as discussed by Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2009), Mathis et al. (Reference Mathis, Monty, Hutchins and Marusic2009b ), Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009) and Ng et al. (Reference Ng, Monty, Hutchins, Chong and Marusic2011). The differences between the internal flow (channel or pipe flows) and ZPG TBL are traced by Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2009) to an excess of production of the streamwise turbulent energy in the outer part of the boundary layer.