4.1 Effect of the streamwise domain

We first clarify the effect of $L_{x}$ on the total energy dissipation $E$ . Distributions of $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle$ and $\langle \overline{{\it\varepsilon}}\rangle$ normalized by $U_{{\it\tau}}^{3}$ are shown in figure 1 as a function of $L_{x}/h$ for both $h^{+}=395$ and 1020. Also included in figure 1(a) are the data of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) at $h^{+}=4179$ with $L_{x}=2{\rm\pi}h$ and Lee & Moser (Reference Lee and Moser2015) at $h^{+}=1000$ and 5186 with $L_{x}=8{\rm\pi}h$ since $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ is constant for $h^{+}\geqslant 300$ (see figure 3 a) and the departure from constancy corresponds to the $L_{x}$ effect. It is shown that the viscous energy dissipation rate $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ increases with decreasing $L_{x}$ when $L_{x}<2h$ . This is caused by the thickened buffer region as $L_{x}$ decreases (see figure 2 a where the distributions of $(\text{d}\bar{U}^{+}/\text{d}y^{+})^{2}$ are shown). $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ is however essentially unaffected by $L_{x}/h$ , whereas $\langle \overline{{\it\varepsilon}}_{11}\rangle /U_{{\it\tau}}^{3}$ (the streamwise component of the dissipation) is affected in a similar manner to $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ . This is closely associated with the energy redistribution where the pressure strain rate and pressure diffusion terms, viz.

and

Figure 1.

Distributions of $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ and $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ in a channel flow as a function of $L_{x}/h$ : (a) $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ ; (b) $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ .

Figure 2.

Distributions of $(\text{d}\bar{U}^{+}/\text{d}y^{+})^{2}$ , ${\it\phi}_{ij}^{+}$ and ${\it\varphi}_{22}^{+}$ in a channel flow for $h^{+}=1020$ : (a) $(\text{d}\bar{U}^{+}/\text{d}y^{+})^{2}$ ; (b) ${\it\phi}_{ij}^{+}$ and ${\it\varphi}_{22}^{+}$ .

play an important role. Distributions of ${\it\phi}_{ij}^{+}$ and ${\it\varphi}_{ij}^{+}$ are shown in figure 2(b) for $h^{+}=1020$ . The data of Lee & Moser (Reference Lee and Moser2015) for $h^{+}=1000$ with $L_{x}=8{\rm\pi}h$ are also plotted. For $L_{x}<6.4h$ , the magnitudes of ${\it\phi}_{11}^{+}$ , ${\it\phi}_{22}^{+}$ and ${\it\phi}_{33}^{+}$ are attenuated significantly due to the impaired global nature of the pressure, i.e. the turbulent kinetic energy is not redistributed properly to $\overline{v^{+}v^{+}}$ and $\overline{w^{+}w^{+}}$ . In particular, the rate of decrease in ${\it\phi}_{11}^{+}$ and ${\it\phi}_{33}^{+}$ with decreasing $L_{x}/h$ is large, which makes $\overline{u^{+}u^{+}}$ and $\overline{w^{+}w^{+}}$ more and less energetic, respectively. This result implies that the inactive motion is affected more significantly by $L_{x}$ than the active motion. Note that the inactive motion refers to the large-scale motion (or structure) and the irrotational pressure fluctuations of the outer region (Townsend Reference Townsend1961; Bradshaw Reference Bradshaw1967) which contribute to the Reynolds normal stress (mostly to the low-wavenumber components of $\overline{u^{+}u^{+}}$ and $\overline{w^{+}w^{+}}$ as described by Townsend’s (Reference Townsend1976) attached eddy model) in the inner region, while the active motion contributes almost exclusively to the Reynolds shear stress (see also § 6 of Antonia et al. (Reference Antonia, Teitel, Kim and Browne1992)) in this region. $\overline{{\it\varepsilon}}_{11}$ (the streamwise component of the dissipation) and $\overline{{\it\varepsilon}}_{33}$ (the spanwise component of the dissipation) become therefore larger and smaller with decreasing $L_{x}/h$ . This leads to an increase in $\langle \overline{{\it\varepsilon}}_{11}\rangle /U_{{\it\tau}}^{3}$ and decrease in $\langle \overline{{\it\varepsilon}}_{33}\rangle /U_{{\it\tau}}^{3}$ (figure 1 b), and hence a small dependence on $L_{x}/h$ of $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ since the sum of $\langle \overline{{\it\varepsilon}}_{11}\rangle /U_{{\it\tau}}^{3}$ , $\langle \overline{{\it\varepsilon}}_{22}\rangle /U_{{\it\tau}}^{3}$ , $\langle \overline{{\it\varepsilon}}_{33}\rangle /U_{{\it\tau}}^{3}$ is equal to $2\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ . This is the reason why the $L_{x}$ effect does not appear in $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ explicitly. The present criterion of $L_{x}=6.4h$ is identical with the finding of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) that correct one-point statistics are obtained when $L_{x}$ is larger than $2{\rm\pi}h$ . This criterion is most likely associated with the streamwise extent of the large-scale structures (Balakumar & Adrian (Reference Balakumar and Adrian2007) reported 2– $3{\it\delta}$ ), but not very large-scale structures (Monty et al. (Reference Monty, Stewart, Williams and Chong2007) reported long meandering features up to $25{\it\delta}$ long in the channel and pipe) since the latter are very long structures and exhibit a shallow angle to the wall so that they may be thought of as the wall-parallel mode (i.e. the zero mode in the energy spectra). The required $L_{x}$ in the DNS is obviously shorter than that in the experiment since the DNS is started with the fully developed turbulent state. In the DNS, once the velocity field reaches the statistically steady state (viz. a linear profile of the total shear stress), the Navier–Stokes equations are integrated further in time to obtain various turbulence statistics.

Figure 3.

Distributions of $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ and $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ in a channel flow as a function of $h^{+}$ : (a) $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ ; (b) $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ .

4.2 Reynolds number dependence

We now quantify the $h^{+}$ dependence of the total energy dissipation rate by considering the DNS data in which $L_{x}\geqslant 2{\rm\pi}h$ . The normalized values of $\langle \overline{{\it\varepsilon}}\rangle$ and $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle$ are plotted in figure 3 (on linear–log scales) in terms of $h^{+}$ using the present and existing DNS data. The viscous energy dissipation rate $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ (figure 3 a) is essentially constant for $h^{+}\geqslant 300$ ; the value of the constant ( $=$ 9.13) is the same as that reported by Laadhari (Reference Laadhari2007), who noted that this constancy applies for $h^{+}\geqslant 500$ , when $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ exceeds $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ . The value of $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ seems to be identical with the values obtained in either a pipe or a boundary layer (see figure 3 a where the data of Wu & Moin (Reference Wu and Moin2008) in a pipe flow and Schlatter & Örlü (Reference Schlatter and Örlü2010) in a zero-pressure-gradient boundary layer are included in terms of $R^{+}$ ( $R$ is the pipe radius) and ${\it\delta}_{99}^{+}$ ( ${\it\delta}_{99}$ is the 99 % boundary layer thickness), respectively). This is consistent with the good collapse in the near-wall distribution of $\bar{U}^{+}$ in terms of $y^{+}$ for these three flows; the inner layer similarity of $\bar{U}^{+}$ leads to the constancy of $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ . Klebanoff (Reference Klebanoff1954) also noted that the viscous energy dissipation rate is negligible above $y^{+}\approx 30$ for the three flows (see also figure 2 a where the distribution of $(\text{d}\bar{U}^{+}/\text{d}y^{+})^{2}$ is shown at $h^{+}=1020$ ). On the other hand, $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ (figure 3 b) increases logarithmically with increasing $h^{+}$ over the same range of $h^{+}$ (viz. $h^{+}\geqslant 300$ ). This increase is well described by

which is obtained by substituting the relation for $U_{b}^{+}$ (4.6) and the constant value ( $=$ 9.13) for $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ into (2.7). Viscosity affects $\langle \overline{{\it\varepsilon}}\rangle$ and $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle$ significantly below $h^{+}=300$ since there is no clear separation between the inner and outer regions.

To ensure the accuracy of $\langle \overline{{\it\varepsilon}}\rangle$ and $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle$ , the ratios

and

should be equal to 1 (see also (2.6) and (2.7)). Several DNS data, including the present data, indicate that the departure from 1 of the ratios ${\it\xi}$ and ${\it\psi}$ is almost negligible (figure 4), thus confirming the accuracy of these simulations. From a historical perspective, it is of interest to note that the earliest attempt to test $\overline{{\it\varepsilon}_{iso\,2}}$ , measured in a turbulent channel flow, was made in Taylor’s (Reference Taylor1935) seminal paper where the isotropic expressions $\overline{{\it\varepsilon}_{iso}}=15{\it\nu}\overline{(\partial u/\partial x)^{2}}$ and $\overline{{\it\varepsilon}_{iso\,2}}=7.5{\it\nu}\overline{(\partial u/\partial y)^{2}}$ were first derived. In essence, Taylor (Reference Taylor1935) was testing (2.6), except that his integration was started at the centre of the channel and the upper limit of the integration did not quite go to the wall (the measurement location closest to the wall was $y/h\approx 0.08$ ). By extrapolating the data to the wall, we estimate that $\langle P_{k}\rangle /\langle \overline{{\it\varepsilon}_{iso2}}\rangle$ could be somewhat greater than 2. This is, perhaps surprisingly, not an implausible outcome since the contribution from $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle$ is likely to be comparable to that of $\langle \overline{{\it\varepsilon}}\rangle$ at this Reynolds number ( $h^{+}\approx 450$ ), $\langle \overline{{\it\varepsilon}_{iso\,2}}\rangle$ is likely to overestimate $\langle \overline{{\it\varepsilon}}\rangle$ (inset of figure 4 a) and the aspect ratio of the channel was only 4. In experiments, data for $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ and $\langle {\it\nu}(\text{d}\bar{U}/\text{d}y)^{2}\rangle /U_{{\it\tau}}^{3}$ are not usually available. A possible formula for $U_{b}^{+}$ can however be established from the experimental data.

Figure 4.

Distributions of ${\it\xi}$ and ${\it\psi}$ in a channel flow as a function of $h^{+}$ : (a) ${\it\xi}$ ; (b) ${\it\psi}$ .

Distributions of $U_{b}^{+}$ in a channel flow for all available DNS data and several sets of experimental data are shown in figures 5(a) and (b), respectively. A least squares fit to the DNS data in figure 5(a) over the range $h^{+}\geqslant 300$ yields

or, equivalently,

This relation represents a good fit to nearly all the experimental data for $U_{b}^{+}$ over more than one decade of $h^{+}$ (see figure 5 b). Also included in figure 5 are relations for $U_{b}^{+}$ , originally established for the skin friction coefficient $C_{f}$ by Dean (Reference Dean1978) and Zanoun et al. (Reference Zanoun, Nagib and Durst2009), which can be written as

and

respectively. Dean’s (Reference Dean1978) relation (4.8) appears to represent the data adequately over almost the complete range of $h^{+}$ . However, the concave curvature exhibited by (4.8) is not discernible in the overall trend of the data and implies that the difference between the logarithmic dependence (4.6) and the power-law relation (4.8) will become more perceptible as $h^{+}$ extends beyond $10^{4}$ .

Figure 5.

Distributions of $U_{b}^{+}$ in a channel flow as a function of $h^{+}$ : (a) DNS; (b) experiment.

Note that among the experimental data included in figure 5(b), $U_{{\it\tau}}$ was most likely underestimated by Comte-Bellot (Reference Comte-Bellot1963) ( $AR=13.3$ , $L=133h$ ) (see also Abe & Antonia Reference Abe and Antonia2011). For the latter data, correction was hence made for $U_{{\it\tau}}$ using (4.7); the resulting values of $U_{b}^{+}$ lie much closer to relation (4.6) than the original values (see figure 5 b). For more recent experiments, special care was taken in determining $U_{{\it\tau}}$ . Zanoun et al. (Reference Zanoun, Durst and Nagib2003) (also Zanoun et al. Reference Zanoun, Nagib and Durst2009) obtained close agreement between ${\it\tau}_{w}$ determined from the static pressure distribution along the fully developed part of the channel (aspect ratio $AR=12$ and length $L=260h$ ) and that estimated using oil film interferometry. Their hot-wire data covered the range $1167<h^{+}<4783$ . Fischer, Jovanović & Durst (Reference Fischer, Jovanović and Durst2001) determined $U_{{\it\tau}}$ from a polynomial fit to the mean velocity in the viscous sublayer and the lower part of the buffer region. Their LDA measurements in a water channel ( $AR=18$ , $L=200h$ ), acquired with good spatial and temporal resolution, extended up to $h^{+}\approx 480$ . Earlier, Durst et al. (Reference Durst, Fischer, Jovanović and Kimura1998) reported results in the same channel over the range $87<h^{+}<293$ . Shah (Reference Shah1988) ( $AR=18.1$ , $L=349h$ ) and Hussain & Reynolds (Reference Hussain and Reynolds1975) ( $AR=16.4$ , $L=460h$ ) conducted measurements with long channels for $h^{+}=187{-}1357$ and 637–1383, respectively, to ensure $\text{d}\bar{P}/\text{d}x=\text{const.}$ and hence accuracy of $U_{{\it\tau}}$ . Bakken et al. (Reference Bakken, Krogstad, Ashrafian and Andersson2005) made measurements in both smooth and rough walls ( $AR=14$ , $L=100h$ ). In their smooth wall case, $h^{+}$ ranges from 360 to 3300; $U_{{\it\tau}}$ was determined from both the mean pressure gradient and the Reynolds shear stress using the total shear stress relation. Monty (Reference Monty2005) obtained $U_{{\it\tau}}$ using the mean pressure gradient in a channel ( $AR=11.7$ , $L=205h$ ) for $h^{+}\approx 380{-}4000$ . Monty’s data for $U_{b}^{+}$ are in reasonable agreement with those of Zanoun et al. (Reference Zanoun, Durst and Nagib2003) (also Zanoun et al. Reference Zanoun, Nagib and Durst2009) when $h^{+}$ exceeds approximately 2000. Schultz & Flack (Reference Schultz and Flack2013) also obtained $U_{{\it\tau}}$ using (1.1) in their LDV measurements ( $AR=8$ , $L=248h$ ) for $h^{+}=1010{-}5900$ . These experimental data indeed provide strong support for relation (4.6) in figure 5(b). The logarithmic variation of $U_{b}^{+}$ , established from DNS data up to $h^{+}=5000$ , can be extrapolated to values of $h^{+}$ that extend to approximately  $10^{4}$ .

It should also be noted that the logarithmic relation (4.6) is a better fit to both the DNS and experimental data than (4.9) for $h^{+}<2000$ , possibly because the present $U_{b}^{+}$ relation (4.6) is based on (2.7), whereas Zanoun et al. (Reference Zanoun, Nagib and Durst2009) obtained the $U_{b}^{+}$ relation (4.9) from the velocity log law (1.4). This difference cannot be dismissed given that, unlike the velocity log law, the skin friction law (4.6) is established unambiguously for $h^{+}>300$ .

4.3 Extension to a pipe flow

Given that the sum of the integrals of the mean and turbulent energy dissipation rates is equal to the non-dimensional energy input from the mean pressure gradient $(U_{b}^{+})$ in internal flows (i.e. (2.7)) and that as in a channel (i.e. (4.3)), $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ exhibits the $2.54\ln (R^{+})$ dependence in a pipe (see § 4.5), it would be straightforward to check if the slope of $U_{b}^{+}$ (viz. $\ln (h^{+})$ versus $h^{+}$ ) obtained in the channel (viz. the slope of (4.6)) also applies to the pipe. In the latter flow, Prandtl (see Schlichting Reference Schlichting1979) established the well-known law for the friction factor ${\it\lambda}(=4C_{f})$ based on Nikuradse’s (Reference Nikuradse1932) data, viz.

This equation can be rewritten as

The present relation, based on the Reynolds number dependence of $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ , is written as

which is close to Prandtl’s relation. Clearly, (4.12) gives a reasonable fit to the DNS and experimental data in the range $R^{+}=300{-}10^{4}$ (see figure 6). The magnitude of $U_{b}^{+}$ is approximately 5 % larger in the channel than in the pipe. On the other hand, while there is reasonable inner layer similarity in the distributions of $\bar{U}^{+}$ , the magnitude of $\bar{U}^{+}$ is larger in the pipe than in the channel in the outer region (see Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009; see also figure 12 where $\overline{{\it\varepsilon}}{\it\delta}/U_{{\it\tau}}^{3}$ ( ${\it\delta}$ denotes either $h$ , $R$ or ${\it\delta}_{99}$ ) shows the same trend as $\bar{U}^{+}$ ). The difference in $U_{b}^{+}$ is thus most likely due to a difference in geometry between these two flows.

Figure 6.

Distributions of $U_{b}^{+}$ in a pipe flow as a function of $R^{+}$ .

It is worthwhile mentioning that the present skin friction relation for the pipe (4.12) is quite close to that obtained by Furuichi et al. (Reference Furuichi, Terao, Wada and Tsuji2015) using LDV measurements in the range $4\times 10^{2}<R^{+}<2.7\times 10^{5}$ , viz.

The corresponding expression for $U_{b}^{+}$ is

Both (4.12) and (4.14) give a reasonable fit to the data for lower Reynolds numbers than those at which the velocity log law is established.

Figure 7.

Distributions of $\overline{U}$ and $\overline{{\it\varepsilon}}$ with the inner and outer scalings: (a) $\overline{U}^{+}$ ; (b)  $\overline{{\it\varepsilon}}{\it\nu}/U_{{\it\tau}}^{4}$ ; (c) $(U_{0}-\overline{U})/U_{{\it\tau}}$ ; (d) $\overline{{\it\varepsilon}}h/U_{{\it\tau}}^{3}$ . Note that $U_{0}$ denotes the centreline mean velocity.

4.4 Scaling laws of $\overline{{\it\varepsilon}}$ and matching argument

Here, we focus on the scaling of $\overline{{\it\varepsilon}}$ in different regions of the channel. There have been a number of observations indicating that the classical inner scaling,

does not hold, the normalized distributions of $\overline{{\it\varepsilon}}^{+}$ in the wall region exhibiting a non-negligible dependence on $h^{+}$ (Antonia, Kim & Browne Reference Antonia, Kim and Browne1991; Hoyas & Jiménez Reference Hoyas and Jiménez2008). On the other hand, there is adequate evidence that the outer scaling,

is satisfied closely in the outer region of the channel (see Hoyas & Jiménez Reference Hoyas and Jiménez2008; Abe & Antonia Reference Abe and Antonia2011). Distributions of $\overline{{\it\varepsilon}}$ with inner and outer scalings are shown in figure 7 together with those of $\overline{U}$ . The magnitude of $\overline{{\it\varepsilon}}^{+}$ increases with $h^{+}$ close to the wall due to the effect of the inactive motion (see Hoyas & Jiménez Reference Hoyas and Jiménez2008), while $\overline{{\it\varepsilon}}^{+}$ seems to collapse for $y^{+}\geqslant 20$ provided $h^{+}\geqslant 300$ (figure 7 b). In contrast to the mean velocity, viscous effects are unlikely to affect the energy dissipation rate significantly for $y^{+}>20$ . This is associated with Bradshaw’s (Reference Bradshaw1967) observation that the only noticeable effect of the inactive motion is an increased dissipation of kinetic energy into heat in the viscous sublayer. For $y^{+}>20$ , $-\overline{uv}=\text{const.}$ is also satisfied reasonably well when $h^{+}$ is large enough (see figure 10 b). On the other hand, unlike $\overline{U}$ , $\overline{{\it\varepsilon}}$ collapses almost perfectly on $U_{{\it\tau}}^{3}$ and $h$ in the region $20/h^{+}<y/h<1$ (figure 7 d; see also figure 9 b where the location $20/h^{+}$ is identified by the dashed vertical lines). In this context, McKeon & Morrison (Reference McKeon and Morrison2007) indicated that the similarity in $\overline{{\it\varepsilon}}$ is obtained for $R^{+}>5000$ in a pipe by analysing the superpipe data, in which estimates for $\overline{{\it\varepsilon}}$ are made by invoking $P_{k}=\overline{{\it\varepsilon}}$ . The present results indicate that the outer layer similarity for $\overline{{\it\varepsilon}}$ is more convincing than that for $\overline{U}$ even at small $h^{+}$ ( $h^{+}\geqslant 300$ ) (see figure 7 c,d).

We now apply a matching argument to $\overline{{\it\varepsilon}}$ . Here, we assume that $h^{+}$ is large enough to have a clear distinction between inner and outer regions, and that there is a region where relations (4.15) and (4.16) overlap so that the gradient of $\overline{{\it\varepsilon}}$ should coincide, viz.

where $y^{\ast }\equiv y/h$ . After multiplying by $y^{2}$ , the equality between the second and third members of (4.17) becomes

This is satisfied if

Equation (4.18) indicates that $\overline{{\it\varepsilon}}$ should indeed scale with $U_{{\it\tau}}^{3}$ and $y$ in the overlap region. After integrating (4.19a,b ), we obtain

Here, we adopt a small parameter ${\it\gamma}=1/h^{+}$ and an outer variable $y^{\ast }={\it\gamma}y^{+}$ as was done by Afzal (Reference Afzal1976) for the mean velocity gradient and obtain $D_{1}=-{\it\gamma}c$ and $D_{2}=-c$ . Equation (4.20a,b ) is then rewritten as

where $c$ is a constant. After the normalization, it follows from (4.21a,b ) that the overlap scaling may be written as

and

in inner and outer coordinates, respectively, where $D=-1/{\it\kappa}_{{\it\varepsilon}}$ and ${\it\kappa}_{{\it\varepsilon}}$ is a constant. The matching argument highlights that the overlap scaling of $\overline{{\it\varepsilon}}$ requires neither the existence of a velocity log law nor energy equilibrium $(P_{k}=\overline{{\it\varepsilon}})$ . It does however require the Reynolds number to be large enough ( $h^{+}\approx 300$ ) to allow the overlap region, where the relevant length scale is the distance from the wall, $y$ to be distinguished unambiguously. In the overlap region, when $-\overline{uv}\simeq \text{const.}$ (this constancy is plausible only when $h^{+}$ is large enough) (see figure 10 b), $P_{k}y/U_{{\it\tau}}^{3}\approx y\text{d}\bar{U}^{+}/\text{d}y$ (see figures 10 c and 11 a) and $P_{k}/\overline{{\it\varepsilon}}\approx 1$ (see figure 10 a). Equations (4.22) and (4.23) are analogous to the generalized velocity log law for the channel flow proposed by Jiménez & Moser (Reference Jiménez and Moser2007) using the asymptotic matching of the mean velocity gradient,

and

in inner and outer coordinates, respectively, with $1/{\it\kappa}=2.49$ , ${\it\alpha}=1.0$ and ${\it\beta}=150$ . While both the present study and that of Jiménez & Moser (Reference Jiménez and Moser2007) use a matching argument, there are discernible differences between (4.22) and (4.23) and (4.24) and (4.25) in terms of the constants (i.e. ${\it\kappa}_{{\it\varepsilon}}$ and ${\it\kappa}$ ) and in the limits of applicability of the overlap expressions (i.e. $30/h^{+}\leqslant y/h\leqslant 0.2$ in (4.22) and (4.23) (this is discussed below) and $300/h^{+}<y/h<0.45$ in (4.24) and (4.25)).

Figure 8.

Distributions of $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ in a channel, pipe and boundary layer: (a) channel (in terms of $y^{+}$ ); (b) boundary layer (in terms of $y^{+}$ ); (c) channel (in terms of $y/h$ ); (d) pipe (in terms of $y/R$ ). In (c), line patterns are the same as in (a).

In (4.22) and (4.23), the second terms on the right-hand sides are responsible for the finite Reynolds number effect, i.e. $-c(y^{+}/h^{+})$ (the second term of (4.22)) goes to zero as $h^{+}\rightarrow \infty$ , while $-c(y/h)$ (the second term of (4.23)) does not depend on $h^{+}$ but may enhance the outer limit of the overlap scaling. When the finite Reynolds number effect disappears, equations (4.22) and (4.23) reduce to $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}=1/{\it\kappa}_{{\it\varepsilon}}$ analogous to the classical scaling based on the velocity log law $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}=1/{\it\kappa}$ (see, for example, Townsend Reference Townsend1976). Note that ${\it\kappa}_{{\it\varepsilon}}$ is identical with ${\it\kappa}$ only if the velocity log law and energy equilibrium conditions are satisfied. Hoyas & Jiménez (Reference Hoyas and Jiménez2008) found that the magnitudes of all the terms in the budgets of $\overline{u_{i}u_{j}}$ and $k$ in the channel decrease almost proportionally to $y^{-1}$ in the region $y^{+}=50$ to $y/h=0.4$ for $h^{+}=2003$ . Figure 8(a) highlights that the largest magnitude of $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ ( $=$ 2.54) is attained at $y^{+}=30{-}40$ . This value is the same for the channel, pipe and boundary layer due to the wall similarity of the constant stress region (Townsend Reference Townsend1976). It depends on the Reynolds number in the channel and pipe, but not in the boundary layer (see figure 8). This finite Reynolds number effect comes from the $\text{d}\overline{P}/\text{d}x$ effect, which reduces the magnitude of $-\overline{uv}$ and hence $P_{k}$ and $\overline{{\it\varepsilon}}$ when $h^{+}$ is small. On the other hand, this effect is negligible when $h^{+}\approx 5000$ , which allows us to determine $1/{\it\kappa}_{{\it\varepsilon}}$ unambiguously. Indeed, $1/{\it\kappa}_{{\it\varepsilon}}=2.54$ is identified asymptotically for $h^{+}\geqslant 5000$ in the channel (figure 8 a), while this value is obtained for ${\it\delta}_{99}^{+}\geqslant 300$ in the zero-pressure-gradient boundary layer due to the absence of the $\text{d}\overline{P}/\text{d}x$ effect (figure 8 b). A fit to the DNS data over $30/{\it\delta}^{+}\leqslant y/{\it\delta}\leqslant 0.2$ then yields $c=2.6$ , 1.3 and 0 in the channel, pipe and boundary layer. Indeed, equations (4.22) and (4.23) reproduce the $h^{+}$ dependence of the peak magnitude of $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ successfully for $h^{+}\geqslant 300$ and provide a good fit to the data in the region $30/h^{+}\leqslant y/h\leqslant 0.2$ (see figure 8 c). While the classical overlap argument based on $\bar{U}$ (Millikan Reference Millikan1938) strictly holds only at large $h^{+}$ , the overlap region for $\overline{{\it\varepsilon}}$ is established at small $h^{+}$ ( ${\approx}$ 300) (see also figure 8 c in which the DNS data for $h^{+}=395$ follow the present overlap scaling very well).

Figure 9.

Distributions of $\overline{{\it\varepsilon}}{\it\nu}/U_{{\it\tau}}^{4}$ and $\overline{{\it\varepsilon}}h/U_{{\it\tau}}^{3}$ in a channel flow with log–log coordinates: (a) $\overline{{\it\varepsilon}}{\it\nu}/U_{{\it\tau}}^{4}$ ; (b) $\overline{{\it\varepsilon}}h/U_{{\it\tau}}^{3}$ . – ⋅ – ⋅ –, $h^{+}=180$ ; $\cdots \cdots$ , $h^{+}=395$ ; – – – –, $h^{+}=640$ ; ——, $h^{+}=1020$ ; - ⋅ - ⋅ - ⋅ -, $h^{+}=1440$ (Hu et al. Reference Hu, Morfey and Sandham2006); — ⋅ ⋅ — ⋅ ⋅ —, $h^{+}=2003$ (Hoyas & Jiménez Reference Hoyas and Jiménez2008); — —, $h^{+}=4079$ (Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014); - - - - - -, $h^{+}=5186$ (Lee & Moser Reference Lee and Moser2015);

, (4.26)–(4.27).

Figure 10.

Distributions of $P_{k}/\overline{{\it\varepsilon}}$ , $-\overline{u^{+}v^{+}}$ , ${\it\tau}^{+}$ and $y^{+}(\text{d}\bar{U}^{+}/\text{d}y^{+})$ in a channel flow: (a)  $P_{k}/\overline{{\it\varepsilon}}$ ; (b) $-\overline{u^{+}v^{+}}$ and ${\it\tau}^{+}$ ; (c) $y^{+}(\text{d}\bar{U}^{+}/\text{d}y^{+})$ . In (b), line patterns are the same as in (a).

In the outer region ( $y/h>0.2$ ), there is still a decrease in $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ towards the centreline (see figure 11 b). After the fit to the DNS data in figure 7(d), the decrease is approximated closely by the outer scaling, viz.

with $d=1.7$ . Figure 9(b) underlines that (4.26) is a good fit in the region $20/h^{+}<y/h<1$ (see the thick lines in figure 9 b). Equation (4.26) also applies to the pipe reasonably well albeit with a different value of $d$ . In the boundary layer, it applies only below $y/{\it\delta}_{99}\approx 0.4$ , with yet another value of $d$ (see figure 12 where $d=0.9$ and $-1.0$ for the pipe and boundary layer, respectively). The different magnitudes of $d$ are most likely associated with the different flow characteristics (e.g. effects of the mean pressure gradient in the channel and pipe, curvature in the pipe and the presence of the turbulent/non-turbulent interface in the boundary layer). The present results underline that, as for the mean velocity, there is a discernible departure from similarity in the energy dissipation rate for the channel, pipe and boundary layer.

The excellent agreement with (4.26) in both the channel and pipe is due to the $\text{d}\bar{P}/\text{d}x$ effect. This latter effect seems to be responsible for yielding nearly the same slope (i.e. 2.45) as in the overlap scaling (i.e. $1/{\it\kappa}_{{\it\varepsilon}}=2.54$ ) so that a reasonable collapse on the outer scaling is obtained over a wider range $20/h^{+}<y/h<1$ (see figure 9 b) than for the velocity log law, usually between $y^{+}\simeq 30$ and $y/h=0.2$ . A nearly identical observation was made for $\bar{U}$ by Monty et al. (Reference Monty, Stewart, Williams and Chong2007) who noted that the pipe and channel mean velocity profiles lie closer to the log law than in the boundary layer beyond $y/{\it\delta}=0.15$ . In the region $20<y^{+}<h^{+}$ , there is satisfactory collapse on inner scaling with the same slope (i.e. 2.45) as for outer scaling, viz.

(see figure 9 a). Note that since it seems difficult to make a clear distinction between the overlap and outer regions of $\overline{{\it\varepsilon}}$ in the channel (see figure 9 where the inner and outer scalings (4.26), (4.27) hold reasonably well in the region $20/h^{+}<y/h<1$ ), we assume that the upper bound of the overlap region of $\overline{{\it\varepsilon}}$ is $y/h=0.2$ , as is normally assumed for the velocity log law. In the present work, the overlap region of $\overline{{\it\varepsilon}}$ is thus defined by the region $30/h^{+}\leqslant y/{\it\delta}\leqslant 0.2$ , in which (4.22) and (4.23) hold quite well.

Figure 11.

Distributions of $P_{k}y/U_{{\it\tau}}^{3}$ and $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ in a channel, pipe and boundary layer at ${\it\delta}^{+}\approx 1000$ : (a) $P_{k}y/U_{{\it\tau}}^{3}$ ; (b) $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ .

Figure 12.

Distributions of $\overline{{\it\varepsilon}}{\it\delta}/U_{{\it\tau}}^{3}$ in a channel, pipe and boundary layer at ${\it\delta}^{+}\approx 1000$ .

4.5 Fractional contributions to $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$

It is of importance to clarify if the integral of $\overline{{\it\varepsilon}}$ over the overlap region yields the logarithmic $h^{+}$ dependence of $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ . In the present study, we follow the same approach as in Sreenivasan (Reference Sreenivasan, Deshpande, Prabhu, Sreenivasan and Viswanath1995), viz.

where the limits for the second integral in (4.28) correspond to the extent of the overlap region of $\overline{{\it\varepsilon}}$ , i.e. $y^{+}\simeq 30$ and $y/h=0.2$ . Values of $C_{i}$ , $C_{log}$ and $C_{o}$ obtained from the DNS data in the channel are shown in figure 13 and are compared with the corresponding values in the pipe and boundary layer. Clearly, there is a $\ln (h^{+})$ dependence for $C_{log}$ (figure 13 b). This dependence is obtained by integrating (4.22) or (4.23), viz.

in which the last term of (4.29), the finite Reynolds number effect or the $\text{d}\bar{P}/\text{d}x$ effect, cannot be dismissed when $h^{+}$ is small. On the other hand, the last term of (4.29) is negligible in the zero-pressure-gradient boundary layer due to the absence of the $\text{d}\bar{P}/\text{d}x$ effect (see figure 13 b). Given that $C_{o}$ is essentially constant $({\sim}1/{\it\kappa}_{{\it\varepsilon}})$ (figure 13 c) but the magnitude of $C_{i}$ increases slowly with $h^{+}$ (figure 13 a), we integrate $\overline{{\it\varepsilon}}$ from $y=0$ to $0.2h$ (viz. $C_{i}+C_{log}$ ) (figure 13 b). The resulting integral is described adequately by

for $h^{+}\geqslant 300$ with $C_{2}=9.28$ . Note that the sum of (4.30) and $C_{o}$ ( ${\approx}$ 2.6) is identical to (4.3). This implies that the more appropriate expression for the logarithmic dependence of $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ for the channel requires integration from $y=0$ to $0.2h$ , viz. the contribution of $C_{i}$ ( $h^{+}$ ) cannot be ignored. The same trend is observed for the pipe. The slightly different magnitude of the integral from $y=0$ to $0.2{\it\delta}$ (viz. $C_{i}+C_{log}$ ) between the three flows is attributed to the finite Reynolds number effect or the $\text{d}\bar{P}/\text{d}x$ effect, represented well by the terms $-c(y^{+}/h^{+})$ and $-c(y/h)$ in (4.22) and (4.23). This effect however does not affect the $\ln ({\it\delta}^{+})$ dependence of $C_{i}+C_{log}$ (see figure 13 b where the $2.54\ln ({\it\delta}^{+})$ dependence is valid for the three flows when ${\it\delta}^{+}\geqslant 300$ ). The present results highlight that the slope of 2.54 in (4.30) can be identified with $1/{\it\kappa}_{{\it\varepsilon}}$ as inferred from the overlap scaling of $\overline{{\it\varepsilon}}$ , and that $1/{\it\kappa}_{{\it\varepsilon}}$ is identical with the slope for the $\ln (h^{+})$ dependence of $U_{b}^{+}$ .

Figure 13.

Distributions of piecewise contributions to $\langle \overline{{\it\varepsilon}}\rangle /U_{{\it\tau}}^{3}$ in a channel, pipe and boundary layer as a function of ${\it\delta}^{+}$ : (a) from $y^{+}=0$ to 30 $(C_{i})$ ; (b) from $y^{+}=30$ to $y/{\it\delta}=0.2$ $(C_{log})$ and from $y/{\it\delta}=0$ to 0.2 $(C_{i}+C_{log})$ ; (c) from $y/{\it\delta}=0.2$ to 1 $(C_{o})$ .

The Reynolds number dependence of $C_{i}$ is more significant for the channel/pipe than for the boundary layer (figure 13 a) owing to the finite Reynolds number effect (viz. the $\text{d}\bar{P}/\text{d}x$ effect) in the former two flows. Schlatter & Örlü (Reference Schlatter and Örlü2010) observed a difference in ${\it\tau}_{1\,rms}^{+}=(\partial u^{+\prime }/\partial y^{+})_{w}$ , ${\it\tau}_{1\,rms}^{+2}$ being one of the major contributors to $\overline{{\it\varepsilon}}^{+}$ at the wall, between the channel and boundary layer at low Reynolds numbers and attributed it to the presence of $\text{d}\bar{P}/\text{d}x$ in the channel. On the other hand, $C_{o}$ differs significantly between the three flows (figure 13 c) due to the different external flow characteristics (see figure 12 which shows distributions of $\overline{{\it\varepsilon}}{\it\delta}/U_{{\it\tau}}^{3}$ in the three flows). The constancy of $C_{o}$ for the channel and pipe is associated with the excellent outer scaling we observe for these two flows, viz. integrating (4.26) from $y=0.2{\it\delta}$ to ${\it\delta}$ yields 2.58 and 1.53 for the channel and pipe, respectively, these two values being reasonable estimates of $C_{o}$ (see figure 13 c).

The dependence on ${\it\delta}^{+}$ of $C_{i}$ tends to become small in all three flows as the Reynolds number increases (see figure 13 a), implying a decreasing contribution of $C_{i}$ to the $2.54\ln (h^{+})$ dependence as $h^{+}$ increases. When $h^{+}\rightarrow \infty$ , the overlap region should contribute exclusively to the $2.54\ln (h^{+})$ dependence of the integrated turbulent energy dissipation rate. The present logarithmic $h^{+}$ dependence of $U_{b}^{+}$ is essentially linked to the excellent overlap region we observe for $\overline{{\it\varepsilon}}$ even at small $h^{+}$ . Note that ${\it\kappa}_{{\it\varepsilon}}=0.39$ defined in (4.22) and (4.23) is not identical with ${\it\kappa}$ obtained from the velocity log law (1.6) for the Reynolds numbers examined (see figures 8 a and 10 c). This is because the constancy of $1/{\it\kappa}\equiv y^{+}(\text{d}\bar{U}^{+}/\text{d}y^{+})$ is apparent only above $h^{+}=5000$ due to the non-negligible viscous effect on $\bar{U}$ (figure 10 c). Also, there is a discernible departure from unity of $P_{k}/\overline{{\it\varepsilon}}$ in the region normally referred to as the velocity log law (figure 10 a). The universality of ${\it\kappa}$ obtained from the velocity log law has been questioned (see the review of Smits, McKeon & Marusic (Reference Smits, McKeon and Marusic2011)); its magnitude increases slowly with increasing $h^{+}$ in the channel ( ${\it\kappa}=0.37$ at $h^{+}=2000$ by Zanoun et al. (Reference Zanoun, Durst and Nagib2003); ${\it\kappa}=0.38$ at $h^{+}=5200$ by Lee & Moser (Reference Lee and Moser2015); see also figure 10 c). Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) also reported that experiments support a universal logarithmic behaviour of the mean velocity and streamwise turbulence intensities in the region $3{\it\delta}^{+^{1/2}}<y^{+}<0.15{\it\delta}^{+}$ of the laboratory boundary layer, pipe and atmospheric surface layer with ${\it\kappa}=0.39$ at extremely large Reynolds number $(2\times 10^{4}<{\it\delta}^{+}<6\times 10^{5})$ . The present value of ${\it\kappa}_{{\it\varepsilon}}$ may be reconcilable with the value of ${\it\kappa}$ (obtained at very large $h^{+}$ ) if one recognizes that the outer layer similarity of $\overline{{\it\varepsilon}}$ is established at a much smaller $h^{+}$ than for $\bar{U}^{+}$ . Indeed, this appears to be adequately supported by the available DNS data (see figure 7).