2.1 Homogeneous dilations in the log layer

Substituting (2.2) into (2.1a )–(2.1e ) and requiring the same proportional coefficients for all the terms in the equations (needed to obtain invariant transformed equations) yield the following relations for the correlations of random dilation variables $\{\unicode[STIX]{x1D706}_{i}\}$ :

(2.4a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D706}_{U}/q_{2}^{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}/q_{2}=1/\unicode[STIX]{x1D706}_{R}, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}\unicode[STIX]{x1D706}_{U}/q_{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}/q_{1}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}/q_{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}/q_{1}^{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}/q_{2}^{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}/q_{3}^{2}, & \displaystyle\end{eqnarray}$$

(2.4c ) $$\begin{eqnarray}\displaystyle & \overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}/q_{2}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}/q_{2}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}/q_{1}^{2}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}/q_{2}^{2}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}/q_{3}^{2}, & \displaystyle\end{eqnarray}$$

(2.4d ) $$\begin{eqnarray}\displaystyle & \overline{\unicode[STIX]{x1D706}_{3}\unicode[STIX]{x1D706}_{p}}/q_{3}=\overline{\unicode[STIX]{x1D706}_{3}\unicode[STIX]{x1D706}_{3}\unicode[STIX]{x1D706}_{2}}/q_{2}=\overline{\unicode[STIX]{x1D706}_{3}\unicode[STIX]{x1D706}_{3}}/q_{1}^{2}=\overline{\unicode[STIX]{x1D706}_{3}\unicode[STIX]{x1D706}_{3}}/q_{2}^{2}=\overline{\unicode[STIX]{x1D706}_{3}\unicode[STIX]{x1D706}_{3}}/q_{3}^{2}, & \displaystyle\end{eqnarray}$$

(2.4e ) $$\begin{eqnarray}\displaystyle & \overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}\unicode[STIX]{x1D706}_{U}/q_{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}/q_{2}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}/q_{1}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}/q_{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}/q_{1}^{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}/q_{2}^{2}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}/q_{3}^{2}.\qquad & \displaystyle\end{eqnarray}$$

$q_{1}=q_{2}=q_{3}=q$

(2.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{U}=1/q,\quad \unicode[STIX]{x1D706}_{R}=q^{3},\end{eqnarray}$$

(2.5c ) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=1/q^{2},\end{eqnarray}$$

(2.5d ) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=1/q^{3}.\end{eqnarray}$$

Accordingly, the Reynolds stresses $W_{ij}^{+}$ are dilated by a factor of $\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j}}=1/q^{2}$ (for $i,j,=1,2$ ), and the mean shear $S$ by a factor of $\unicode[STIX]{x1D706}_{U}/q=1/q^{2}$ , so that the length functions, according to definitions in (2.3a,b ), are dilated as

(2.6) $$\begin{eqnarray}\ell _{ij}^{+\ast }=\sqrt{\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j}}}/(\unicode[STIX]{x1D706}_{U}/q)\ell _{ij}^{+}=q\ell _{ij}^{+}.\end{eqnarray}$$

In this case, the dilation invariants consisting of $y^{+}$ and $\ell _{ij}^{+}$ are

(2.7) $$\begin{eqnarray}I_{ij}^{log}=\ell _{ij}^{+}/y^{+}.\end{eqnarray}$$

Following Part 1, we assume constant dilation invariants (i.e. $I_{ij}=\text{const.}$ ) to obtain possible (or candidate) solutions of the balance equations, i.e.

(2.8) $$\begin{eqnarray}\ell _{ij}^{+}=I_{ij}^{log}y^{+},\quad I_{ij}^{log}=\text{const.}\end{eqnarray}$$

This is the standard (canonical) case where no direction is preferred, and all the lengths $\ell _{ij}$ scale as the wall distance ( $y$ ) – also a simple consequence of dimensional analysis.

In particular, the well-known $W_{12}^{+}\approx 1$ gives rise to $S^{+}\approx 1/(I_{12}^{log}y^{+})$ and hence the mean velocity $U^{+}\approx (1/I_{12}^{log})\log y^{+}+B$ ; this shows that the Kármán constant is just one of our dilation invariants: $I_{12}^{log}$ . Note that the dilation invariance of $W_{12}^{+}$ actually is broken since $W_{12}^{+}\approx 1$ against $W_{12}^{+\ast }=W_{12}^{+}/q^{2}$ (unless $q=1$ , which would indicate no dilation), but the length function $\ell _{12}$ still holds its dilation invariance: $\ell _{12}^{+\ast }=q\ell _{12}^{+}$ (this is why $\ell _{ij}$ , not $W_{ij}$ , is chosen as the similarity variable). This is also found in other layers that we study below.

More interestingly, from (2.8) and (2.3c,d ), $W_{ii}^{+}=W_{12}^{+}(I_{ii}^{log}/I_{12}^{log})^{2}$ , indicating that all the Reynolds normal stresses are constants in the log layer (since $W_{12}^{+}\approx 1$ ). These constants are parts of the so-called ‘inertial-range’ description, widely used in engineering models (Wilcox Reference Wilcox2006). The present analysis shows that they arise from a dilation symmetry homogeneous in all directions (when all terms in the balance equations are under the same dilation). Note that the constancy of $W_{11}^{+}$ and $W_{33}^{+}$ are against Townsend’s log law of inactive motions in (1.3) and (1.4), where more discussions on this issue will be presented later.

Also note that the above analysis shows that dilations of $W_{11}$ , $W_{22}$ and $W_{12}$ are linked, because of the production and pressure–strain terms in the corresponding equations. In contrast, dilation on $W_{33}$ seems to be independent of those on $W_{11}$ or $W_{22}$ (hence is free), since no term in the spanwise kinetic energy equation is coupled with other equations. Here, invoking the isotropic turbulence assumption, $\ell _{33}$ satisfies the same dilation as $\ell _{11}$ or $\ell _{22}$ ; this argument also applies to the core layer. However, near the wall, the isotropic condition is invalid, and dilations on these lengths should be different. In fact, the homogeneous dilation symmetry is partially broken near the wall since some terms in the balance equations are unimportant. We explore such possibilities below. The analysis will first treat $\ell _{ij}$ (for $i,j=1,2$ ) and then focus on  $\ell _{33}$ .

2.2 Directional dilations in the viscous sublayer

Near the wall, compared to $\unicode[STIX]{x2202}_{y}^{2}$ , the $\unicode[STIX]{x2202}_{x}^{2}$ and $\unicode[STIX]{x2202}_{z}^{2}$ terms can be ignored (i.e. the boundary layer approximation). This enables us to drop the $\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j}}/q_{1}^{2}$ and $\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{j}}/q_{3}^{2}$ terms in (2.4), and hence all $q_{i}$ are not needed to be equal. In this case, let us choose $\unicode[STIX]{x1D706}_{U}$ and $q_{2}$ as two independent dilation parameters, then all the other dilation parameters are determined from the remnant of (2.4):

(2.9a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{R}=q_{2}^{2}/\unicode[STIX]{x1D706}_{U},\quad \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\unicode[STIX]{x1D706}_{U}/q_{2},\quad \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}=\unicode[STIX]{x1D706}_{U}^{2},\quad \overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=1/q_{2}^{2},\quad q_{1}=1/\unicode[STIX]{x1D706}_{U}. & & \displaystyle\end{eqnarray}$$

(2.9b ) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=\unicode[STIX]{x1D706}_{U}/q_{2}^{2},\quad \overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=1/q_{2}^{3},\quad \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\unicode[STIX]{x1D706}_{U}^{2}/q_{2}.\end{eqnarray}$$

(2.10a-c ) $$\begin{eqnarray}\ell _{11}^{+\ast }=q_{2}\ell _{11}^{+},\quad \ell _{12}^{+\ast }=\sqrt{q_{2}/\unicode[STIX]{x1D706}_{U}}\ell _{12}^{+},\quad \ell _{22}^{+\ast }=(1/\unicode[STIX]{x1D706}_{U})\ell _{22}^{+}.\end{eqnarray}$$

Since $\ell _{12}^{+\ast }=q_{2}^{3/2}\ell _{12}^{+}$ was obtained in the viscous sublayer due to the scaling $\ell _{12}^{+}\propto \,y^{+3/2}$ (Part 1), we thus have $\unicode[STIX]{x1D706}_{U}=q_{2}^{-2}$ from the second equation of (2.10), and hence $\ell _{22}^{+\ast }=q_{2}^{2}\ell _{22}^{+}$ . Then, the corresponding dilation invariants are

(2.11a-c ) $$\begin{eqnarray}I_{11}^{sub}=\ell _{11}^{+}/y^{+},\quad I_{12}^{sub}=\ell _{12}^{+}/y^{+3/2},\quad I_{22}^{sub}=\ell _{22}^{+}/y^{+2}.\end{eqnarray}$$

Following the first invariant ansatz (i.e. assuming constant invariants) in Part 1, local power laws in the viscous sublayer are obtained as

(2.12a-c ) $$\begin{eqnarray}\ell _{11}^{+}=I_{11}^{sub}y^{+},\quad \ell _{12}^{+}=I_{12}^{sub}y^{+3/2},\quad \ell _{22}^{+}=I_{22}^{sub}y^{+2}.\end{eqnarray}$$

This shows that by random dilation of approximate balance equations, we can obtain different power laws for the length functions, in contrast to the uniform linear scaling (2.8) for all the lengths in log layer.

2.3 Directional dilations in the buffer layer

The dominant balancing mechanisms in the buffer layer are different from those in the viscous sublayer. While (2.1a ), (2.1c ) and (2.1d ) remain the same as in the viscous sublayer (i.e. ignoring $\unicode[STIX]{x2202}_{x}^{2}$ and $\unicode[STIX]{x2202}_{z}^{2}$ in dissipations), both (2.1e ) and (2.1b ) are different. For the Reynolds shear stress equation (2.1e ), diffusion and dissipation are very small (Pope Reference Pope2000), and hence ignored. For the streamwise stress equation (2.1b ), a special consideration is given below.

Recall that in Part 1, we defined an eddy shear length $\ell _{\unicode[STIX]{x1D708}}^{+}$ :

(2.13) $$\begin{eqnarray}\ell _{\unicode[STIX]{x1D708}}^{+}=(W_{12}^{+}/S^{+})^{3/4}/\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D708}}^{1/4},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D708}}=(\overline{|\unicode[STIX]{x1D735}u^{\prime }|^{+}}^{2}-\unicode[STIX]{x2202}^{2}\overline{u^{\prime }u^{\prime }}/\unicode[STIX]{x2202}y^{+2})$ is the net viscous effect (a sum of viscous diffusion and dissipation) and $\unicode[STIX]{x1D708}_{T}^{+}=W_{12}^{+}/S^{+}$ is the eddy viscosity. In the viscous sublayer, $W_{12}^{+}/S^{+}\propto y^{+3}$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D708}}\propto y^{+}$ (since the leading order expansions of $\overline{|\unicode[STIX]{x1D735}u^{\prime }|^{+}}^{2}$ and $\unicode[STIX]{x2202}^{2}(\overline{u^{\prime }u^{\prime }})/\unicode[STIX]{x2202}y^{+2}$ – which are non-zero values – exactly balance each other (see figure 1 a), leaving the first expansion term as a order of $y^{+}$ ). Thus, the expansion for the eddy shear length is $\ell _{\unicode[STIX]{x1D708}}^{+}\approx c_{\unicode[STIX]{x1D708}}y^{+2}+h.o.t.$ (where $h.o.t.$ denotes higher-order terms.) Here, using the same buffer layer approximation $\ell _{\unicode[STIX]{x1D708}}^{+}\approx c_{\unicode[STIX]{x1D708}}y^{+2}$ as in Part 1, the net viscous effect is approximated as

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D708}}=\frac{W_{12}^{+3}}{S^{+3}\ell _{\unicode[STIX]{x1D708}}^{+4}}=\frac{W_{12}^{+3}}{S^{+3}(c_{\unicode[STIX]{x1D708}}y^{+2}+h.o.t.)^{4}}\approx \frac{W_{12}^{+3}}{S^{+3}(c_{\unicode[STIX]{x1D708}}y^{+2})^{4}}.\end{eqnarray}$$

Substituting (2.14) into (2.1b ) yields

(2.15) $$\begin{eqnarray}0\approx S^{+}W_{12}^{+}-\overline{u^{\prime }\unicode[STIX]{x2202}_{x}p^{\prime }}^{+}-\unicode[STIX]{x2202}_{y^{+}}\left(\frac{1}{2}\overline{u^{\prime }u^{\prime }v^{\prime }}^{+}\right)-\frac{W_{12}^{+3}}{S^{+3}(c_{\unicode[STIX]{x1D708}}y^{+2})^{4}},\end{eqnarray}$$

which will replace (2.1b ) to be the approximate streamwise stress equation for the dilation analysis (explained later).

Applying (2.2) to the newly obtained approximated equations yields the following constraints on the correlations of dilation parameters:

(2.16a ) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{R}=q_{2}^{2}/\unicode[STIX]{x1D706}_{U},\quad \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\unicode[STIX]{x1D706}_{U}/q_{2},\quad \unicode[STIX]{x1D706}_{U}=1/q_{2}^{3},\quad q_{1}=q_{2}^{2}\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}/\unicode[STIX]{x1D706}_{U}.\end{eqnarray}$$

(2.16b ) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=\unicode[STIX]{x1D706}_{U}\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}},\quad \overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=q_{2}(\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}})^{2},\quad \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\unicode[STIX]{x1D706}_{U}^{2}/q_{2}.\end{eqnarray}$$

(2.17a-c ) $$\begin{eqnarray}\ell _{11}^{+\ast }=q_{11}\ell _{11}^{+},\quad \ell _{12}^{+\ast }=q_{2}^{2}\ell _{12}^{+},\quad \ell _{22}^{+\ast }=q_{22}\ell _{22}^{+},\end{eqnarray}$$

where $q_{11}=q_{2}\sqrt{\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}}/\unicode[STIX]{x1D706}_{U}$ and $q_{22}=q_{2}\sqrt{\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}}/\unicode[STIX]{x1D706}_{U}$ are two free parameters, since $\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}$ and $\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}$ are free.

Under the constant dilation ansatz, equation (2.17) further leads to

(2.18a-c ) $$\begin{eqnarray}\ell _{11}^{+}=I_{11}^{buf}y^{+\unicode[STIX]{x1D6FD}_{11}},\quad \ell _{12}^{+}=I_{12}^{buf}y^{+2},\quad \ell _{22}^{+}=I_{22}^{buf}y^{+\unicode[STIX]{x1D6FD}_{22}},\end{eqnarray}$$

where $q_{11}=q_{2}^{\unicode[STIX]{x1D6FD}_{11}}$ and $q_{22}=q_{2}^{\unicode[STIX]{x1D6FD}_{22}}$ are substituted in. Note that $\ell _{12}^{+}=I_{12}^{buf}y^{+2}$ in the second equation of (2.18) has been obtained in Part 1. Further arguments are invoked to determine the values of $\unicode[STIX]{x1D6FD}_{11}$ and $\unicode[STIX]{x1D6FD}_{22}$ . Consider the following ratios:

(2.19a,b ) $$\begin{eqnarray}\ell _{11}^{+}/\ell _{12}^{+}=(I_{11}^{buf}/I_{12}^{buf})y^{+(\unicode[STIX]{x1D6FD}_{11}-2)},\quad \ell _{22}^{+}/\ell _{12}^{+}=(I_{22}^{buf}/I_{12}^{buf})y^{+(\unicode[STIX]{x1D6FD}_{22}-2)}.\end{eqnarray}$$

From (2.12), $\ell _{11}^{+}/\ell _{12}^{+}\propto 1/y^{+1/2}$ and $\ell _{22}^{+}/\ell _{12}^{+}\propto y^{+1/2}$ in the viscous sublayer. We assume that the two ratios representing the geometrical feature (e.g. a two-dimensional shape in the $x{-}y$ plane) of near-wall eddies remain invariant from the sublayer to the buffer layer – so that $\unicode[STIX]{x1D6FD}_{11}=3/2$ and $\unicode[STIX]{x1D6FD}_{22}=5/2$ in (2.19). The mechanism for these invariant scalings is not clear yet, and is to be studied in future.

Note that the free parameter $q_{11}$ originates from the approximation (2.15). Without this approximation, repeating the same analysis would yield $\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}=\unicode[STIX]{x1D706}_{U}^{2}$ , $q_{11}=q_{2}$ , and hence $\ell _{11}^{+}\propto y^{+}$ – the same scaling as in the viscous sublayer (2.12). However, this contradicts the plot in figure 1(b), where $\ell _{11}^{+}$ shows a distinct scaling transition from the viscous sublayer to the buffer layer. Thus, the approximation (2.14) (hence (2.15)) is necessary.

2.4 Homogeneous dilations in the core layer

Similar to Part 1, we change the independent variable from the wall distance $y^{+}$ to the distance from the centreline $r=1-y^{+}/Re_{\unicode[STIX]{x1D70F}}$ , with the corresponding outer scales in the other two directions, namely, $\tilde{x}=x^{+}/Re_{\unicode[STIX]{x1D70F}}$ and $\tilde{z}=z^{+}/Re_{\unicode[STIX]{x1D70F}}$ . Also, we introduce a velocity defect $U_{d}^{+}=U_{c}^{+}-U^{+}$ (subscript $c$ indicates centreline value) to replace the mean velocity in the balance equations, so that the centreline condition $U_{d}^{+}=0$ – in analogy to $U^{+}=0$ at the wall – remains unchanged under dilation. Note that if one applies (2.2) (also changing $U^{+}$ to $U_{d}^{+}$ , $x^{+}$ to $\tilde{x}$ , $y^{+}$ to $r$ and $z^{+}$ to $\tilde{z}$ ) to the balance equations scaled in outer units, one would obtain uniform dilations $\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{i}}=q_{i}$ and hence the same power law for all the lengths. However, this contradicts the centre asymptotes:

(2.20a,b ) $$\begin{eqnarray}\ell _{12}\propto 1/\sqrt{r};\quad \ell _{ii}\propto 1/r\end{eqnarray}$$

( $i=1,2,3$ ), explained as follows. In the limit $r\rightarrow 0$ , using Taylor expansion, $U_{d}^{+}\sim r^{2}$ (due to the mirror symmetry $\unicode[STIX]{x2202}_{r}U^{+}=0$ at $r=0$ ), and $W_{12}^{+}\approx r$ , thus $\ell _{12}^{+}=\sqrt{W_{12}^{+}}/S^{+}\propto 1/\sqrt{r}$ ; on the other hand, the well-known fact $\overline{u_{i}^{\prime }u_{i}^{\prime }}\rightarrow C_{i}>0$ leads to $\ell _{ii}^{+}=\sqrt{W_{ii}^{+}}/S^{+}\propto 1/r$ near the centre. Hence, we obtain (2.20), where $\ell _{ij}=\ell _{ij}^{+}/Re_{\unicode[STIX]{x1D70F}}$ .

To obtain (2.20) from symmetry analysis, additional physical considerations are needed. As we know, when $r\rightarrow 0$ , mean shear $S^{+}$ is much smaller than $W_{12}^{+}$ and hence can be ignored in the mean momentum equation; meanwhile, for the stress equations, production and viscous diffusion terms approach zero and can be ignored also, whilst turbulent transport, pressure–strain and dissipation are the dominant terms. Note that for $W_{12}^{+}$ , dissipation is much smaller than pressure–strain and turbulent transport (Pope Reference Pope2000) hence also ignored. Thus, approximate balance equations for the core layer are obtained as

(2.21a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}W_{12}^{+}}{\unicode[STIX]{x2202}r}+1\approx 0, & \displaystyle\end{eqnarray}$$

(2.21b ) $$\begin{eqnarray}\displaystyle & \displaystyle 0\approx -\overline{u^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}\tilde{x}}}^{+}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{2}\overline{u^{\prime }u^{\prime }v^{\prime }}^{+}\right)-\frac{1}{Re_{\unicode[STIX]{x1D70F}}}\overline{|\unicode[STIX]{x1D735}u^{\prime }|^{+}}^{2}, & \displaystyle\end{eqnarray}$$

(2.21c ) $$\begin{eqnarray}\displaystyle & \displaystyle 0\approx -\overline{v^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}r}}^{+}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{2}\overline{v^{\prime }v^{\prime }v^{\prime }}^{+}\right)-\frac{1}{Re_{\unicode[STIX]{x1D70F}}}\overline{|\unicode[STIX]{x1D735}v^{\prime }|^{+}}^{2}, & \displaystyle\end{eqnarray}$$

(2.21d ) $$\begin{eqnarray}\displaystyle & \displaystyle 0\approx -\overline{w^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}\tilde{z}}}^{+}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{2}\overline{w^{\prime }w^{\prime }v^{\prime }}^{+}\right)-\frac{1}{Re_{\unicode[STIX]{x1D70F}}}\overline{|\unicode[STIX]{x1D735}w^{\prime }|^{+}}^{2}, & \displaystyle\end{eqnarray}$$

(2.21e ) $$\begin{eqnarray}\displaystyle & \displaystyle 0\approx \frac{1}{2}\overline{u^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}r}}^{+}-\frac{1}{2}\overline{v^{\prime }\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}\tilde{x}}}^{+}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{2}\overline{u^{\prime }v^{\prime }v^{\prime }}^{+}\right). & \displaystyle\end{eqnarray}$$

(2.22a ) $$\begin{eqnarray}\displaystyle q_{1}=q_{2}=q_{3}=q=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}, & & \displaystyle\end{eqnarray}$$

(2.22b ) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}},\quad \overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{i}}/(q\unicode[STIX]{x1D706}_{R}). & & \displaystyle\end{eqnarray}$$

(2.23a,b ) $$\begin{eqnarray}\ell _{12}^{\ast }=(\sqrt{q^{3}}/\unicode[STIX]{x1D706}_{U})\ell _{12},\quad \ell _{ii}^{\ast }=\left(q\sqrt{\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{i}}}/\unicode[STIX]{x1D706}_{U}\right)\ell _{ii}.\end{eqnarray}$$

Now, we show that the result (2.23) is consistent with the expansion solution (2.20) in the core layer. Note that as $r\rightarrow 0$ , $U_{d}^{+}\sim r^{2}$ , thus $\unicode[STIX]{x1D706}_{U}=q^{2}$ . Moreover, $\overline{u_{i}^{\prime }u_{i}^{\prime }}\rightarrow C_{i}>0$ , which gives $\overline{\unicode[STIX]{x1D706}_{i}\unicode[STIX]{x1D706}_{i}}=1$ . Further, with $\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=q$ in (2.22a ) (consistent with $W_{12}^{+}\sim r$ ), equation (2.23) leads to $\ell _{12}^{\ast }=(1/\sqrt{q})\ell _{12}$ , $\ell _{ii}^{\ast }=(1/q)\ell _{ii}$ , and hence $\ell _{12}\sim r^{-1/2}$ , $\ell _{ii}\sim r^{-1}$ . Therefore, we have the following power laws for the core layer ( $i=1,2,3$ ):

(2.24a,b ) $$\begin{eqnarray}\ell _{12}=I_{12}^{core}/\sqrt{r},\quad \ell _{ii}=I_{ii}^{core}/r.\end{eqnarray}$$

2.5 Homogeneous dilations in the bulk layer

Unlike in the core layer, productions in the bulk are the dominant terms in the stress equations and hence cannot be ignored. Thus, by including $\overline{u^{\prime }v^{\prime }}^{+}\unicode[STIX]{x2202}_{r}U^{+}$ in (2.21b ) and $\overline{v^{\prime }v^{\prime }}^{+}\unicode[STIX]{x2202}_{r}U^{+}$ in (2.21e ), and applying (2.2) to the new balance equations, one obtains

(2.25a ) $$\begin{eqnarray}\displaystyle q_{1}=q_{2}=q_{3}=q=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}},\quad \unicode[STIX]{x1D706}_{R}=1/(q\unicode[STIX]{x1D706}_{U}), & & \displaystyle\end{eqnarray}$$

(2.25b ) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{p}}=\overline{\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{2}}=\overline{\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}}=q\unicode[STIX]{x1D706}_{U}.\end{eqnarray}$$

(2.26) $$\begin{eqnarray}\ell _{ij}^{\ast }=(q^{3/2}/\unicode[STIX]{x1D706}_{U})\ell _{ij}\end{eqnarray}$$

for $i,j=1,2$ . Note that dilation on $\ell _{33}$ is the same as $\ell _{ij}$ in (2.26) under the isotropic turbulence assumption.

Denote $\unicode[STIX]{x1D706}_{U}=q^{3/2-\unicode[STIX]{x1D6FC}}$ so that lengths obey the following dilations in the bulk flow:

(2.27a,b ) $$\begin{eqnarray}r^{\ast }=\text{e}^{\unicode[STIX]{x1D700}}r,\quad \ell _{ij}^{\ast }=\text{e}^{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D700}}\ell _{ij}.\end{eqnarray}$$

Now, following the second ansatz in Part 1, the differential invariants of the lengths are assumed to be constants in the bulk layer, leading to

(2.28) $$\begin{eqnarray}(\text{d}/\text{d}r)\ell _{ij}=I_{ij}r^{\unicode[STIX]{x1D6FC}-1},\end{eqnarray}$$

(where $I_{ij}$ are constants). After integration with respect to $r$ , equation (2.28) leads to

(2.29) $$\begin{eqnarray}\ell _{ij}=b_{ij}-\frac{I_{ij}r^{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

Furthermore, in the limit $r\rightarrow 1$ (to the wall), all lengths go to zero, then we readily obtain (for $i=1,2,3$ )

(2.30a,b ) $$\begin{eqnarray}\ell _{12}=I_{12}^{\prime }(1-r^{\unicode[STIX]{x1D6FC}});\quad \ell _{ii}=I_{ii}^{\prime }(1-r^{\unicode[STIX]{x1D6FC}}).\end{eqnarray}$$

Note that all the lengths in (2.30) asymptotes to the linear $y$ scaling (since $1-r^{\unicode[STIX]{x1D6FC}}=1-(1-y)^{\unicode[STIX]{x1D6FC}}\rightarrow \unicode[STIX]{x1D6FC}y$ ), hence matching the linear scaling (2.8) in the log layer. Also, as explained in Part 1 and Chen, Hussain & She (Reference Chen, Hussain and She2016b ), $\unicode[STIX]{x1D6FC}=4$ for channels and 5 for pipes.

2.6 A summary of canonical dilation structure

Table 1.

A summary of the canonical scaling in the four layers of turbulent channels. Note that the log layer is contained in the bulk layer since $1-r^{4}\propto y$ as $r=1-y\rightarrow 1$ (towards the wall). For pipe flows, the scalings are the same as for channels except for $\ell _{ij}\propto 1-r^{5}$ in the bulk layer of pipes.

The above scalings obtained in each layer for $\ell _{12}$ , $\ell _{11}$ and $\ell _{22}$ are summarized in table 1. Then, following exactly the same matching procedure in Part 1 (using the third ansatz and the multiplicative rule; see appendix A), we readily obtain the following multilayer formula of lengths for channels:

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \ell _{12}^{+}=c_{12}y^{+3/2}\left(\!1+\left(\frac{y^{+}}{y_{sub}^{+}}\right)^{4}\right)^{1/8}\left(\!1+\left(\frac{y^{+}}{y_{buf}^{+}}\right)^{4}\right)^{-1/4}\frac{1-r^{4}}{4(1-r)}\,\left(\!1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/4},\qquad & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \ell _{11}^{+}=c_{11}y^{+}\left(\!1+\left(\frac{y^{+}}{y_{sub}^{+}}\right)^{4}\right)^{1/8}\left(\!1+\left(\frac{y^{+}}{y_{buf}^{+}}\right)^{4}\right)^{-1/8}\frac{1-r^{4}}{4(1-r)}\left(\!1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/2},\qquad & \displaystyle\end{eqnarray}$$

(2.33) $$\begin{eqnarray}\displaystyle & \displaystyle \ell _{22}^{+}=c_{22}y^{+2}\!\left(\!1+\left(\frac{y^{+}}{y_{sub}^{+}}\right)^{4}\right)^{1/8}\!\left(\!1+\left(\frac{y^{+}}{y_{buf}^{+}}\right)^{4}\right)^{-3/8}\frac{1-r^{4}}{4(1-r)}\left(\!1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/2}. & \displaystyle\end{eqnarray}$$

Note that (2.31) has been obtained in Part 1, and the lengths in the above equations (2.31)–(2.33) are the same for both channels and pipes – except for the bulk factor being $(1-r^{5})/5(1-r)$ in pipes. Here, $c_{ij}$ ( $i,j=1,2$ ) are the global proportionality coefficients (represented by Kármán-like constants $\unicode[STIX]{x1D705}_{ij}$ defined later), which may have a moderate $Re$ dependence, and are responsible for the finite $Re$ -scaling for the peak magnitude of Reynolds stresses (see detailed analysis below).

The above equations therefore lead to

(2.34) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{11}=\frac{\ell _{11}^{+}}{\ell _{12}^{+}}=\frac{\sqrt{W_{11}^{+}}}{\sqrt{W_{12}^{+}}}=c_{1}\left(1+\left(\frac{y_{buf}^{+}}{y^{+}}\right)^{4}\right)^{1/8}\left(1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/4}, & \displaystyle\end{eqnarray}$$

(2.35) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{22}=\frac{\ell _{22}^{+}}{\ell _{12}^{+}}=\frac{\sqrt{W_{22}^{+}}}{\sqrt{W_{12}^{+}}}=c_{2}\left(1+\left(\frac{y_{buf}^{+}}{y^{+}}\right)^{4}\right)^{-1/8}\left(1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/4}, & \displaystyle\end{eqnarray}$$

where $c_{1}=c_{11}/(c_{12}y_{buf}^{+1/2})$ and $c_{2}=c_{22}y_{buf}^{+1/2}/c_{12}$ . Note that $\unicode[STIX]{x1D6FC}_{11}$ and $\unicode[STIX]{x1D6FC}_{22}$ have been introduced and studied for compressible turbulent boundary layers (Morkovin Reference Morkovin and Favre1962; Bradshaw Reference Bradshaw1977; Zhang et al. Reference Zhang, Bi, Hussain, Li and She2012). The two ratios can be interpreted to describe the relative sizes of turbulent eddies associated with different fluctuation components, which have simple scaling properties as follows. In the near-wall region, equation (2.34) yields $\unicode[STIX]{x1D6FC}_{11}\propto 1/\sqrt{y^{+}}$ as $y^{+}\rightarrow 0$ , because $W_{11}^{+}\propto y^{+2}$ and $W_{12}^{+}\propto y^{+3}$ . In the core layer, $W_{12}^{+}\approx r$ and $W_{11}^{+}=\text{const.}$ , hence $\unicode[STIX]{x1D6FC}_{11}\propto 1/\sqrt{r}$ . In the overlap region, i.e. $y^{+}\gg 1$ and $r\rightarrow 1$ , fluctuations are nearly isotropic; hence $\unicode[STIX]{x1D6FC}_{11}\approx \text{const.}$ (an anomalous scaling modification will be introduced for the emergence of outer peak of $\overline{u^{\prime }u^{\prime }}$ , as explained later). Similarly, for $\unicode[STIX]{x1D6FC}_{22}$ , equation (2.35) yields $\unicode[STIX]{x1D6FC}_{22}\propto \sqrt{y^{+}}$ as $y^{+}\rightarrow 0$ , due to $v^{\prime }\propto y^{2}$ ; the overlap region and core layer scaling of $\unicode[STIX]{x1D6FC}_{22}$ are the same as $\unicode[STIX]{x1D6FC}_{11}$ . Also note that the exponent 2 of the core layer factor $(r_{core}/r)^{2}$ in (2.34) and (2.35) leads to a parabolic profile for each Reynolds normal stress, i.e. $W_{ii}^{+}-W_{iic}^{+}\propto r^{2}$ , hence satisfying mirror symmetry at the centreline.

The above analysis of $\unicode[STIX]{x1D6FC}_{11}$ and $\unicode[STIX]{x1D6FC}_{22}$ also applies to $\unicode[STIX]{x1D6FC}_{33}$ ( $\equiv \sqrt{W_{33}^{+}/W_{12}^{+}}$ ). As $W_{33}^{+}\propto y^{+2}$ and $W_{12}^{+}\propto y^{+3}$ when $y^{+}\rightarrow 0$ , one has $\unicode[STIX]{x1D6FC}_{33}\propto 1/\sqrt{y^{+}}$ . Near the centre, $\unicode[STIX]{x1D6FC}_{33}\propto 1/\sqrt{r}$ due to a non-zero value of $W_{33}^{+}$ at $r=0$ . In the overlap region, the isotropic turbulence assumption implies the same scaling for all quantities, and hence a constant $\unicode[STIX]{x1D6FC}_{33}$ (an anomalous scaling modification will be introduced later). Since all the scalings in these three layers are the same as those of $\unicode[STIX]{x1D6FC}_{11}$ , we expect that $\unicode[STIX]{x1D6FC}_{33}$ takes the same functional form as $\unicode[STIX]{x1D6FC}_{11}$ in (2.34). However, there is a significant difference between $W_{11}^{+}$ and $W_{33}^{+}$ in their balance mechanisms: the (net) production–dissipation balance of $W_{11}^{+}$ occurs until above the buffer layer thickness (hence the overlap region) – in contrast to the pressure–strain–dissipation balance of $W_{33}^{+}$ just above the sublayer thickness. Because of this difference, the scaling transition to the overlap region occurs earlier at $y_{sub}^{+}$ for $\unicode[STIX]{x1D6FC}_{33}$ , in contrast to at $y_{buf}^{+}$ for $\unicode[STIX]{x1D6FC}_{11}$ . Thus, the expression for $\unicode[STIX]{x1D6FC}_{33}$ is

(2.36) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{33}=\frac{\ell _{33}^{+}}{\ell _{12}^{+}}=\frac{\sqrt{W_{33}^{+}}}{\sqrt{W_{12}^{+}}}=c_{3}\left(1+\left(\frac{y_{sub}^{+}}{y^{+}}\right)^{4}\right)^{1/8}\left(1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/4}.\end{eqnarray}$$

Multiplying $\unicode[STIX]{x1D6FC}_{33}$ with $\ell _{12}^{+}$ yields

(2.37) $$\begin{eqnarray}\ell _{33}^{+}=c_{33}y^{+}\left(1+\left(\frac{y^{+}}{y_{sub}^{+}}\right)^{4}\right)^{1/4}\left(1+\left(\frac{y^{+}}{y_{buf}^{+}}\right)^{4}\right)^{-1/4}\frac{1-r^{4}}{4(1-r)}\left(1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/2},\end{eqnarray}$$

where $c_{33}=c_{3}c_{12}y_{sub}^{+1/2}$ . The local scaling of (2.37) is also summarized in table 1. Figure 1(b) shows the compensated plot for all lengths divided by $1-r^{4}$ , where a typical four-layer structure is clearly visible.

In the sections below, we will compare above formulae with data. It turns out that two normal fluctuation-related components, i.e. $W_{12}^{+}$ and $W_{22}^{+}$ (related to active motions), follow exactly the canonical four-layer description. The two other components $W_{11}^{+}$ and $W_{33}^{+}$ (related to inactive motions) show notable departures from the four-layer descriptions (which lead to constant $W_{11}^{+}$ and $W_{33}^{+}$ in the bulk), but are readily improved by the aforementioned, slight modifications of the scalings in the log layer.

3.1 Location of maximum production

Near the wall, the mean momentum equation (3.1) leads to $S^{+}+W_{12}^{+}\approx 1$ . Thus, the production $S^{+}W_{12}^{+}\approx S^{+}(1-S^{+})$ has its maximum value of $1/4$ , located at the $y_{max}^{+}$ where $S^{+}=W_{12}^{+}=1/2$ . Then $\ell _{12}^{+}=\sqrt{W_{12}^{+}}/S^{+}=\sqrt{2}$ at $y_{max}^{+}$ .

On the other hand, as the wall is approached, i.e. $y^{+}\rightarrow 0$ , equation (2.31) asymptotes to its sublayer expression:

(3.4) $$\begin{eqnarray}\ell _{12}^{+}=\ell _{0}(y^{+}/y_{sub}^{+})^{3/2}\approx (y^{+}/y_{sub}^{+})^{3/2}.\end{eqnarray}$$

Then, letting $\ell _{12}^{+}=\sqrt{2}$ in (3.4), one has

(3.5) $$\begin{eqnarray}y_{max}^{+}=\sqrt[3]{2}y_{sub}^{+}\approx 12.2,\end{eqnarray}$$

which is very close to the documented value of approximately 12 (Kim, Moin & Moser Reference Kim, Moin and Moser1987; Pope Reference Pope2000). The result $y_{max}^{+}\approx 12.2$ can be inverted to measure the sublayer thickness $y_{sub}^{+}\approx 9.7$ .

Note that $W_{12}^{+}$ predicted by (3.4) also agrees with data in most of the flow domain (not shown here) – except for the buffer layer. This is because: (1) for the outer region, equation (3.4) yields $\ell _{12}^{+}\gg 1$ , which leads to $W_{12}^{+}\approx r$ – a good approximation in the entire outer flow; (2) for the inner region, equation (3.4) gives an accurate description of $W_{12}^{+}$ in the viscous sublayer. However, for the buffer layer, the simple sublayer formula (3.4) is insufficient and needs to be extended to predict how $W_{12}^{+}$ approaches its peak value, addressed below.

3.2 Scaling for the peak of Reynolds shear stress

To determine the peak of $W_{12}$ , equation (3.2) is approximated to $W_{12}^{+}\approx r-\sqrt{r}/\ell _{12}^{+}$ (since $\ell _{12}^{+}\gg 1$ as $y^{+}\gg y_{sub}^{+}$ ). Further, with $r=1-y^{+}/Re_{\unicode[STIX]{x1D70F}}$ , we have:

(3.6) $$\begin{eqnarray}W_{12}^{+}\approx 1-y^{+}/Re_{\unicode[STIX]{x1D70F}}-1/\ell _{12}^{+}\end{eqnarray}$$

(the $1/Re_{\unicode[STIX]{x1D70F}}$ term in $\sqrt{r}$ is small and ignored). Letting $\unicode[STIX]{x2202}_{y^{+}}W_{12}^{+}=0$ using (3.6), one has the following relation at the peak location (i.e. $y_{12p}^{+}$ ) of $W_{12}^{+}$ :

(3.7) $$\begin{eqnarray}Re_{\unicode[STIX]{x1D70F}}^{-1}=\dot{\ell }_{12}^{+}/\ell _{12}^{+2}.\end{eqnarray}$$

Here, the superscript dot indicates derivative with respect to $y^{+}$ .

By substituting the stress length (2.31) into (3.7) we can solve for $y_{12p}^{+}$ , which is located in the buffer layer for small $Re$ , but in the log layer for large $Re$ . Since $\ell _{12}^{+}$ from (2.31) has the following two simple asymptotes:

(3.8a ) $$\begin{eqnarray}\displaystyle & \ell _{12}^{+}\approx (\unicode[STIX]{x1D705}/y_{buf}^{+})y^{+2},\quad \text{for }y_{sub}^{+}\ll y_{12p}^{+}\lesssim y_{buf}^{+}, & \displaystyle\end{eqnarray}$$

(3.8b ) $$\begin{eqnarray}\displaystyle & \ell _{12}^{+}\approx \unicode[STIX]{x1D705}y^{+},\quad \text{for }y_{buf}^{+}\ll y_{12p}^{+}\ll Re_{\unicode[STIX]{x1D70F}}, & \displaystyle\end{eqnarray}$$

$y_{12p}^{+}$

(3.9a ) $$\begin{eqnarray}\displaystyle & y_{12p}^{+}=Re_{\unicode[STIX]{x1D70F}}^{1/3}(2y_{buf}^{+}/\unicode[STIX]{x1D705})^{1/3}\approx 5.7Re_{\unicode[STIX]{x1D70F}}^{1/3}, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & y_{12p}^{+}=Re_{\unicode[STIX]{x1D70F}}^{1/2}/\sqrt{\unicode[STIX]{x1D705}}\approx 1.5Re_{\unicode[STIX]{x1D70F}}^{1/2}. & \displaystyle\end{eqnarray}$$

$Re_{\unicode[STIX]{x1D70F}}^{1/2}$

$Re_{\unicode[STIX]{x1D70F}}^{1/3}$

$Re$

$Re_{\unicode[STIX]{x1D70F}}^{1/3}$

$Re_{\unicode[STIX]{x1D70F}}^{0.372}$

$Re_{\unicode[STIX]{x1D70F}}^{1/3}$

$Re_{\unicode[STIX]{x1D70F}}^{1/2}$

$Re_{\unicode[STIX]{x1D70F}}=4\unicode[STIX]{x1D705}y_{buf}^{+2}\approx 3000$

Figure 3.

Scaling for the peak of Reynolds shear stress: (a) location; (b) magnitude defect. Solid lines denote (3.9a ) and (3.10a ) for low $Re$ , and dashed lines denote (3.9b ) and (3.10b ) for high $Re$ . Symbols are DNS data: squares from Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) for $Re_{\unicode[STIX]{x1D70F}}=300,400$ and 650 (channel); circles from Hoyas & Jimenez (Reference Hoyas and Jimenez2006) for $Re_{\unicode[STIX]{x1D70F}}=940$ and 2000 (channel); stars from Wu & Moin (Reference Wu and Moin2008) for $Re_{\unicode[STIX]{x1D70F}}=1142$ (pipe); cross from Ahn et al. (Reference Ahn, Lee, Lee, Kang and Sung2015) for $Re_{\unicode[STIX]{x1D70F}}=3008$ (pipe); and diamonds from Lee & Moser (Reference Lee and Moser2015) for $Re_{\unicode[STIX]{x1D70F}}=550$ , 1000 and 5200 (channel).

Also, the small departure of the peak value of $W_{12}^{+}$ from unity shows a scaling transition (figure 3 b), explained as follows. Substituting (3.8) and (3.9) back into (3.6) yields the departure:

(3.10a ) $$\begin{eqnarray}\displaystyle & 1-W_{12p}^{+}=3(y_{buf}^{+}/4\unicode[STIX]{x1D705})^{1/3}Re_{\unicode[STIX]{x1D70F}}^{-2/3}\approx 8.5Re_{\unicode[STIX]{x1D70F}}^{-2/3}, & \displaystyle\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle & 1-W_{12p}^{+}=(2/\sqrt{\unicode[STIX]{x1D705}})Re_{\unicode[STIX]{x1D70F}}^{-1/2}\approx 3.0Re_{\unicode[STIX]{x1D70F}}^{-1/2}. & \displaystyle\end{eqnarray}$$

$Re$

$B_{12}$

$B_{12}=W_{12p}^{+}$

$W_{12}^{+}$

4.1 Measurement of $\unicode[STIX]{x1D705}_{22}$

Let us consider the scaling of the peak value of $W_{22}^{+}$ for asymptotically large $Re$ . Let $c_{22}=c_{22}^{\prime }/\sqrt{1+r_{22core}^{2}}$ in (2.33) so that $\ell _{22}^{+}\approx c_{22}^{\prime }y^{+2}$ for $y^{+}\rightarrow 0$ . On the other hand, for $y^{+}\gg y_{buf}^{+}$ , we obtain the linear scalings: $\ell _{12}^{+}\approx \unicode[STIX]{x1D705}y^{+}$ and $\ell _{22}^{+}\approx \unicode[STIX]{x1D705}_{22}y^{+}$ , where $\unicode[STIX]{x1D705}_{22}=c_{22}^{\prime }y_{buf}^{+3/2}/y_{sub}^{+1/2}$ . Accordingly, $W_{22}^{+}\approx W_{12}^{+}(\unicode[STIX]{x1D705}_{22}/\unicode[STIX]{x1D705})^{2}$ . Thus, its peak value is

(4.2) $$\begin{eqnarray}W_{22p}^{+}\approx W_{12p}^{+}\unicode[STIX]{x1D705}_{22}^{2}/\unicode[STIX]{x1D705}^{2}.\end{eqnarray}$$

Therefore,

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{22}\approx \unicode[STIX]{x1D705}\sqrt{W_{22p}^{+}/W_{12p}^{+}}.\end{eqnarray}$$

This yields a method to measure the new constant $\unicode[STIX]{x1D705}_{22}$ . For $Re_{\unicode[STIX]{x1D70F}}=5200$ , with $\unicode[STIX]{x1D705}=0.45$ , we predict $W_{12p}^{+}\approx 0.96$ from (3.10b ). Then, with the measured $W_{22p}^{+}\approx 1.29$ from data at $Re_{\unicode[STIX]{x1D70F}}=5200$ (Lee & Moser Reference Lee and Moser2015), we obtain $\unicode[STIX]{x1D705}_{22}\approx 0.52\approx 1.16\unicode[STIX]{x1D705}$ .

Figure 5.

Wall-normal Reynolds stress profiles by (4.1) compared with data. Also included are predictions at $Re_{\unicode[STIX]{x1D70F}}=50\,000$ . The departure between data and our theory (i.e. $W_{22}^{+Theory}-W_{22}^{+DNS}$ ) is shown in (c,d), which is bounded within $\pm 0.1$ . For legend, see figure 2.

Note that according to $W_{22}^{+}\approx W_{12}^{+}(\unicode[STIX]{x1D705}_{22}/\unicode[STIX]{x1D705})^{2}$ , $W_{22}^{+}$ should reach its peak at the same location as the peak of $W_{12}^{+}$ for large $Re$ ; i.e. $y_{22p}^{+}=\sqrt{Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705}}$ . However, $Re$ for the current data is not high enough to show such a scaling, and higher- $Re$ data are needed to test it.

4.2 Measurement of $r_{22core}$

Near the centreline (as $r\rightarrow 0$ ), $\ell _{12}^{+}\approx (\unicode[STIX]{x1D705}/4)(1+r_{core}^{2}/r^{2})^{1/4}$ from (2.31), and $\ell _{22}^{+}\approx (\unicode[STIX]{x1D705}_{22}/4)(1+r_{22core}^{2}/r^{2})^{1/2}$ from (2.33). Further, with $W_{12}^{+}\approx r$ , the centreline value is given as

(4.4) $$\begin{eqnarray}W_{22c}^{+}=W_{12}^{+}(\ell _{22}^{+}/\ell _{12}^{+})^{2}\approx \unicode[STIX]{x1D705}_{22}^{2}r_{22core}^{2}/(\unicode[STIX]{x1D705}^{2}r_{core}),\end{eqnarray}$$

which leads to

(4.5) $$\begin{eqnarray}r_{22core}\approx (W_{22c}^{+}r_{core})^{1/2}(\unicode[STIX]{x1D705}/\unicode[STIX]{x1D705}_{22}).\end{eqnarray}$$

Thus, with $r_{core}=0.27$ , $\unicode[STIX]{x1D705}=0.45$ , as well as $\unicode[STIX]{x1D705}_{22}\approx 0.52$ and $W_{22c}^{+}\approx 0.47$ from DNS data (Lee & Moser Reference Lee and Moser2015), we have $r_{22core}\approx 0.3$ . This also results in $\unicode[STIX]{x1D705}_{22}\approx 0.78W_{22c}^{+1/2}$ . Note that if one imposes $r_{22core}=r_{core}=0.27$ , the resulting $W_{22c}^{+}$ from (4.4) would underpredict the data by 20 % at $Re_{\unicode[STIX]{x1D70F}}=5200$ . For the smallest $Re_{\unicode[STIX]{x1D70F}}$ ( $=300$ ), DNS data shows $W_{22c}^{+}\approx 0.39$ (Iwamoto et al. Reference Iwamoto, Suzuki and Kasagi2002), so that (4.5) leads to $r_{22core}\approx 0.28$ . Therefore, $r_{22core}$ varies mildly with increasing $Re_{\unicode[STIX]{x1D70F}}$ (for a moderate $Re_{\unicode[STIX]{x1D70F}}$ range).

Substituting $\unicode[STIX]{x1D705}_{22}\approx 0.52$ and $r_{22core}\approx 0.3$ into (2.33), where $c_{22}=\unicode[STIX]{x1D705}_{22}y_{sub}^{+1/2}/[y_{buf}^{+3}(1+r_{22core}^{2})]^{1/2}$ , the resulting $W_{22}^{+}$ profiles from (4.1) are shown in figure 5. The agreement is quite good, with the departure bounded within $\pm 0.1$ . Note that for the smallest $Re_{\unicode[STIX]{x1D70F}}(=300)$ , there is a slight overestimation of the peak magnitude. This small departure can be easily rectified by $\unicode[STIX]{x1D705}_{22}\approx 0.50$ at $Re_{\unicode[STIX]{x1D70F}}=300$ ; hence $\unicode[STIX]{x1D705}_{22}$ slightly changes for moderate $Re$ .

5.1 Anomalous scaling and invariant peak location

Note that the deviation in figure 6(a) occurs in the log layer where $\ell _{33}^{+}\approx \unicode[STIX]{x1D705}_{33}y^{+}$ . However, as shown by the ratio $\unicode[STIX]{x1D6FC}_{33}=\ell _{33}^{+}/\ell _{12}^{+}$ (see figure 6 b), the linear scaling of $\ell _{33}^{+}$ is invalid since $\unicode[STIX]{x1D6FC}_{33}\propto 1/y^{+0.07}$ – otherwise $\unicode[STIX]{x1D6FC}_{33}$ would be a constant equalling $\unicode[STIX]{x1D705}_{33}/\unicode[STIX]{x1D705}$ . This deviation thus suggests an anomalous scaling modification of $\ell _{33}$ – via a change of $(1+(y^{+}/y_{buf}^{+})^{4})^{-1/4}$ in (2.37) to $(1+(y^{+}/y_{buf}^{+})^{4})^{-(1+\unicode[STIX]{x1D6FE})/4}$ – so that $\ell _{33}^{+}\propto y^{+(1-\unicode[STIX]{x1D6FE})}$ , and hence $\unicode[STIX]{x1D6FC}_{33}\propto 1/y^{+\unicode[STIX]{x1D6FE}}$ in the log layer ( $y^{+}\gg y_{buf}^{+}$ ). In other words, $\ell _{33}$ is modified to

(5.2) $$\begin{eqnarray}\ell _{33}^{\prime }=c_{33}y^{+}\left(1+\left(\frac{y^{+}}{y_{sub}^{+}}\right)^{4}\right)^{1/4}\left(1+\left(\frac{y^{+}}{y_{buf}^{+}}\right)^{4}\right)^{-(1+\unicode[STIX]{x1D6FE})/4}\frac{1-r^{4}}{4(1-r)}\left(1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/2},\end{eqnarray}$$

and hence

(5.3) $$\begin{eqnarray}W_{33}^{+}=W_{12}^{+}\ell _{33}^{\prime 2}/\ell _{12}^{+2}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6FE}$ is the so-called anomalous scaling exponent, and its value 0.07 – shown in figure 6(b) – can be theoretically determined by the peak location of $W_{33}^{+}$ . Note that if $\unicode[STIX]{x1D6FE}=0$ (i.e. no anomalous scaling), $W_{33}^{+}\approx W_{12}^{+}(\unicode[STIX]{x1D705}_{33}/\unicode[STIX]{x1D705})^{2}$ in the log layer, and thus the peak location of $W_{33}^{+}$ (i.e. $y_{33p}^{+}$ ) should follow that of $W_{12}^{+}$ (i.e. $y_{12p}^{+}$ ) – both $y_{33p}^{+}$ and $y_{12p}^{+}$ increasing rapidly with increasing $Re$ . However, a careful examination of DNS data (from $Re_{\unicode[STIX]{x1D70F}}=650$ to $5200$ ) shows that $y_{33p}^{+}$ remains almost constant, i.e. $y_{33p}^{+}\approx 46$ , in sharp contrast to increasing $y_{12p}^{+}$ in figure 3. This constancy of $y_{33p}^{+}$ is actually due to the existence of the anomalous scaling $\unicode[STIX]{x1D6FE}\neq 0$ , and enables the determination of $\unicode[STIX]{x1D6FE}\approx 0.07$ , as below.

In fact, for $y^{+}\simeq y_{buf}^{+}$ where $y_{33p}^{+}$ is located, we have $\ell _{12}^{+}\approx (\unicode[STIX]{x1D705}y^{+2}/y_{buf}^{+})[1+(y^{+}/y_{buf}^{+})^{4}]^{-1/4}$ from (2.31), and hence an approximation of $W_{12}^{+}$ is obtained from (3.6), i.e. $W_{12}^{+}\approx 1-y^{+}/Re_{\unicode[STIX]{x1D70F}}-y_{buf}^{+}/(\unicode[STIX]{x1D705}y^{+2})$ (note that the $(y^{+}/y_{buf}^{+})^{4}$ term is very small for $W_{12}^{+}$ and hence ignored). Similarly, $\ell _{33}^{\prime }\approx (\unicode[STIX]{x1D705}_{33}y^{+2}/y_{buf}^{+})[1+(y^{+}/y_{buf}^{+})^{4}]^{-(1+\unicode[STIX]{x1D6FE})/4}$ from (5.2), and thus we have

(5.4) $$\begin{eqnarray}W_{33}^{+}=W_{12}^{+}(\ell _{33}^{\prime }/\ell _{12}^{+})^{2}\approx (\unicode[STIX]{x1D705}_{33}/\unicode[STIX]{x1D705})^{2}\left(1-\frac{y^{+}}{Re_{\unicode[STIX]{x1D70F}}}-\frac{y_{buf}^{+}}{\unicode[STIX]{x1D705}y^{+2}}\right)\left(1+\left(\frac{y^{+}}{y_{buf}^{+}}\right)^{4}\right)^{-\unicode[STIX]{x1D6FE}/2}.\end{eqnarray}$$

Then, solving $\unicode[STIX]{x2202}_{y^{+}}W_{33}^{+}=0$ at $y_{33p}^{+}$ using (5.4) yields the following result:

(5.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\frac{[2y_{buf}^{+}/(\unicode[STIX]{x1D705}y_{33p}^{+3})-1/Re_{\unicode[STIX]{x1D70F}}](y_{buf}^{+4}+y_{33p}^{+4})}{2y_{33p}^{+3}[1-y_{33p}^{+}/Re_{\unicode[STIX]{x1D70F}}-y_{buf}^{+}/(\unicode[STIX]{x1D705}y_{33p}^{+2})]}\approx 0.05\unicode[STIX]{x1D70E}^{-2}(1+\unicode[STIX]{x1D70E}^{-4}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}=y_{33p}^{+}/y_{buf}^{+}$ , and $y_{buf}^{+}=41$ and $\unicode[STIX]{x1D705}=0.45$ are substituted in (and the $1/Re_{\unicode[STIX]{x1D70F}}$ term is small and ignored). As explained before, if $y_{33p}^{+}$ follows the scaling of $y_{12p}^{+}$ , then $\unicode[STIX]{x1D70E}=\sqrt{Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705}}/y_{buf}^{+}$ , and (5.5) leads to $\unicode[STIX]{x1D6FE}\approx 0.05\unicode[STIX]{x1D705}^{2}y_{buf}^{+2}/Re_{\unicode[STIX]{x1D70F}}\rightarrow 0$ (e.g. $\unicode[STIX]{x1D6FE}\approx 0.003$ at $Re_{\unicode[STIX]{x1D70F}}=5200$ ), hence indicating no anomalous scaling. In contrast, DNS data show an invariant peak location $y_{33p}^{+}\approx 46$ , then $\unicode[STIX]{x1D70E}\approx 1.12$ , and (5.5) yields a constant $\unicode[STIX]{x1D6FE}\approx 0.07$ – the value shown in figure 6(b). In order words, the invariant peak location leads to the existence of the anomalous scaling. It would be intriguing to test if $\unicode[STIX]{x1D6FE}=0.07$ remains invariant at higher $Re$ values.

Now, the only unknown parameter in (5.2) is $c_{33}$ , which is $c_{33}=y_{sub}^{+}\unicode[STIX]{x1D705}_{33}/(\sqrt{1+r_{core}^{2}}y_{buf}^{+})$ . Below, $\unicode[STIX]{x1D705}_{33}$ is determined, hence yielding $c_{33}$ .

5.2 Scaling of $\unicode[STIX]{x1D705}_{33}$ and of the $W_{33}^{+}$ peak value

Figure 7.

(a) Peak value $W_{33p}^{+}$ of DNS data compared with prediction by (5.10) (line) showing good agreement. Symbols are the same as in figure 3. (b) Diagnostic function $\unicode[STIX]{x1D6E4}_{33}=-y\text{d}W_{33}^{+}/\text{d}y$ showing an approximate plateau $\unicode[STIX]{x1D6E4}_{33}=0.41\pm 0.04$ for $0.02\leqslant y\leqslant 0.14$ . Symbols are DNS data at $Re_{\unicode[STIX]{x1D70F}}=5200$  (Lee & Moser Reference Lee and Moser2015). Solid line indicates (5.12), whose minimum value $\unicode[STIX]{x1D6E4}_{33min}\approx 0.41$ obtained in (5.15) – dashed line – represents well the plateau value.

Note that for $y^{+}\gg y_{buf}^{+}$ , $W_{12}^{+}\approx r$ , then (5.3) leads to a simple expression for the outer profile of $W_{33}^{+}$ :

(5.6) $$\begin{eqnarray}W_{33Outer}^{+}\approx r\ell _{33}^{\prime 2}/\ell _{12}^{+2}\approx \left(\frac{\unicode[STIX]{x1D705}_{33}}{\unicode[STIX]{x1D705}}\right)^{2}\left(\frac{y_{buf}^{+}}{y^{+}}\right)^{2\unicode[STIX]{x1D6FE}}\left(\frac{r^{2}+r_{core}^{2}}{1+r_{core}^{2}}\right)^{1/2}.\end{eqnarray}$$

According to Townsend’s $Re$ -similarity hypothesis (Townsend Reference Townsend1976) (namely, mean properties in the outer flow region are self-similar, independent of viscous effects, and hence $Re$ ), one should expect that $W_{33Outer}^{+}$ depends only on $r$ , with no dependence on $Re$ . Then, requiring a zero exponent of $Re_{\unicode[STIX]{x1D70F}}$ for $W_{33Outer}^{+}$ in (5.6), we readily obtain $\unicode[STIX]{x1D705}_{33}\propto Re_{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D6FE}}$ .

We denote $\unicode[STIX]{x1D705}_{33}=\unicode[STIX]{x1D705}_{33}^{\prime }Re_{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D6FE}}$ , where $\unicode[STIX]{x1D705}_{33}^{\prime }$ is a $Re$ -independent constant, which can be measured by $W_{33}^{+}$ at the centreline. Letting $r=0$ , equation (5.6) becomes

(5.7) $$\begin{eqnarray}W_{33c}^{+}\approx (\unicode[STIX]{x1D705}_{33}^{\prime }/\unicode[STIX]{x1D705})^{2}y_{buf}^{+2\unicode[STIX]{x1D6FE}}r_{core}/\sqrt{1+r_{core}^{2}}.\end{eqnarray}$$

Therefore,

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{33}^{\prime }=\unicode[STIX]{x1D705}W_{33c}^{+1/2}(1+r_{core}^{2})^{1/4}/(y_{buf}^{+\unicode[STIX]{x1D6FE}}r_{core}^{1/2}).\end{eqnarray}$$

Using $\unicode[STIX]{x1D705}=0.45$ , $y_{buf}^{+}=41$ , $r_{core}=0.27$ , $\unicode[STIX]{x1D6FE}=0.07$ and the centreline value $W_{33c}^{+}\approx 0.4$ by current DNS data (Hoyas & Jimenez Reference Hoyas and Jimenez2006; Lee & Moser Reference Lee and Moser2015), we thus have $\unicode[STIX]{x1D705}_{33}^{\prime }\approx 0.43$ , and hence

(5.9) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{33}=\unicode[STIX]{x1D705}_{33}^{\prime }Re_{\unicode[STIX]{x1D70F}}^{\unicode[STIX]{x1D6FE}}\approx 0.43Re_{\unicode[STIX]{x1D70F}}^{0.07}.\end{eqnarray}$$

Furthermore, substituting $\unicode[STIX]{x1D705}=0.45$ , $y_{buf}^{+}=41$ , $\unicode[STIX]{x1D6FE}=0.07$ , $y^{+}=y_{33p}^{+}=\unicode[STIX]{x1D70E}y_{buf}^{+}$ (with $\unicode[STIX]{x1D70E}=1.12$ ) and (5.9) into (5.4), one readily obtains the peak value of $W_{33}^{+}$ as

(5.10) $$\begin{eqnarray}\displaystyle W_{33p}^{+} & = & \displaystyle (\unicode[STIX]{x1D705}_{33}^{\prime }/\unicode[STIX]{x1D705})^{2}Re_{\unicode[STIX]{x1D70F}}^{2\unicode[STIX]{x1D6FE}}(1+\unicode[STIX]{x1D70E}^{4})^{-\unicode[STIX]{x1D6FE}/2}[1-41\unicode[STIX]{x1D70E}/Re_{\unicode[STIX]{x1D70F}}-1/(41\unicode[STIX]{x1D705}\unicode[STIX]{x1D70E}^{2})]\nonumber\\ \displaystyle & {\approx} & \displaystyle 0.84Re_{\unicode[STIX]{x1D70F}}^{0.14}(1-48/Re_{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

To emphasize, in (5.10), $\unicode[STIX]{x1D705}_{33}^{\prime }$ only affects the magnitude of $W_{33p}^{+}$ , while the $Re_{\unicode[STIX]{x1D70F}}$ scaling of $W_{33p}^{+}$ is due to the anomalous scaling exponent $\unicode[STIX]{x1D6FE}$ . Figure 7(a) shows that (5.10) agrees well with the data. Note that the downward curve in the figure 7(a) is due to the $48/Re_{\unicode[STIX]{x1D70F}}$ term in (5.10), which becomes negligible at large $Re$ , when $W_{33p}^{+}\approx 0.84Re_{\unicode[STIX]{x1D70F}}^{0.14}$ . Thus, although the peak location $y_{33p}^{+}$ is invariant, the peak value $W_{33p}^{+}$ increases with increasing $Re$ .

5.3 Townsend’s log profile for $W_{33}^{+}$

Note that the outer profile (5.6) has a power-law asymptote towards the wall, i.e.

(5.11) $$\begin{eqnarray}W_{33}^{+}\approx y_{buf}^{+2\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D705}_{33}^{\prime }/\unicode[STIX]{x1D705})^{2}/y^{2\unicode[STIX]{x1D6FE}},\quad y=1-r.\end{eqnarray}$$

This power law is conceptually different from the Townsend’s log law (Townsend Reference Townsend1976) in (1.4). Reminiscent of the debate on the mean velocity scaling (Barenblatt Reference Barenblatt1993), a new debate between the power law and log law on the $W_{33}^{+}$ scaling may arise. However, as both (5.11) and (1.4) represent data in limited but different flow domains, such a debate is really not meaningful and may not be resolved due to their unspecified $y$ -ranges. Here, since we already have a global expression (5.6) which extends (5.11) to the centreline and describes $W_{33}^{+}$ for the entire outer flow domain, it is more interesting to see if the present formalism (5.6) can yield the log law with its two coefficients quantitatively determined.

This is achieved via the diagnostic function of the log profile, $\unicode[STIX]{x1D6E4}_{33}=-yd_{y}W_{33}^{+}$ , which is calculated using (5.6) as

(5.12) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{33}=\frac{y_{buf}^{+2\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D705}_{33}^{\prime }/\unicode[STIX]{x1D705})^{2}}{\sqrt{1+r_{core}^{2}}}\left(\frac{2\unicode[STIX]{x1D6FE}\sqrt{(1-y)^{2}+r_{core}^{2}}}{y^{2\unicode[STIX]{x1D6FE}}}+\frac{(1-y)y^{1-2\unicode[STIX]{x1D6FE}}}{\sqrt{(1-y)^{2}+r_{core}^{2}}}\right).\end{eqnarray}$$

Note that as $y\ll 1$ in the log region, equation (5.12) is further approximated to

(5.13) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{33}\approx \frac{y_{buf}^{+2\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D705}_{33}^{\prime }/\unicode[STIX]{x1D705})^{2}}{\sqrt{1+r_{core}^{2}}}y^{-2\unicode[STIX]{x1D6FE}}\left(2\unicode[STIX]{x1D6FE}\sqrt{1+r_{core}^{2}}+\frac{y}{\sqrt{1+r_{core}^{2}}}\right).\end{eqnarray}$$

Solving $\text{d}\unicode[STIX]{x1D6E4}_{33}/\text{d}y=0$ using (5.13), we get the $y$ -location of the minimum $\unicode[STIX]{x1D6E4}_{33}$ , i.e.

(5.14) $$\begin{eqnarray}y_{33min}\approx (1+r_{core}^{2})(2\unicode[STIX]{x1D6FE})^{2}\approx 0.021,\end{eqnarray}$$

and the corresponding minimum value $\unicode[STIX]{x1D6E4}_{33min}$ is

(5.15) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{33min}\approx 2\unicode[STIX]{x1D6FE}\left(1+2\unicode[STIX]{x1D6FE}\right)y_{buf}^{+2\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x1D705}_{33}^{\prime }/\unicode[STIX]{x1D705})^{2}(1-y_{33min})y_{33min}^{-2\unicode[STIX]{x1D6FE}}\approx 0.412.\end{eqnarray}$$

This calculation shows that physical quantities such as $y_{buf}^{+}$ , $\unicode[STIX]{x1D705}_{33}^{\prime }$ , $\unicode[STIX]{x1D705}$ , $\unicode[STIX]{x1D6FE}$ and $r_{core}$ contribute to the slope of the log law. It is interesting to observe that both $y_{33min}$ and $\unicode[STIX]{x1D6E4}_{33min}$ (i.e. log-law slope) are independent of $Re$ . Note that this $Re$ -independent value is meaningful only in the log layer because of (5.6), which requires that $y_{33min}^{+}>y_{buf}^{+}(=41)$ . This implies that only for $Re_{\unicode[STIX]{x1D70F}}>y_{buf}^{+}/y_{33min}\approx 2000$ does the log-law behaviour of $W_{33}^{+}$ become notable, as the data confirm (see figure 8 b). Figure 7(b) shows that data agree well with our analytic expressions (5.12), (5.14) and (5.15); in particular, a plateau $\unicode[STIX]{x1D6E4}_{33}=0.41\pm 0.04$ appears in figure 7(b) in the range $0.018\leqslant y\leqslant 0.14$ , very close to our prediction of $\unicode[STIX]{x1D6E4}_{33min}=0.412$ . Then, it is demonstrated that (5.6) indeed implies an (locally approximate) log law with its slope $A_{33}=\unicode[STIX]{x1D6E4}_{33min}\approx 0.41$ .

Figure 8.

Validation of (5.3) – solid lines – for spanwise Reynolds stress profiles compared with data. Also included is the prediction of $W_{33}^{+}$ at $Re_{\unicode[STIX]{x1D70F}}=50\,000$ . Note the log profile (1.4) – dotted line in (b) – agreeing well with data and with (5.3); its slope $A_{33}\approx 0.41$ and the additive constant $B_{33}\approx 1.00$ are theoretically determined in (5.16). The departure between data and our theory (i.e. $W_{33}^{+Theory}-W_{33}^{+DNS}$ ) is shown in (c,d), which is bounded within $\pm 0.15$ . For legend, see figure 2.

Furthermore, the additive constant $B_{33}$ can also be estimated by matching (5.6) with (1.4) at the position $y=y_{33min}$ . That is, substituting $y=y_{33\unicode[STIX]{x1D6E4}}=0.021$ into (5.6) yields $W_{33\unicode[STIX]{x1D6E4}}^{+}\approx 2.58$ , and hence the additive constant $B_{33}=W_{33\unicode[STIX]{x1D6E4}}^{+}+A_{33}\ln y_{33\unicode[STIX]{x1D6E4}}$ . Therefore, equation (5.6) theoretically leads to

(5.16a,b ) $$\begin{eqnarray}A_{33}=\unicode[STIX]{x1D6E4}_{33min}\approx 0.41,\quad B_{33}=W_{33\unicode[STIX]{x1D6E4}}^{+}+A_{33}\ln y_{33\unicode[STIX]{x1D6E4}}\approx 1.00,\end{eqnarray}$$

which are very close to the empirical fitting values $A_{33}=0.387$ and $B_{33}\approx 1.08$ by Lee & Moser (Reference Lee and Moser2015). Thus, from (5.6) we obtain a (approximate) log law, hence presenting an alternative derivation of Townsend’s log law.

Figure 8 shows the resulting $W_{33}^{+}$ profiles by (5.2) – using (5.9) – and (5.3), in good agreement with data. The departure is bounded within $\pm 0.15$ . To emphasize, for all the predictions in figures 7 and 8, quite surprisingly, only two new parameters are introduced for $\ell _{33}$ , i.e. $\unicode[STIX]{x1D6FE}=0.07$ in (5.5) (or the peak location $y_{33p}^{+}\approx 46$ ) and $\unicode[STIX]{x1D705}_{33}^{\prime }=0.43$ in (5.8), both fixed for all $Re$ . To our knowledge, this is the first time an analytical $W_{33}^{+}$ profile for the entire flow domain has been obtained. Moreover, the log profile (1.4) with its slope and additive constant determined in (5.16) is also plotted in figure 8(b), which agrees not only with DNS data, but also with (5.3). A prediction at $Re_{\unicode[STIX]{x1D70F}}=50\,000$ is included in the figure, awaiting future measurements for validation.

6.1 Inner profile of $W_{11}^{+}$

As in the preceding sections, we let $c_{11}=c_{11}^{\prime }/\sqrt{1+r_{core}^{2}}$ in (2.32) so that $\ell _{11}^{+}\approx c_{11}^{\prime }y^{+}$ as $y^{+}\rightarrow 0$ , and let $c_{11}^{\prime }=\unicode[STIX]{x1D705}_{11}\sqrt{y_{sub}^{+}/y_{buf}^{+}}$ so that $\ell _{11}^{+}\approx \unicode[STIX]{x1D705}_{11}y^{+}$ as $y^{+}\gg y_{buf}^{+}$ . Since (3.4) shows that $\ell _{12}^{+}\approx (y^{+}/y_{sub}^{+})^{3/2}$ , we have $W_{12}^{+}\approx 1-2/[1+\sqrt{1+4(y^{+}/y_{sub}^{+})^{3}}]$ (where $y^{+}/Re_{\unicode[STIX]{x1D70F}}$ is small and ignored). Thus, $W_{11}^{+}$ for the inner flow region is approximated as

(6.2) $$\begin{eqnarray}\displaystyle W_{11Inner}^{+}=W_{12}^{+}\ell _{11}^{+2}/\ell _{12}^{+2}\approx \frac{22.3\unicode[STIX]{x1D705}_{11}^{2}}{y^{\ast }}\left[1-\frac{2}{1+\sqrt{1+4(y^{\ast })^{3}}}\right], & & \displaystyle\end{eqnarray}$$

where $y_{sub}^{+3}/y_{buf}^{+}\approx 22.3$ and $y^{\ast }\equiv y^{+}/y_{sub}^{+}=y^{+}/9.7$ have been substituted into (6.2). Solving $\text{d}W_{11}^{+}/\text{d}y^{\ast }=0$ using (6.2), the inner peak location $y_{c}^{\ast }$ is then obtained:

(6.3) $$\begin{eqnarray}\displaystyle y_{c}^{\ast }\approx \sqrt[3]{2}, & & \displaystyle\end{eqnarray}$$

and hence $y_{11pi}^{+}=\sqrt[3]{2}y_{sub}^{+}$ (scaled in viscous units). This estimation of $y_{11pi}^{+}$ , coinciding with the location of maximum production in (3.5), is independent of $\unicode[STIX]{x1D705}_{11}$ and close to the value $y^{+}\approx 14$ observed from data (Pope Reference Pope2000; Lee & Moser Reference Lee and Moser2015).

Moreover, the inner peak value is found to be determined by $\unicode[STIX]{x1D705}_{11}$ by substituting (6.3) into (6.2), i.e.

(6.4a,b ) $$\begin{eqnarray}\displaystyle W_{11pi}^{+}\approx 8.8\unicode[STIX]{x1D705}_{11}^{2},\quad \text{or}\quad \unicode[STIX]{x1D705}_{11}=\sqrt{W_{11pi}^{+}/8.8}. & & \displaystyle\end{eqnarray}$$

Thus, $\unicode[STIX]{x1D705}_{11}$ can be obtained from the measured inner peak value of $W_{11pi}^{+}$ . Figure 10(a) shows the measured $\unicode[STIX]{x1D705}_{11}$ based on the DNS data and the Princeton pipe data (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012). It seems that $\unicode[STIX]{x1D705}_{11}$ saturates to a constant value around 1.02 (due to $W_{11pi}^{+}\approx 9.2$ ) for $Re_{\unicode[STIX]{x1D70F}}\gtrsim 5000$ of the Princeton pipe. However, if $W_{11pi}^{+}$ continues to grow with increasing $Re$ , as that observed in TBL (Degraaff & Eaton Reference Degraaff and Eaton2000; Metzger et al. Reference Metzger, Klewicki, Bradshaw and Sadr2001; Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009; Sillero et al. Reference Sillero, Jimenez and Moser2013), so also will $\unicode[STIX]{x1D705}_{11}$ – according to (6.4). Issues related to this saturation are further discussed below.

Note that current channel flow DNS data show a logarithmic growth of $W_{11pi}^{+}$ as $Re_{\unicode[STIX]{x1D70F}}$ increases (Lozano-Duran & Jimenez Reference Lozano-Duran and Jimenez2014; Lee & Moser Reference Lee and Moser2015), which was also observed in TBL (Klewicki & Falco Reference Klewicki and Falco1990; Degraaff & Eaton Reference Degraaff and Eaton2000; Metzger et al. Reference Metzger, Klewicki, Bradshaw and Sadr2001; Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009; Sillero et al. Reference Sillero, Jimenez and Moser2013; Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015). Using the empirical formula $W_{11pi}^{+}=3.63+0.65\ln Re_{\unicode[STIX]{x1D70F}}$ by Lozano-Duran & Jimenez (Reference Lozano-Duran and Jimenez2014) or $3.66+0.642\ln Re_{\unicode[STIX]{x1D70F}}$ by Lee & Moser (Reference Lee and Moser2015), the resultant $\unicode[STIX]{x1D705}_{11}$ from (6.4) is indicated by the solid line in figure 10(a), agreeing well with data at moderate $Re$ . However, for $Re_{\unicode[STIX]{x1D70F}}>5000$ , Princeton pipe data (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012) shows an invariant inner peak $W_{11pi}^{+}\approx 9.2$ , and hence the saturated $\unicode[STIX]{x1D705}_{11}\approx 1.02$ . This saturation is in contrast to Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017), where they report that the inner peak $W_{11pi}^{+}$ keeps increasing in their $Re$ range for $Re_{\unicode[STIX]{x1D70F}}$ from 5000 to 40 000, and $W_{11pi}^{+}\approx 9.8$ at $Re_{\unicode[STIX]{x1D70F}}=40\,000$ . The growth of the inner peak at finite $Re_{\unicode[STIX]{x1D70F}}$ (up to 20 000) is attributed to the inner–outer interaction; see Marusic, Baars & Hutchins (Reference Marusic, Baars and Hutchins2017) for more details.

Figure 10.

(a) Measurement of $\unicode[STIX]{x1D705}_{11}$ by the inner peak value $W_{11pi}^{+}$ using (6.4). Dashed line denotes $\unicode[STIX]{x1D705}_{11}=1.02$ resulting from $W_{11pi}^{+}=9.2$ for Princeton pipe; solid line results from the empirical observation $W_{11pi}^{+}=3.66+0.642\ln Re_{\unicode[STIX]{x1D70F}}$ (Lee & Moser Reference Lee and Moser2015). Data: hexagon – Princeton pipe (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012); other symbols are the same as in figure 3 for DNS data. (b) Predictions of outer peak location $y_{11po}^{+}$ (left axis) using (6.13a ) – dashed line – and of peak value $W_{11po}^{+}$ (right axis) using (6.13b ) – solid line; circles are Princeton pipe data and triangles are CICLoPE data; open – $y_{11po}^{+}$ , solid – $W_{11po}^{+}$ .

Here, to match Princeton pipe data (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012), $\unicode[STIX]{x1D705}_{11}=1.02$ is adopted for $Re_{\unicode[STIX]{x1D70F}}>5000$ . As shown in figure 11(a), the inner profile given by (6.2) agrees well with high- $Re$ pipe data, where $\unicode[STIX]{x1D705}_{11}=1.02$ is the only new parameter. Moreover, figure 11(b) shows the similar comparison for the CICLoPE data, where $\unicode[STIX]{x1D705}_{11}=1.04$ is adopted for the current $Re$ values (the inner profiles for $Re_{\unicode[STIX]{x1D70F}}$ above 20 000 are not shown because of data uncertainty on the peak value). Note that in figure 11(b), $y^{\ast }=y^{+}/12$ (instead of $y^{\ast }=y^{+}/9.7$ ) is redefined in (6.2) – to match a larger $y_{11pi}^{+}\approx 15{-}16$ of the CICLoPE data. One can check that the inner peak is still located at (6.3), and hence the invariance of (6.4); but the new definition of $y^{\ast }$ now leads to $y_{11pi}^{+}=12y_{c}^{\ast }\approx 15$ , hence matching the CICLoPE data. A slight difference between Princeton pipe and CICLoPE data is also found in the outer flow region, which is addressed below.

6.2 Outer profile with the mesolayer modification

Note that according to the canonical four-layer descriptions (2.32) and (2.31), $W_{11}^{+}$ should exhibit a constant plateau in the log layer, namely

(6.5) $$\begin{eqnarray}W_{11}^{+}(r)=W_{12}^{+}(\ell _{11}^{+}/\ell _{12}^{+})^{2}\approx \unicode[STIX]{x1D705}_{11}^{2}/\unicode[STIX]{x1D705}^{2}.\end{eqnarray}$$

However, this contrasts the recently observed outer peak (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Willert et al. Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017). To characterize this peak, a mesolayer model was introduced by us (Chen et al. Reference Chen, Wei, Hussain and She2015) based on an anomalous scaling consideration, which further leads to a notable improvement of the $k-\unicode[STIX]{x1D714}$ model (Wilcox Reference Wilcox2006) in predicting the entire $W_{11}^{+}$ profile (Chen, Hussain & She Reference Chen, Hussain and She2016a ). Note that the mesolayer concept has been proposed before (Afzal Reference Afzal1982; Wosnik et al. Reference Wosnik, Castillo and George2000; Hultmark Reference Hultmark2012), and we define it as the region above the buffer layer and ending at the peak location of the Reynolds shear stress (i.e. $y_{12p}^{+}$ in (3.9)).

Figure 11.

Inner profile by (6.2) – dash-dotted line – and the outer profile by (6.9) – solid line – compared with high $Re$ data in (a) Princeton pipe (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012) and (b) CICLoPE (Willert et al. Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017). Note that $y^{\ast }$ in (6.2), and $c$ and $\unicode[STIX]{x1D6FE}_{m}$ in (6.9) are slightly different between Princeton and CICLoPE datasets. Also included in (a) are log profiles of (1.3) – dashed lines – with $A_{11}=1.25$ and $B_{11}=1.63$ theoretically determined in (6.17).

Our definition of mesolayer yields an phenomenological explanation of the critical $Re$ (i.e. $Re_{\unicode[STIX]{x1D70F}}^{c}\approx 10^{4}$ ) above which the outer peak emerges. Note that $y_{12p}^{+}$ was defined as the location of the ‘critical layer’ by Sreenivasan (Reference Sreenivasan1988), and as the demarcation of the inner–outer region by Klewicki et al. (Reference Klewicki, Chin, Blackburn, Ooi and Marusic2012). Accordingly, we consider $y_{12p}^{+}$ ( $=\sqrt{Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705}}$ ) as a typical height of near-wall coherent structures (or wall-attached eddies). As $Re$ increases, $y_{12p}^{+}$ will increase, indicating that such attached eddies move away from wall and finally enter the quasi-balance region (balance between production and dissipation of total kinetic energy) – the near-wall boundary is $y_{Q}^{+}\approx 150$ according to DNS data (Del Alamo & Jimenez Reference Del Alamo and Jimenez2006). Hence, $y_{12p}^{+}=y_{Q}^{+}$ yields a critical $Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D705}y_{Q}^{+2}\approx 10\,100$ . For small $Re$ , these eddies are outside the quasi-balance zone; until $Re_{\unicode[STIX]{x1D70F}}>10\,100$ , they fully enter the quasi-balance zone, which may suppress local dissipation and result in the outer peak of $W_{11}^{+}$ – in analogy to the coherent vortex filaments introducing scaling anomaly in the cascade of isotropic turbulence (She & Leveque Reference She and Leveque1994). It is still a speculation, which may be complementary to the effect of LSM or VLSM (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2013; Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013; Vallikivi et al. Reference Vallikivi, Ganapathisubramani and Smits2015). Also, the non-equilibrium feature of dissipation (Vassilicos Reference Vassilicos2015) in the region of the outer peak is an interesting issue to pursue, as the non-local energy flux from the near-wall production peak may be important. A study of the two-point statistics (such as the velocity structure function) would provide evidence of the complex interplay between the non-local spatial and spectral energy fluxes, which requires more studies involving both the Reynolds stress equations and the spectral transfer equations.

Based on the mesolayer concept, Chen et al. (Reference Chen, Wei, Hussain and She2015) postulate an anomalous scaling model for the ratio between $\ell _{11}$ and $\ell _{12}$ , i.e.

(6.6) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{meso}=c(y^{+})^{\unicode[STIX]{x1D6FE}_{b}}[1+(y^{+}/y_{meso}^{+})^{2}]^{(\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b})/2},\end{eqnarray}$$

where $y_{meso}^{+}=y_{12p}^{+}=\sqrt{Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705}}$ . It readily leads to the outer profile of $W_{11}^{+}$ :

(6.7) $$\begin{eqnarray}\displaystyle W_{11meso}^{+}=W_{12}^{+}\unicode[STIX]{x1D703}_{meso}^{2}=c^{2}r(y^{+})^{2\unicode[STIX]{x1D6FE}_{b}}[1+(y^{+}/y_{meso}^{+})^{2}]^{\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b}}. & & \displaystyle\end{eqnarray}$$

Compared to models for the outer $W_{11}^{+}$ profile by Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015) and Laval et al. (Reference Laval, Vassilicos, Foucaut and Stanislas2017) (with eight parameters), equation (6.7) agrees equally well with data using much fewer parameters (only three). These three are: (1) the scaling exponent $\unicode[STIX]{x1D6FE}_{b}=0.05$ quantifying the growth of $W_{11}^{+}$ before reaching its outer peak; (2) the second exponent $\unicode[STIX]{x1D6FE}_{m}=-0.09$ quantifying the decay of $W_{11}^{+}$ after its outer peak; and (3) the global proportionality coefficient $c$ determining the magnitude of outer profile. The two exponents are believed to be related to intermittent cascade dynamics in the bulk developed (more isotropic) turbulence. Presently, they are measured from experimental data (Chen et al. Reference Chen, Wei, Hussain and She2015), which seem to be invariant with $Re$ – to be further elucidated in the future.

On the other hand, the value of $c$ for each $Re$ is determined by the following procedure. Note that (6.7) implies $W_{11c}^{+}=0$ at the centreline $r=0$ , due to a lack of the centre core layer factor in (6.6), which can be readily rectified by incorporating the core layer asymptote $\ell _{11}/\ell _{12}\propto (1+(r_{core}/r)^{2})^{1/4}$ as $r\rightarrow 0$ from (2.34). This yields

(6.8) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{outer}=c(y^{+})^{\unicode[STIX]{x1D6FE}_{b}}\left(1+\left(\frac{y^{+}}{y_{meso}^{+}}\right)^{2}\right)^{(\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b})/2}\frac{(1+(r_{core}/r)^{2})^{1/4}}{(1+r_{core}^{2})^{1/4}},\end{eqnarray}$$

where the factor $(1+r_{core}^{2})^{1/4}$ guarantees that $\unicode[STIX]{x1D703}_{outer}\rightarrow \unicode[STIX]{x1D703}_{meso}$ as $r\rightarrow 1$ . Therefore, the expression for the entire outer profile of $W_{11}^{+}$ is

(6.9) $$\begin{eqnarray}\displaystyle W_{11Outer}^{+}=W_{12}^{+}\unicode[STIX]{x1D703}_{outer}^{2}=c^{2}(y^{+})^{2\unicode[STIX]{x1D6FE}_{b}}\left(1+\left(\frac{y^{+}}{y_{meso}^{+}}\right)^{2}\right)^{\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b}}\frac{\sqrt{r^{2}+r_{core}^{2}}}{\sqrt{1+r_{core}^{2}}}. & & \displaystyle\end{eqnarray}$$

One can check as $r\rightarrow 1$ , $W_{11Outer}^{+}\rightarrow W_{11meso}^{+}$ (hence (6.9) $\rightarrow$ (6.7)). Moreover, at $r=0$ (i.e. $y^{+}=Re_{\unicode[STIX]{x1D70F}}$ ), we obtain a non-zero centreline value of $W_{11}^{+}$ from (6.9):

(6.10) $$\begin{eqnarray}\displaystyle W_{11c}^{+}=c^{2}Re_{\unicode[STIX]{x1D70F}}^{2\unicode[STIX]{x1D6FE}_{m}}y_{12p}^{+2(\unicode[STIX]{x1D6FE}_{b}-\unicode[STIX]{x1D6FE}_{m})}\frac{r_{core}}{\sqrt{1+r_{core}^{2}}}. & & \displaystyle\end{eqnarray}$$

Now, similar to (5.6), we invoke Townsends’ $Re$ -similarity hypothesis, namely, $W_{c11}^{+}$ is expected to be independent of $Re$ , which yields $c\propto Re_{\unicode[STIX]{x1D70F}}^{-(\unicode[STIX]{x1D6FE}_{b}+\unicode[STIX]{x1D6FE}_{m})/2}$ . Denote $c=c^{\prime }Re_{\unicode[STIX]{x1D70F}}^{-(\unicode[STIX]{x1D6FE}_{b}+\unicode[STIX]{x1D6FE}_{m})/2}$ and substitute this $c$ and $y_{12p}^{+}$ into (6.10), yielding

(6.11) $$\begin{eqnarray}\displaystyle c^{\prime }=W_{c11}^{+1/2}(1+r_{core}^{2})^{1/4}\unicode[STIX]{x1D705}^{\unicode[STIX]{x1D6FE}_{b}-\unicode[STIX]{x1D6FE}_{m}}/\sqrt{r_{core}}. & & \displaystyle\end{eqnarray}$$

Using $r_{core}=0.27$ , $\unicode[STIX]{x1D705}=0.45$ , and the centreline measurement $W_{11c}^{+}\approx 0.86$ from the EXP pipe at $Re_{\unicode[STIX]{x1D70F}}=10\,481$ (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012), we obtain $c^{\prime }\approx 1.72$ , which yields

(6.12) $$\begin{eqnarray}c=c^{\prime }Re_{\unicode[STIX]{x1D70F}}^{-(\unicode[STIX]{x1D6FE}_{b}+\unicode[STIX]{x1D6FE}_{m})/2}\approx 1.72Re_{\unicode[STIX]{x1D70F}}^{0.02}.\end{eqnarray}$$

Figure 11(a) shows that (6.9) agrees well with the high- $Re$ data of the Princeton pipe. Recall the model of Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015) and Laval et al. (Reference Laval, Vassilicos, Foucaut and Stanislas2017) for the outer profile of $W_{11}^{+}$ ; it is built on a match of two piecewise functions which are obtained by the integration of a (presumed) four-layer power-law spectrum (including a $k_{x}^{-1}$ spectrum corresponding to the logarithmic profile by Townsend). However, it is difficult for such a spectral model to locate precisely the log region (particularly for small $Re$ ). In contrast, equation (6.9) describes the simple multilayer structure, needing no spectral description, and involving far fewer parameters. Figure 11 shows that (6.9) characterizes well both the smooth variation of the data across the outer peak and the gradual deviation of the data from the logarithmic distribution near the centreline, which were not described yet in Laval et al. (Reference Laval, Vassilicos, Foucaut and Stanislas2017).

The comparison between (6.9) and the revised CICLoPE data is shown in figure 11(b). Note that only a small part of $W_{11}^{+}$ is measured in the outer flow region of CICLoPE, which differs from the Princeton pipe data slightly. In particular, beyond the outer peak, $W_{11}^{+}$ declines more slowly in CICLoPE, hence suggesting a larger $\unicode[STIX]{x1D6FE}_{m}=-0.07$ (compared to $-0.09$ ) for the four profiles shown in figure 11(b). Moreover, for $Re_{\unicode[STIX]{x1D70F}}=27\,789$ and 39 652, the outer peak value of CICLoPE is slightly larger than the Princeton pipe at similar values of $Re_{\unicode[STIX]{x1D70F}}$ , thus implying a larger magnitude parameter $c$ for these two profiles (i.e. $c=1.75Re_{\unicode[STIX]{x1D70F}}^{0.02}$ ). Except for these two changes, all the other parameters in (6.9) stay the same as for the Princeton pipe, and the resulting profiles agree closely with the CICLoPE data. As a reminder, according to the Princeton data in figure 11(a), $\unicode[STIX]{x1D6FE}_{m}=-0.09$ and $c=1.72Re_{\unicode[STIX]{x1D70F}}^{0.02}$ match well with data at even higher $Re$ (i.e. $Re_{\unicode[STIX]{x1D70F}}=37\,450$ and 98 187). It is thus unknown whether the small changes of $c$ and $\unicode[STIX]{x1D6FE}_{m}$ are physically necessary, or simply due to data uncertainty (an issue that has been emphasized for CICLoPE by Willert et al. (Reference Willert, Soria, Stanislas, Klinner, Amili, Eisfelder, Cuvier, Bellani, Fiorini and Talamelli2017)).

6.3 Scaling of the $W_{11}^{+}$ outer peak

Now, using the three parameters ( $\unicode[STIX]{x1D6FE}_{b}=0.05$ , $\unicode[STIX]{x1D6FE}_{m}=-0.09$ and $c^{\prime }=1.72$ ), let us predict the $Re$ -scaling of the outer peak value $W_{11po}^{+}$ and its $y$ -location $y_{11po}^{+}$ . Solving $\unicode[STIX]{x2202}_{y^{+}}W_{11}^{+}=0$ using (6.9), which actually is $\unicode[STIX]{x2202}_{y^{+}}\unicode[STIX]{x1D703}_{meso}\approx 0$ since $r\approx 1$ , one has the outer peak location:

(6.13a ) $$\begin{eqnarray}y_{11po}^{+}=\sqrt{-\unicode[STIX]{x1D6FE}_{b}/\unicode[STIX]{x1D6FE}_{m}}y_{meso}^{+}\approx 1.1Re_{\unicode[STIX]{x1D70F}}^{1/2}.\end{eqnarray}$$

Moreover, substituting (6.13a ), $y_{meso}^{+}=\sqrt{Re_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D705}}$ and (6.12) into (6.7) yields the outer peak value:

(6.13b ) $$\begin{eqnarray}W_{11po}^{+}={c^{\prime }}^{2}(-\unicode[STIX]{x1D6FE}_{b}/\unicode[STIX]{x1D6FE}_{m})^{\unicode[STIX]{x1D6FE}_{b}}(1-\unicode[STIX]{x1D6FE}_{b}/\unicode[STIX]{x1D6FE}_{m})^{\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b}}Re_{\unicode[STIX]{x1D70F}}^{-\unicode[STIX]{x1D6FE}_{m}}\approx 2.8Re_{\unicode[STIX]{x1D70F}}^{0.09}.\end{eqnarray}$$

Predictions by (6.13a ) and (6.13b ) compared with data are shown in figure 10(b). The agreement is quite good, though the triangles (CICLoPE data) at the two largest $Re$ are a little higher than the circles (Princeton pipe data) – due to the aforementioned larger $\unicode[STIX]{x1D6FE}_{m}$ and $c$ for CICLoPE.

To emphasize, $y_{11po}^{+}\propto Re_{\unicode[STIX]{x1D70F}}^{1/2}$ in (6.13a ) is obtained by the scaling of $y_{meso}^{+}$ , and this scaling is independent of the values of $c^{\prime }$ , $\unicode[STIX]{x1D6FE}_{b}$ and $\unicode[STIX]{x1D6FE}_{m}$ . This is self-consistent with the preceding suggested scenario that $y_{meso}^{+}$ (i.e. typical height of near-wall coherent structures) entering the quasi-balance zone causes the anomalous dissipation, and hence the outer peak. This $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling is also different from the $Re_{\unicode[STIX]{x1D70F}}^{2/3}$ scaling by McKeon & Sharma (Reference McKeon and Sharma2010), where they argue that the outer peak corresponds to a specific mode whose propagation velocity equals the local mean velocity at the position where $U^{+}=2U_{c}^{+}/3$ . Currently, both scalings agree with data (within data uncertainty) in the present $Re$ range, while further higher- $Re$ data are needed to indicate the preferred of the two scalings (note that $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ scaling is better for TBL data of Mathis, Hutchins & Marusic (Reference Mathis, Hutchins and Marusic2009)).

According to (6.13b ), the outer peak will exceed the inner peak – taking the aforementioned constant value 9.2 – at $Re_{\unicode[STIX]{x1D70F}}=(9.2/2.8)^{1/0.09}\approx 5.5\times 10^{5}$ . (However, if the inner peak continues to increase, this $Re_{\unicode[STIX]{x1D70F}}$ will be larger.) Prediction of this specific $Re_{\unicode[STIX]{x1D70F}}$ , approximately six times larger than the maximum $Re_{\unicode[STIX]{x1D70F}}=98\,187$ in the Princeton pipe data (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012), awaits verification when data become available.

6.4 Townsend’s log profile for $W_{11}^{+}$

Another important issue concerns the theoretical explanation of the recently observed logarithmic distribution (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012), i.e. $W_{11}^{+}=-1.25\ln y+1.61$ . As introduced before, previous theories of the log profile build on the $k_{x}^{-1}$ spectrum (Townsend Reference Townsend1976; Perry et al. Reference Perry, Henbest and Chong1986), which is, however, not fully supported by experimental pipe data (Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). Alternatively, Hultmark (Reference Hultmark2012) suggested investigating the diagnostic function $\unicode[STIX]{x1D6E4}_{11}=-yd_{y}W_{11}^{+}$ to check the log profile. Here, similar to (5.12), we can calculate $\unicode[STIX]{x1D6E4}_{11}$ from (6.9), and hence theoretically estimate the values of the log-law slope $A_{11}$ and the additive constant $B_{11}$ , presented below.

Since the log profile region is beyond the outer peak ( $y^{+}\gg y_{meso}^{+}$ and $r\approx 1$ ), equation (6.9) is approximated as

(6.14) $$\begin{eqnarray}W_{11}^{+}={c^{\prime }}^{2}\unicode[STIX]{x1D705}^{\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b}}ry^{2\unicode[STIX]{x1D6FE}_{m}},\end{eqnarray}$$

from which we have

(6.15) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{11}={c^{\prime }}^{2}\unicode[STIX]{x1D705}^{\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b}}y^{2\unicode[STIX]{x1D6FE}_{m}}[1-(1+2\unicode[STIX]{x1D6FE}_{m})r].\end{eqnarray}$$

Again, the value of the approximate plateau of our theoretical $\unicode[STIX]{x1D6E4}_{11}$ can be represented by its minimum value $\unicode[STIX]{x1D6E4}_{11min}$ , which can be obtained by solving $\text{d}\unicode[STIX]{x1D6E4}_{11}/\text{d}y=0$ using (6.15). The results are

(6.16a ) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{11min}={c^{\prime }}^{2}\unicode[STIX]{x1D705}^{\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b}}\left(\frac{2\unicode[STIX]{x1D6FE}_{m}}{1+2\unicode[STIX]{x1D6FE}_{m}}\right)^{1+4\unicode[STIX]{x1D6FE}_{m}}\approx 1.25,\end{eqnarray}$$

which is located at

(6.16b ) $$\begin{eqnarray}y_{11\unicode[STIX]{x1D6E4}}=\left(\frac{2\unicode[STIX]{x1D6FE}_{m}}{1+2\unicode[STIX]{x1D6FE}_{m}}\right)^{2}\approx 0.048.\end{eqnarray}$$

Repeating the calculation for $B_{33}$ in (5.16), the logarithmic additive constant $B_{11}$ can be obtained by matching (6.14) with (1.3) at the specific location $y=y_{11\unicode[STIX]{x1D6E4}}$ . Substituting $y=y_{11\unicode[STIX]{x1D6E4}}\approx 0.048$ into (6.14) yields $W_{11\unicode[STIX]{x1D6E4}}^{+}\approx 5.43$ ; therefore

(6.17a,b ) $$\begin{eqnarray}A_{11}=\unicode[STIX]{x1D6E4}_{11min}\approx 1.25,\quad B_{11}=W_{11\unicode[STIX]{x1D6E4}}^{+}+A_{11}\ln y_{11\unicode[STIX]{x1D6E4}}\approx 1.63,\end{eqnarray}$$

which are very close to the empirical values of $A_{11}=1.25$ and $B_{11}=1.61$ obtained by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012). Figure 11 shows our theoretical log law with parameters in (6.17), in good agreement with both (6.9) and data. Thus, (6.9) leads to a local approximate log law, hence presenting an alternative derivation of the Townsend’s log profile.

6.5 Composite expression for the entire flow

Now, we incorporate $\unicode[STIX]{x1D703}_{outer}$ (or $\unicode[STIX]{x1D703}_{meso}$ ) with $\unicode[STIX]{x1D6FC}_{11}$ (and hence $\ell _{11}$ ) to form a composite expression for the entire flow, without introducing any additional parameter. Since $\unicode[STIX]{x1D6FC}_{11}\propto 1/\sqrt{y^{+}}$ as $y^{+}\rightarrow 0$ , we now modify the $y^{+\unicode[STIX]{x1D6FE}_{b}}$ factor in $\unicode[STIX]{x1D703}_{outer}$ to match the near-wall scaling $1/\sqrt{y^{+}}$ and obtain the following improved $\unicode[STIX]{x1D6FC}_{11}$ (explained below):

(6.18) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{11}^{\prime }=\frac{c_{1}y_{buf}^{+1/2}}{\sqrt{y^{+}}}\left(1+\left(\frac{y^{+}}{y_{B}^{+}}\right)^{4}\right)^{(1/2+\unicode[STIX]{x1D6FE}_{b})/4}\left(1+\left(\frac{y^{+}}{y_{meso}^{+}}\right)^{2}\right)^{(\unicode[STIX]{x1D6FE}_{m}-\unicode[STIX]{x1D6FE}_{b})/2}\left(1+\left(\frac{r_{core}}{r}\right)^{2}\right)^{1/4},\end{eqnarray}$$

and hence

(6.19a,b ) $$\begin{eqnarray}\ell _{11}^{\prime }=\ell _{12}^{+}\unicode[STIX]{x1D6FC}_{11}^{\prime },\quad \text{and}\quad W_{11}^{+}=W_{12}^{+}(\ell _{11}^{\prime }/\ell _{12})^{2}=W_{12}^{+}(\unicode[STIX]{x1D6FC}_{11}^{\prime })^{2}.\end{eqnarray}$$

It can be easily checked that $\unicode[STIX]{x1D6FC}_{11}^{\prime }=\unicode[STIX]{x1D6FC}_{11}$ for $y^{+}\ll y_{B}^{+}$ , and $\unicode[STIX]{x1D6FC}_{11}^{\prime }=\unicode[STIX]{x1D703}_{outer}$ for $y^{+}\gg y_{B}^{+}$ ; note that the second condition implies $c_{1}y_{buf}^{+1/2}/y_{B}^{+(1/2+\unicode[STIX]{x1D6FE}_{b})}=c/(1+r_{core}^{2})^{1/4}$ , which leads to the definition of $y_{B}^{+}$ :

(6.20a ) $$\begin{eqnarray}y_{B}^{+}=[c_{1}y_{buf}^{+1/2}(1+r_{core}^{2})^{1/4}/c]^{2/(1+2\unicode[STIX]{x1D6FE}_{b})}.\end{eqnarray}$$

Substituting the global coefficients $c_{1}=c_{11}/(c_{12}y_{buf}^{+1/2})$ , $c_{11}=\unicode[STIX]{x1D705}_{11}y_{sub}^{+1/2}/[y_{buf}^{+}(1+r_{core}^{2})]^{1/2}$ , $c_{12}=\unicode[STIX]{x1D705}y_{sub}^{+1/2}/[y_{buf}^{+}(1+r_{core}^{2})^{1/4}]$ and $c=c^{\prime }Re_{\unicode[STIX]{x1D70F}}^{-(\unicode[STIX]{x1D6FE}_{m}+\unicode[STIX]{x1D6FE}_{b})/2}$ into (6.20a ) yields

(6.20b ) $$\begin{eqnarray}y_{B}^{+}=[\unicode[STIX]{x1D705}_{11}y_{buf}^{+1/2}/(c^{\prime }\unicode[STIX]{x1D705})]^{2/(1+2\unicode[STIX]{x1D6FE}_{b})}Re_{\unicode[STIX]{x1D70F}}^{(\unicode[STIX]{x1D6FE}_{b}+\unicode[STIX]{x1D6FE}_{m})/(1+2\unicode[STIX]{x1D6FE}_{b})}\approx 48Re_{\unicode[STIX]{x1D70F}}^{-0.0364}.\end{eqnarray}$$

Here, the values of $\unicode[STIX]{x1D705}=0.45$ , $y_{buf}^{+}=41$ , $\unicode[STIX]{x1D705}_{11}=1.02$ , $c^{\prime }=1.72$ , $\unicode[STIX]{x1D6FE}_{b}=0.05$ and $\unicode[STIX]{x1D6FE}_{m}=-0.09$ are used.

Figure 12.

(a) EXP pipe data (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012) denoted by symbols compared with (6.19) denoted by solid lines for $Re_{\unicode[STIX]{x1D70F}}=10\,481$ (downward triangle), 37 450 (leftward triangle) and 98 187 (square) – showing good agreement. Our log profile $W_{11}^{+}=-1.25\ln (y^{+}/Re_{\unicode[STIX]{x1D70F}})+1.63$ from (6.17) is marked in dash-dotted line. Also included are predictions for $Re_{\unicode[STIX]{x1D70F}}=3\times 10^{5}$ and $10^{6}$ , awaiting future data for validation. (b) The logarithmic diagnostic function $\unicode[STIX]{x1D6E4}_{11}=-yd_{y}W_{11}^{+}$ for data (symbol) and using (6.19) (solid line) showing good agreement in the entire flow domain. The dashed line marks the log profile slope $-1.25$ given in (6.17); also included is the prediction of $\unicode[STIX]{x1D6E4}_{11}$ at $Re_{\unicode[STIX]{x1D70F}}=3\times 10^{5}$ .

Substituting (6.20b ) into (6.18) and hence (6.19), the resulting $W_{11}^{+}$ profiles are shown in figure 12(a), in good agreement with data for the entire flow domain with a departure less than 1. Predictions for two larger $Re$ values at $Re_{\unicode[STIX]{x1D70F}}=3\times 10^{5}$ and $10^{6}$ are also plotted in the figure, awaiting future measurement for validation. The diagnostic function $\unicode[STIX]{x1D6E4}_{11}$ for the entire flow domain is also calculated from data and from (6.19), as shown in figure 12(b); the plateau value of the data in the bulk flow is well represented by $\unicode[STIX]{x1D6E4}_{min}=1.25$ in (6.16a ). That is why the log profile with $A_{11}=1.25$ and $B_{11}=1.63$ agrees well with the data in figure 12(a).

Figure 13.

$W_{11}^{+}$ obtained by anomalous scaling modification compared with data of (a) DNS (each profile has been vertically shifted by one unit for better display) and (b) Princeton pipe experiment (EXP) (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012). The departure between data and our theory is shown in (c) (for DNS) and (d) (for EXP) – all bounded within $\pm 1$ . In addition to the DNS channel data in figure 2, DNS pipe data are also included in (a,c) at $Re_{\unicode[STIX]{x1D70F}}=1142$ by Wu & Moin (Reference Wu and Moin2008) (diamonds) and at $Re_{\unicode[STIX]{x1D70F}}=3008$ by Ahn et al. (Reference Ahn, Lee, Lee, Kang and Sung2015) (leftward triangles).

It is interesting to observe that for the two larger $Re$ values ( $Re_{\unicode[STIX]{x1D70F}}=3\times 10^{5}$ and $10^{6}$ ), a more extended logarithmic scaling is predicted by (6.18), and hence (6.19). This is in sharp contrast to the power-law formula by Barenblatt (Reference Barenblatt1993) and Barenblatt & Chorin (Reference Barenblatt and Chorin2004), where their power-law asymptotes to a log law due to a decreasing scaling exponent with increasing $Re$ . Here, (6.18) predicts the (approximate) log profile for $W_{11}^{+}$ based on three $Re$ -invariant parameters, i.e. $c^{\prime }=1.72$ , $\unicode[STIX]{x1D6FE}_{b}=0.05$ and $\unicode[STIX]{x1D6FE}_{m}=-0.09$ . Moreover, equation (6.18) extends the log law to the centreline and to the wall, considerably beyond the previous scaling theories valid for a limited flow domain, hence showing the superiority of (6.18).

For small $Re$ values, using $\unicode[STIX]{x1D705}_{11}$ in figure 10(a) and $y_{B}^{+}$ in (6.20b ), the resulting $W_{11}^{+}$ profiles by (6.19) are shown in figure 13. Note that in this figure, $Re_{\unicode[STIX]{x1D70F}}$ covers over two decades from 650 to 98 187. Compared to the canonical descriptions (figure 9 b), the modified $W_{11}^{+}$ profiles agree much better with data (figure 13 a), although there is a slight mismatch in the buffer layer for small $Re$ (figure 13 c). The agreement can be improved by slightly varying $\unicode[STIX]{x1D6FE}_{m}$ (from $-0.09$ to $-0.07$ ). However, to limit the scope and the size of the paper, a detailed discussion of the finite $Re$ -dependence of the parameters is not considered. Still, the improvement indicates that the anomalous scaling modification, although inspired from large- $Re$ data, also applies to moderate $Re$ . The underpinning physics for the successful modifications of small- $Re$ profiles need further studies.

Remember that (6.13b ) predicts that the outer peak will exceed 9.2 (the presumed saturated inner peak value) at $Re_{\unicode[STIX]{x1D70F}}\approx 5.5\times 10^{5}$ . According to (6.20b ), $y_{B}^{+}$ decreases from approximately 41 to 31 for $Re_{\unicode[STIX]{x1D70F}}$ varying from $10^{2}$ to $10^{5}$ , and $y_{B}^{+}\approx 29$ at $Re_{\unicode[STIX]{x1D70F}}=5.5\times 10^{5}$ . For $Re\rightarrow \infty$ , equation (6.20b ) indicates that $y_{B}^{+}\rightarrow 0$ , which seems to be questionable since $y_{B}^{+}$ should saturate to a non-zero value larger than $y_{sub}^{+}=9.7$ (i.e. above the sublayer thickness). In fact, a non-zero $y_{B}^{+}$ as $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$ indicates that $\unicode[STIX]{x1D6FE}_{b}+\unicode[STIX]{x1D6FE}_{m}\rightarrow 0$ . However, $y_{B}^{+}$ decays very slowly with $Re$ , and it is not until $Re_{\unicode[STIX]{x1D70F}}>10^{20}$ that $y_{B}^{+}$ becomes smaller than $y_{sub}^{+}$ – extremely far away from today’s experiments and simulations. Thus, to re-emphasize, equation (6.20b ) is quite robust for finite $Re$ .