2.1 Anisotropy

The traceless anisotropy tensor $\unicode[STIX]{x1D623}_{ij}$ (Lumley Reference Lumley1978) and its invariants (Rivlin Reference Rivlin1955) $\text{II}_{\boldsymbol{b}}$ and $\text{III}_{\boldsymbol{b}}$ (Gerolymos, Lo & Vallet Reference Gerolymos, Lo and Vallet2012a )

(2.1a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D633}_{ij}:=\overline{u_{i}^{\prime }u_{j}^{\prime }};\quad k:=\frac{1}{2}\overline{u_{\ell }^{\prime }u_{\ell }^{\prime }};\quad \unicode[STIX]{x1D623}_{ij}:=\frac{\overline{u_{i}^{\prime }u_{j}^{\prime }}}{2k}-\frac{1}{3}\unicode[STIX]{x1D6FF}_{ij} & & \displaystyle\end{eqnarray}$$

(2.1d-f ) $$\begin{eqnarray}\displaystyle \text{II}_{\boldsymbol{b}}=-{\textstyle \frac{1}{2}}b_{mk}b_{km};\quad \text{III}_{\boldsymbol{b}}={\textstyle \frac{1}{3}}b_{mk}b_{k\ell }b_{\ell m};\quad A:=1+27\text{III}_{\boldsymbol{b}}+9\text{II}_{\boldsymbol{b}} & & \displaystyle\end{eqnarray}$$

describe the local state of the Reynolds-stress tensor whose locus in the $(\text{III}_{\boldsymbol{b}},-\text{II}_{\boldsymbol{b}})$ -plane lies within Lumley’s (Reference Lumley1978) realisability triangle, determined by the fact (Schumann Reference Schumann1977) that the diagonal components ( $r_{xx}:=\overline{u^{\prime 2}}$ , $r_{yy}:=\overline{v^{\prime 2}}$ , and $r_{zz}:=\overline{w^{\prime 2}}$ ) of the Reynolds-stress tensor $\unicode[STIX]{x1D667}$ are positive in any reference frame ( $\unicode[STIX]{x1D667}$ is positive definite). The simplified form of $\text{II}_{\boldsymbol{b}}$ and $\text{III}_{\boldsymbol{b}}$ in (2.1) takes into account that $\unicode[STIX]{x1D657}$ is traceless ( $\text{I}_{\boldsymbol{b}}=\text{tr}\unicode[STIX]{x1D657}\stackrel{(2.1)}{=}0$ ). In (2.1) $A\in [0,1]$ is Lumley’s (Reference Lumley1978) flatness parameter which (Lumley Reference Lumley1978) tends to zero at the two-component limit (Craft & Launder Reference Craft and Launder2001, TCL), i.e. in the wall-turbulence case (Launder & Shima Reference Launder and Shima1989) at the wall (inversely $A\rightarrow 0$ implies TCL).

The mathematical proof (Lumley Reference Lumley1978, pp. 138–140) that the invariants $\{\text{III}_{\boldsymbol{b}},-\text{II}_{\boldsymbol{b}}\}$ (2.1d–f ) must lie within the realizability triangle only uses the condition that the diagonal components of the symmetric Reynolds-stress tensor $\unicode[STIX]{x1D667}$ ( $\unicode[STIX]{x1D633}_{ij}:=\overline{u_{i}^{\prime }u_{j}^{\prime }}$ ) are non-negative for any orientation of the axes of coordinates (Schumann Reference Schumann1977) and hence also for the frame of the principal axes where the symmetric tensor $\unicode[STIX]{x1D667}$ is diagonalized (Aris Reference Aris1962, pp. 25–28) with diagonal components its real eigenvalues. Since the eigenvalues of $\unicode[STIX]{x1D667}$ are real and non-negative, $\unicode[STIX]{x1D667}$ is positive definite (Stewart Reference Stewart1998, Theorem 2.3, p. 186). Therefore, the realizability triangle applies to the anisotropy invariants of any positive-definite symmetric order-two tensor (equivalently any symmetric order-two tensor whose diagonal elements are non-negative for every orientation of the axes of coordinates). From definition (1.2a ), the dissipation tensor $\unicode[STIX]{x1D73A}$ is also positive definite because its diagonal components ( $\unicode[STIX]{x1D700}_{xx}:=2\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x2202}_{x_{\ell }}u^{\prime }\unicode[STIX]{x2202}_{x_{\ell }}u^{\prime }}$ , $\unicode[STIX]{x1D700}_{yy}:=2\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x2202}_{x_{\ell }}v^{\prime }\unicode[STIX]{x2202}_{x_{\ell }}v^{\prime }}$ , and $\unicode[STIX]{x1D700}_{zz}:=2\unicode[STIX]{x1D708}\overline{\unicode[STIX]{x2202}_{x_{\ell }}w^{\prime }\unicode[STIX]{x2202}_{x_{\ell }}w^{\prime }}$ ) are always positive. Therefore, the corresponding traceless anisotropy tensor  $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ and its invariants (Rivlin Reference Rivlin1955) $\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ and $\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$

(2.2a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{ij}\stackrel{(1.2a)}{:=}2\unicode[STIX]{x1D708}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}};\quad \unicode[STIX]{x1D700}\stackrel{(1.2b)}{:=}\frac{1}{2}\unicode[STIX]{x1D700}_{mm};\quad \unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}:=\frac{\unicode[STIX]{x1D700}_{ij}}{2\unicode[STIX]{x1D700}}-\frac{1}{3}\unicode[STIX]{x1D6FF}_{ij} & & \displaystyle\end{eqnarray}$$

(2.2d-f ) $$\begin{eqnarray}\displaystyle \text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}=-{\textstyle \frac{1}{2}}b_{\unicode[STIX]{x1D700}_{mk}}b_{\unicode[STIX]{x1D700}_{km}};\quad \text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}={\textstyle \frac{1}{3}}b_{\unicode[STIX]{x1D700}_{mk}}b_{\unicode[STIX]{x1D700}_{k\ell }}b_{\unicode[STIX]{x1D700}_{\ell m}};\quad A_{\unicode[STIX]{x1D700}}:=1+27\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}+9\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}} & & \displaystyle\end{eqnarray}$$

define the same realizability triangle as $\unicode[STIX]{x1D657}$ (Lumley Reference Lumley1978; Simonsen & Krogstad Reference Simonsen and Krogstad2005). This was first recognized by Lee & Reynolds (Reference Lee, Reynolds, Durst, Launder, Lumley, Schmidt and Whitelaw1987, p. 56), who also state regarding the invariants (2.1d–f ) and (2.2d–f ) that ‘each AIM (anisotropy invariant mapping) has the boundaries first defined by Lumley & Newman (Reference Lumley and Newman1977) for the Reynolds-stress AIM’.

Since isotropy of $\unicode[STIX]{x1D633}_{ij}$ ( $\unicode[STIX]{x1D700}_{ij}$ ) corresponds to $\unicode[STIX]{x1D657}=0$ ( $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}=0$ ), the larger the distance of the components $\unicode[STIX]{x1D623}_{ij}$ ( $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ ) from 0 the higher the anisotropy (Simonsen & Krogstad Reference Simonsen and Krogstad2005).

Table 1.

Asymptotic (as $y^{+}\,\rightarrow \,0$ ) expansions of $\unicode[STIX]{x1D657}$ (2.1a–c ) and of its invariants (2.1d–f ), for general inhomogeneous incompressible (3.2a ) turbulent flow near a plane no-slip (A 1 and A 2)  $xz$ -wall (terms within square brackets $[\cdots ]$ are three-dimensional (3-D) terms which are identically $=0$ for 2-D in the mean flow).

Table 2.

Asymptotic (as $y^{+}\rightarrow 0$ ) expansions of $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ (2.2a–c ) and of its invariants (2.2d–f ), for general inhomogeneous incompressible (3.2a ) turbulent flow near a plane no-slip (A 1 and A 2)  $xz$ -wall (terms within square brackets $[\cdots ]$ are 3-D terms which are identically $=0$ for 2-D in the mean flow).

2.2 Wall asymptotics

Following the asymptotic (truncated Taylor-series) expansions (A 1), the velocity components in the vicinity of the wall are expanded as (Mansour et al. Reference Mansour, Kim and Moin1988)

(2.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle u^{\prime +}\underset{y^{+}\rightarrow 0}{{\sim}}A_{u}^{\prime +}(x^{+},z^{+},t^{+})y^{+}+B_{u}^{\prime +}(x^{+},z^{+},t^{+}){y^{+}}^{2}+C_{u}^{\prime +}(x^{+},z^{+},t^{+}){y^{+}}^{3}+\cdots & \displaystyle\end{eqnarray}$$

(2.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle v^{\prime +}\underset{y^{+}\rightarrow 0}{{\sim}}B_{v}^{\prime +}(x^{+},z^{+},t^{+}){y^{+}}^{2}+C_{v}^{\prime +}(x^{+},z^{+},t^{+}){y^{+}}^{3}+\cdots & \displaystyle\end{eqnarray}$$

(2.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle w^{\prime +}\underset{y^{+}\rightarrow 0}{{\sim}}A_{w}^{\prime +}(x^{+},z^{+},t^{+})y^{+}+B_{w}^{\prime +}(x^{+},z^{+},t^{+}){y^{+}}^{2}+C_{w}^{\prime +}(x^{+},z^{+},t^{+}){y^{+}}^{3}+\cdots & \displaystyle\end{eqnarray}$$

$u_{i}^{\prime }(x,y^{+}=0,z,t)=0$

$\lim _{y^{+}\rightarrow 0}(\unicode[STIX]{x2202}_{y^{+}}v^{\prime +})=0\stackrel{(\text{A}\,1)}{\;\Longrightarrow \;}A_{v}^{\prime +}=0$

$v^{\prime +}\propto {y^{+}}^{2}$

$y^{+}\rightarrow 0$

$xyz$

$\unicode[STIX]{x1D633}_{ij}^{+}$

$\unicode[STIX]{x1D700}_{ij}^{+}$

$\unicode[STIX]{x1D623}_{ij}$

$\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$

$\unicode[STIX]{x1D623}_{ij}$

$\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$

$2$

(2.4a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}\underset{y^{+}\rightarrow 0}{{\sim}}(\unicode[STIX]{x1D623}_{ij})_{w}+2\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D623}_{ij}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}y^{+}+O({y^{+}}^{2}), & & \displaystyle\end{eqnarray}$$

with fundamental differences occurring in the $O({y^{+}}^{2})$ terms, those of $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ containing the spatial gradients $(\unicode[STIX]{x1D735}A_{u}^{\prime })^{+}$ and $(\unicode[STIX]{x1D735}A_{w}^{\prime })^{+}$ , contrary to those of $\unicode[STIX]{x1D623}_{ij}$ . Concerning the wall-normal diagonal components, $b_{yy}$ and $b_{\unicode[STIX]{x1D700}_{yy}}$ , whose linear $O(y^{+})$ terms are $=0$ , notice that (tables 1 and 2) the wall-normal two-derivatives are proportional with a factor 4

(2.4b ) $$\begin{eqnarray}\displaystyle b_{\unicode[STIX]{x1D700}_{yy}}\underset{y^{+}\rightarrow 0}{{\sim}}(b_{yy})_{w}+4\left(\frac{1}{2}\frac{\unicode[STIX]{x2202}^{2}b_{yy}}{\unicode[STIX]{x2202}{y^{+}}^{2}}\right)_{w}{y^{+}}^{2}+O({y^{+}}^{3}). & & \displaystyle\end{eqnarray}$$

Relation (2.4a ) carries over (tables 1 and 2) to the anisotropy invariants (2.1d–f ) and (2.2d–f )

(2.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}\underset{y^{+}\rightarrow 0}{{\sim}}(\text{II}_{\boldsymbol{b}})_{w}+2\left(\frac{\unicode[STIX]{x2202}\text{II}_{\boldsymbol{b}}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}y^{+}+O({y^{+}}^{2}) & \displaystyle\end{eqnarray}$$

(2.4d ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}\underset{y^{+}\rightarrow 0}{{\sim}}(\text{III}_{\boldsymbol{b}})_{w}+2\left(\frac{\unicode[STIX]{x2202}\text{III}_{\boldsymbol{b}}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}y^{+}+O({y^{+}}^{2}). & \displaystyle\end{eqnarray}$$

Consistent with the 2-C componentality of both $\unicode[STIX]{x1D633}_{ij}$ (2.1a–c ) and $\unicode[STIX]{x1D700}_{ij}$ (2.2a–c ) at the wall (Launder & Shima Reference Launder and Shima1989; Hanjalić Reference Hanjalić1994; Craft & Launder Reference Craft and Launder2001), the wall values of the corresponding Lumley flatness parameters (2.1d–f ) and (2.2d–f ) $A_{w}=(A_{\unicode[STIX]{x1D700}})_{w}=0$ (Lumley Reference Lumley1978; Simonsen & Krogstad Reference Simonsen and Krogstad2005). Furthermore, both $A$ and $A_{\unicode[STIX]{x1D700}}$ tend to $0$ quadratically as $y^{+}\rightarrow 0$ , and their wall-normal two-derivatives are proportional by a factor 4 (tables 1 and 2)

(2.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle A\underset{y^{+}\rightarrow 0}{{\sim}}O({y^{+}}^{2}) & \displaystyle\end{eqnarray}$$

(2.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle A_{\unicode[STIX]{x1D700}}\underset{y^{+}\rightarrow 0}{{\sim}}4\left(\frac{1}{2}\frac{\unicode[STIX]{x2202}^{2}A}{\unicode[STIX]{x2202}{y^{+}}^{2}}\right)_{w}{y^{+}}^{2}+O({y^{+}}^{3}) & \displaystyle\end{eqnarray}$$

because, as can be verified by straightforward calculation, the constant and linear terms in the expansion of the invariants $\{\text{II}_{\boldsymbol{b}},\text{III}_{\boldsymbol{b}},\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}\}$ (tables 1 and 2), cancel out

(2.5c ) $$\begin{eqnarray}\displaystyle & \displaystyle 9(\text{II}_{\boldsymbol{b}})_{w}+27(\text{III}_{\boldsymbol{b}})_{w}=9(\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}})_{w}+27(\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}})_{w}=-1 & \displaystyle\end{eqnarray}$$

(2.5d ) $$\begin{eqnarray}\displaystyle & \displaystyle 9\left(\frac{\unicode[STIX]{x2202}\text{II}_{\boldsymbol{b}}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}+27\left(\frac{\unicode[STIX]{x2202}\text{III}_{\boldsymbol{b}}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}=9\left(\frac{\unicode[STIX]{x2202}\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}+27\left(\frac{\unicode[STIX]{x2202}\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}}{\unicode[STIX]{x2202}y^{+}}\right)_{w}=0. & \displaystyle\end{eqnarray}$$

Of course, calculations retaining the $O(y^{+2})$ terms in the expansions of the invariants were required to obtain the result (2.5d ).

Regarding $\unicode[STIX]{x1D623}_{ij}$ (table 1), the new results in the paper, with reference to Hanjalić (Reference Hanjalić1994, Tab. 3, p. 191) are the asymptotic expansions of the invariants $\text{II}_{\boldsymbol{b}}$ and $\text{III}_{\boldsymbol{b}}$ , and in particular of the flatness parameter $A$ . Corresponding expansions were obtained (table 2) for $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ and for its invariants ( $\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ , $\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ , $A_{\unicode[STIX]{x1D700}}$ ). With regard to the relations between $\unicode[STIX]{x1D623}_{ij}$ and $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ , Mansour et al. (Reference Mansour, Kim and Moin1988, p. 32) have ‘point(ed) out that close to the wall, Taylor series expansions of $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ and $\unicode[STIX]{x1D623}_{ij}$ show that they are equal only up to $O(y^{+})$ ’; this remark clearly implies $(\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}})_{w}=(\unicode[STIX]{x1D623}_{ij})_{w}$ . The present results (2.4) and (2.5) show the proportionality relation $(\unicode[STIX]{x2202}_{y^{+}}\unicode[STIX]{x1D623}_{ij})_{w}\stackrel{(2.4a)}{=}((\unicode[STIX]{x2202}_{y^{+}}\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}})_{w})/2$ of the wall-normal gradients and the corresponding relation between the wall-normal gradients of the invariants $(\unicode[STIX]{x2202}_{y^{+}}\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}})_{w}\stackrel{(2.4c)}{=}2(\unicode[STIX]{x2202}_{y^{+}}\text{II}_{\boldsymbol{b}})_{w}$ and $(\unicode[STIX]{x2202}_{y^{+}}\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}})_{w}\stackrel{(2.4d)}{=}2(\unicode[STIX]{x2202}_{y^{+}}\text{III}_{\boldsymbol{b}})_{w}$ on the one hand, and the quadratic asymptotic behaviour of the flatness parameters $A_{\unicode[STIX]{x1D700}}\stackrel{(2.5)}{\underset{y^{+}\rightarrow 0}{{\sim}}}4A+O(y^{+3})\stackrel{(2.5a)}{\underset{y^{+}\rightarrow 0}{\propto }}O(y^{+2})$ on the other hand.

2.3 $Re_{\unicode[STIX]{x1D70F}_{w}}$ influence on anisotropy

The evolution of mean flow and turbulence structure with $Re_{\unicode[STIX]{x1D70F}_{w}}$ (A 3k ) is central in wall-turbulence research (Marusic et al. Reference Marusic, Mckeon, Monkewitz, Nagib, Smits and Sreenivasan2010, Reference Marusic, Monty, Hultmark and Smits2013; Kim Reference Kim2012) and DNS of turbulent plane channel flow at progressively higher $Re_{\unicode[STIX]{x1D70F}_{w}}$ (Moser et al. Reference Moser, Kim and Mansour1999; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lee & Moser Reference Lee and Moser2015) contribute towards answering several open questions on very high $Re$ asymptotics. Turbulence structure is generally represented by non-dimensional ratios of turbulent quantities (Bradshaw Reference Bradshaw1967). In this respect, the anisotropy tensors $\unicode[STIX]{x1D623}_{ij}$ (2.1a–c ) and $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ (2.2a–c ), and their AIMs (Lumley & Newman Reference Lumley and Newman1977; Lumley Reference Lumley1978; Lee & Reynolds Reference Lee, Reynolds, Durst, Launder, Lumley, Schmidt and Whitelaw1987; Simonsen & Krogstad Reference Simonsen and Krogstad2005) are particularly useful in understanding the differences in behaviour between the Reynolds stresses $\unicode[STIX]{x1D633}_{ij}$ (2.1a–c ) and their dissipation rates $\unicode[STIX]{x1D700}_{ij}$ (2.2a–c ), across the channel (in the wall-normal direction $y$ ) and with varying Reynolds number $Re_{\unicode[STIX]{x1D70F}_{w}}$ (A 3k ).

Figure 1.

Anisotropy (2.1, 2.2) of the normal ( $b_{xx}$ , $b_{yy}$ , $b_{zz}$ ) and shear ( $b_{xy}$ ) Reynolds stresses, and of the corresponding dissipation rates ( $b_{\unicode[STIX]{x1D700}_{xx}}$ , $b_{\unicode[STIX]{x1D700}_{yy}}$ , $b_{\unicode[STIX]{x1D700}_{zz}}$ , $b_{\unicode[STIX]{x1D700}_{xy}}$ ), from existing (Moser et al. Reference Moser, Kim and Mansour1999; del Álamo & Jiménez Reference del Álamo and Jiménez2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Lee & Moser Reference Lee and Moser2015) DNS computations of turbulent plane channel flow, in the range $Re_{\unicode[STIX]{x1D70F}_{w}}\in [180,5200]$ , plotted against the inner-scaled (A 3e–g ) wall distance $y^{+}$ (log scale and linear wall zoom).

Figure 2.

Near-wall peaks of the streamwise ( $\max _{y^{+}}b_{xx}$ , $\max _{y^{+}}b_{\unicode[STIX]{x1D700}_{xx}}$ ) and spanwise ( $\min _{y^{+}}b_{zz}$ , $\min _{y^{+}}b_{\unicode[STIX]{x1D700}_{zz}}$ ) components of the anisotropy tensors of the Reynolds stresses $\unicode[STIX]{x1D657}$ (2.1) and of the corresponding dissipation rates $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ (2.2), as a function of the friction Reynolds number (A 3k )  $Re_{\unicode[STIX]{x1D70F}_{w}}$ (log scale) and of its inverse $Re_{\unicode[STIX]{x1D70F}_{w}}^{-1}$ (linear scale), from existing (Moser et al. Reference Moser, Kim and Mansour1999; Hu et al. Reference Hu, Morfey and Sandham2006; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Vreman & Kuerten Reference Vreman and Kuerten2014b , Reference Vreman and Kuerten2016; Lee & Moser Reference Lee and Moser2015) DNS computations of turbulent plane channel flow in the range $Re_{\unicode[STIX]{x1D70F}_{w}}\in [100,5200]$ .

2.3.1 Anisotropy tensors

The strong anisotropy of the Reynolds stresses in wall turbulence is related to the anisotropy of the dissipation-rate tensor $\unicode[STIX]{x1D700}_{ij}$ (Gerolymos et al. Reference Gerolymos, Lo, Vallet and Younis2012b ), and it is established that the behaviour of the anisotropy tensor of the Reynolds stresses $\unicode[STIX]{x1D623}_{ij}$ differs from that of the anisotropy tensor of the dissipation rate $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ (Launder & Reynolds Reference Launder and Reynolds1983; Lai & So Reference Lai and So1990; Jovanović et al. Reference Jovanović, Ye and Durst1995; Jakirlić & Hanjalić Reference Jakirlić and Hanjalić2002). This is obvious by examining (figure 1) existing (Moser et al. Reference Moser, Kim and Mansour1999; del Álamo & Jiménez Reference del Álamo and Jiménez2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Lee & Moser Reference Lee and Moser2015) DNS results which include budgets of $\unicode[STIX]{x1D633}_{ij}$ transport (and hence data for the components of $\unicode[STIX]{x1D700}_{ij}$ ) covering the range $Re_{\unicode[STIX]{x1D70F}_{w}}\in [178,5186]$ . Recall that (§ 2.2), at the wall ( $y^{+}=0$ ), the anisotropy tensors $\unicode[STIX]{x1D623}_{ij}$ and $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ are identical (Mansour et al. Reference Mansour, Kim and Moin1988, p. 32), but the wall-normal gradient of $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ is exactly twice that of $\unicode[STIX]{x1D623}_{ij}$ (2.4a ). This explains the near-wall ( $y^{+}<10$ ) evolution of the streamwise ( $b_{xx}$ , $b_{\unicode[STIX]{x1D700}_{xx}}$ ) and spanwise ( $b_{zz}$ , $b_{\unicode[STIX]{x1D700}_{zz}}$ ) components of the anisotropy tensors (figure 1), viz. that $\{b_{\unicode[STIX]{x1D700}_{xx}},|b_{\unicode[STIX]{x1D700}_{zz}}|\}$ increase much faster with $y^{+}$ , and reach their maxima at $y^{+}\in [3,4]$ , nearer the wall compared to $\{b_{xx},|b_{zz}|\}$ which reach their maxima at $y^{+}\in [7,9]$ . These near-wall maxima of $\{b_{xx},|b_{zz}|,b_{\unicode[STIX]{x1D700}_{xx}},|b_{\unicode[STIX]{x1D700}_{zz}}|\}$ are also global maxima across the channel (figure 1). Notice that at fixed $Re_{\unicode[STIX]{x1D70F}_{w}}$ the maximum of the spanwise component occurs at slightly higher $y^{+}$ than the maximum of the streamwise component, both for $\unicode[STIX]{x1D623}_{ij}$ and $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ (figure 1). Furthermore, at fixed $Re_{\unicode[STIX]{x1D70F}_{w}}$ , the values of the maxima are quite close ( $\max _{y}b_{xx}\approxeq \max _{y}b_{\unicode[STIX]{x1D700}_{xx}}$ and $\max _{y}|b_{zz}|\approxeq \max _{y}|b_{\unicode[STIX]{x1D700}_{zz}}|$ ) but $\unicode[STIX]{x1D700}_{ij}$ is slightly more anisotropic than $\unicode[STIX]{x1D633}_{ij}$ (figure 1). In the near-wall region ( $y^{+}\lessapprox 10$ ), the anisotropy of the streamwise ( $b_{xx}$ , $b_{\unicode[STIX]{x1D700}_{xx}}$ ) and spanwise ( $b_{zz}$ , $b_{\unicode[STIX]{x1D700}_{zz}}$ ) components decreases with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ , first quite sharply ( $Re_{\unicode[STIX]{x1D70F}_{w}}<400$ ) and then more slowly (figure 1). Careful examination of the values of the maxima versus  $Re_{\unicode[STIX]{x1D70F}_{w}}$ (figure 2) suggests that they probably tend to $Re_{\unicode[STIX]{x1D70F}_{w}}$ -independent values as $Re_{\unicode[STIX]{x1D70F}_{w}}$ increases, although these are not reached yet at the highest available $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 5186$ DNS of Lee & Moser (Reference Lee and Moser2015). To alleviate the bias of the computational grid, the values of the extrema were determined by fitting a degree-four interpolating polynomial $p_{I}(y^{+})$ around the on-grid extremum (two neighbours on each side) and solving $p_{I}^{\prime }(y^{+})=0$ by Newton iteration. This generally had no influence on the value of the maximum, but did smooth out variations of its location. Plots (figure 2) of the near-wall peaks of the streamwise ( $\max _{y^{+}}b_{xx}$ , $\max _{y^{+}}b_{\unicode[STIX]{x1D700}_{xx}}$ ) and spanwise ( $\min _{y^{+}}b_{zz}$ , $\min _{y^{+}}b_{\unicode[STIX]{x1D700}_{zz}}$ ) components versus  $Re_{\unicode[STIX]{x1D70F}_{w}}$ (log scale) hint that asymptotic limits are approached, and this is further verified by plotting the data against $Re_{\unicode[STIX]{x1D70F}_{w}}^{-1}$ (figure 2). More sets of data are available for $\unicode[STIX]{x1D633}_{ij}$ than for $\unicode[STIX]{x1D700}_{ij}$ , and two slightly distinct curves appear for the $\unicode[STIX]{x1D623}_{ij}$ -extrema (figure 2), but this only affects the precise value of the asymptotic limits as $Re_{\unicode[STIX]{x1D70F}_{w}}^{-1}\rightarrow 0$ ( ${\sim}2\,\%$ scatter for $\max _{y^{+}}b_{xx}$ and ${\sim}5\,\%$ scatter for $\min _{y^{+}}b_{zz}$ ). The anisotropy of the wall-normal component is of course highest at the 2-C wall $(b_{yy})_{w}=(b_{\unicode[STIX]{x1D700}_{yy}})_{w}=-1/3$ and then becomes less anisotropic with increasing $y^{+}$ (figure 1). Obviously, the coefficient $\unicode[STIX]{x1D700}_{w}^{+-1}\overline{B_{v}^{\prime +2}}$ (tables 1 and 2) of the leading quadratic term in (2.4b ) slightly increases with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ (figure 1). Finally, regarding the shear components, both $b_{xy}$ and $b_{\unicode[STIX]{x1D700}_{xy}}$ seem $Re_{\unicode[STIX]{x1D70F}_{w}}$ independent near the wall ( $y^{+}\lessapprox 10$ ).

In contrast to the near-wall region ( $y^{+}\lessapprox 10$ ), which is dominated by the wall-asymptotic relations (§ 2.2), the behaviour of $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ is completely different from that of $\unicode[STIX]{x1D623}_{ij}$ at higher $y^{+}$ (figure 1). The anisotropy of the streamwise Reynolds-stress component $b_{xx}$ forms, with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ , a plateau corresponding to the log region of the mean velocity profile (Coles Reference Coles1956) followed by a sharp decrease in the wake region. The level of anisotropy $b_{xx}$ in the log region increases with $Re_{\unicode[STIX]{x1D70F}_{w}}$ but may be reaching an asymptotic limit, although simulations at higher $Re_{\unicode[STIX]{x1D70F}_{w}}$ are needed to ascertain this point. The spanwise component $b_{zz}$ becomes more isotropic with increasing $y^{+}$ in the buffer layer ( $y^{+}\in [10,100]$ ), followed by a slight increase in anisotropy in the log region (figure 1), while the wall-normal component $b_{yy}$ becomes more isotropic with increasing $y^{+}$ , quite sharply in the buffer layer and more progressively in the log region. Interestingly, both $b_{yy}$ and $b_{zz}$ also seem to be forming a log-region plateau with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ , but simulations at higher $Re_{\unicode[STIX]{x1D70F}_{w}}$ are required to verify this trend. Both components $b_{yy}$ and $b_{zz}$ become more anisotropic in the log region with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ (figure 1). Finally, in the wake region, both $b_{yy}$ and $b_{zz}$ tend quite sharply to a common centreline value (figure 1).

Contrary to $\unicode[STIX]{x1D623}_{ij}$ , the diagonal components of $\unicode[STIX]{x1D623}_{\unicode[STIX]{x1D700}_{ij}}$ tend quasi-monotonically to a near-isotropic (but not exactly isotropic) state at the centreline, first steeply in the buffer layer and then more gradually in the log region (figure 1). The low- $Re_{\unicode[STIX]{x1D70F}_{w}}$ data exhibit a clear albeit slight increase in the anisotropy of $\{b_{\unicode[STIX]{x1D700}_{xx}},b_{\unicode[STIX]{x1D700}_{yy}},b_{\unicode[STIX]{x1D700}_{zz}}\}$ near the centreline which becomes less pronounced with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ . Unlike $b_{zz}$ , $b_{\unicode[STIX]{x1D700}_{zz}}$ reaches a near-zero value in the log region (figure 1).

The shear components also behave very differently. The Reynolds-stress anisotropy $b_{xy}$ becomes more anisotropic from the wall up to the beginning of the wake region, before sharply going to zero (exactly, because of the symmetry condition $\overline{u^{\prime }v^{\prime }}|_{CL}=0$ ) at the centreline. The shear component $b_{\unicode[STIX]{x1D700}_{xy}}$ is practically $Re_{\unicode[STIX]{x1D70F}_{w}}$ independent up to the end of the buffer layer ( $y^{+}\lessapprox 100$ ), with a wavy shape reaching zero near $y^{+}\approxeq 20$ (figure 1), then being relatively flat in the log region.

Figure 3.

Invariants of the anisotropy tensors (2.1, 2.2) of the Reynolds stresses $\unicode[STIX]{x1D657}$ ( $\text{II}_{\boldsymbol{b}}$ , $\text{III}_{\boldsymbol{b}}$ ) and of the corresponding dissipation rates $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ ( $\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ , $\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ ), and Lumley’s (Reference Lumley1978) flatness parameters ( $A:=1+27\text{III}_{\boldsymbol{b}}+\text{II}_{\boldsymbol{b}}$ and $A_{\unicode[STIX]{x1D700}}:=1+27\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}+\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ ), from existing (Moser et al. Reference Moser, Kim and Mansour1999; del Álamo & Jiménez Reference del Álamo and Jiménez2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Lee & Moser Reference Lee and Moser2015) DNS computations of turbulent plane channel flow, in the range $Re_{\unicode[STIX]{x1D70F}_{w}}\in [180,5200]$ , plotted against the inner-scaled (A 3e–g ) wall distance $y^{+}$ (log scale and linear wall zoom).

2.3.2 Anisotropy invariants

More precise information on the influence of $Re_{\unicode[STIX]{x1D70F}_{w}}$ (A 3k ) on the anisotropy of $\unicode[STIX]{x1D633}_{ij}$ and $\unicode[STIX]{x1D700}_{ij}$ can be obtained by considering the invariants $\{\text{III}_{\boldsymbol{b}},-\text{II}_{\boldsymbol{b}}\}$ (2.1d–f ) and $\{\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},-\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}\}$ (2.2d–f ) whose locus lies inside Lumley’s (Reference Lumley1978) realizability triangle (Mansour et al. Reference Mansour, Kim and Moin1988, figures 17 and 18, p. 32). As the intervals of possible values (Lumley Reference Lumley1978; Simonsen & Krogstad Reference Simonsen and Krogstad2005)

(2.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{II}_{\boldsymbol{b}}\in [-{\textstyle \frac{1}{3}},0]\ni \text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}} & \displaystyle\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{III}_{\boldsymbol{b}}\in [-{\textstyle \frac{1}{108}},{\textstyle \frac{2}{27}}]\ni \text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}} & \displaystyle\end{eqnarray}$$

(2.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle A\in [0,1]\ni A_{\unicode[STIX]{x1D700}} & \displaystyle\end{eqnarray}$$

are rather limited, the invariants are quite sensitive indicators of anisotropy, and this sensitivity is visible in the influence of $Re_{\unicode[STIX]{x1D70F}_{w}}$ on the near-wall peaks of the invariants (figure 3). The evolution of the invariants with $y^{+}$ and $Re_{\unicode[STIX]{x1D70F}_{w}}$ (figure 3) is very similar to what was observed for the diagonal components of the anisotropy tensors (figure 1). Near the wall, $\{\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},-\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}\}$ increase faster with $y^{+}$ compared to $\{\text{III}_{\boldsymbol{b}},-\text{II}_{\boldsymbol{b}}\}$ , in line with (2.4c ) and (2.4d ), reaching their maxima at $y^{+}\in [3,4]$ compared to $y^{+}\in [7,9]$ (figure 3), $\unicode[STIX]{x1D700}_{ij}$ being slightly more anisotropic than $\unicode[STIX]{x1D633}_{ij}$ . Near the wall, $A_{\unicode[STIX]{x1D700}}$ increases much faster with $y^{+}$ compared to $A$ (figure 3), in line with (2.5a ) and (2.5b ). There is noticeable $Re_{\unicode[STIX]{x1D70F}_{w}}$ -influence for $Re_{\unicode[STIX]{x1D70F}_{w}}<400$ both in wall values and in rate of increase with $y^{+}$ for all of the invariants (figure 3). On the other hand, with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ , it would seem that an asymptotic state is approached in the near-wall region, including the buffer layer ( $y^{+}\lessapprox 100$ ), although DNS at higher $Re_{\unicode[STIX]{x1D70F}_{w}}$ are still required to fully substantiate this observation. Regarding the invariants of $\unicode[STIX]{x1D633}_{ij}$ , $\{-\text{II}_{\boldsymbol{b}},\text{III}_{\boldsymbol{b}},A\}$ , a plateau appears with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ (figure 3), marking the log region of the mean velocity profile (Lee & Moser Reference Lee and Moser2015), but again DNS at higher $Re_{\unicode[STIX]{x1D70F}_{w}}$ are needed to determine whether a $Re_{\unicode[STIX]{x1D70F}_{w}}$ -asymptotic value of the level of this plateau exists. On the contrary, the invariants of $\unicode[STIX]{x1D700}_{ij}$ $\{-\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},A_{\unicode[STIX]{x1D700}}\}$ vary monotonically from the wall to centreline, and seem to approach $Re_{\unicode[STIX]{x1D70F}_{w}}$ -independent $y^{+}$ -distributions with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ .

2.3.3 AIM at the wall and at the centreline

The variation with $Re_{\unicode[STIX]{x1D70F}_{w}}$ of the $y^{+}$ -distributions of $\unicode[STIX]{x1D633}_{ij}$ and $\unicode[STIX]{x1D700}_{ij}$ anisotropy (figures 1–3) indicates trends in the evolution of turbulence structure with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ . These trends are better quantified by studying the evolution with $Re_{\unicode[STIX]{x1D70F}_{w}}$ of the anisotropy invariants ( $-\text{II}_{\boldsymbol{b}}$ , $\text{III}_{\boldsymbol{b}}$ , $-\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ , $\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ ) at the wall and at the centreline (figure 4). Several of the available DNS data (Kim et al. Reference Kim, Moin and Moser1987; Moser et al. Reference Moser, Kim and Mansour1999; Hu et al. Reference Hu, Morfey and Sandham2002, Reference Hu, Morfey and Sandham2003, Reference Hu, Morfey and Sandham2006; del Álamo & Jiménez Reference del Álamo and Jiménez2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Vreman & Kuerten Reference Vreman and Kuerten2014a ,Reference Vreman and Kuerten b , Reference Vreman and Kuerten2016; Lee & Moser Reference Lee and Moser2015) acquired since the pioneering work of Kim et al. (Reference Kim, Moin and Moser1987) were considered in this study. These data were obtained by several authors with different computational accuracy indicators (spatio-temporal resolution, box size, observation time and sampling frequency) using a variety of computational methods and/or codes. Notice that for the low $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 180$ case, Vreman & Kuerten (Reference Vreman and Kuerten2014a ) recently reported a detailed study demonstrating consistent convergence of DNS results with increasing computational accuracy. At the centreline, only databases including $\unicode[STIX]{x1D700}_{ij}$ -data (Moser et al. Reference Moser, Kim and Mansour1999; del Álamo & Jiménez Reference del Álamo and Jiménez2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Vreman & Kuerten Reference Vreman and Kuerten2014b , Reference Vreman and Kuerten2016), were used for the $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ -invariants (figure 4), and wall values for the $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ invariants were used when available (Moser et al. Reference Moser, Kim and Mansour1999; del Álamo & Jiménez Reference del Álamo and Jiménez2003; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008). For those databases that did not include wall values (Vreman & Kuerten Reference Vreman and Kuerten2014b , Reference Vreman and Kuerten2016) or $\unicode[STIX]{x1D700}_{ij}$ -data (Hu et al. Reference Hu, Morfey and Sandham2006; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014) wall invariants were estimated by linear extrapolation of the $\unicode[STIX]{x1D657}$ invariants from the first 2 grid points. The very-near-wall ( $y^{+}\lessapprox 0.2$ ) data of Bernardini et al. (Reference Bernardini, Pirozzoli and Orlandi2014) were noisy (presumably because of the extreme near-wall cosine stretching of the wall-normal mesh size) and were not used; corresponding wall values were obtained by extrapolation from the first 2 grid points with $y^{+}\gtrapprox 0.2$ . The $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 4180$ small-box data of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) were only used at the wall. Finally, it was found interesting to include the $Re_{\unicode[STIX]{x1D70F}_{w}}<180$ data of Hu et al. (Reference Hu, Morfey and Sandham2002, Reference Hu, Morfey and Sandham2003, Reference Hu, Morfey and Sandham2006) illustrating low- $Re_{\unicode[STIX]{x1D70F}_{w}}$ asymptotics.

It should be stated from the outset that anisotropy invariants are much more sensitive to $\unicode[STIX]{x1D633}_{ij}$ and $\unicode[STIX]{x1D700}_{ij}$ data uncertainties (Schultz & Flack Reference Schultz and Flack2013, p. 5) because of the cumulative propagation of these uncertainties (Taylor Reference Taylor1997, § 3, pp. 45–91) in the calculation of the anisotropy tensors (2.1a–c ) and (2.2a–c ), and then of their invariants (2.1a–c ) and (2.2a–c ). Nonetheless, although data for some of the invariants exhibit substantial scatter (figure 4), it appears that $Re_{\unicode[STIX]{x1D70F}_{w}}$ -trends can be deduced with reasonable confidence.

Figure 4.

Invariants of the anisotropy tensors (2.1, 2.2) of the Reynolds-stresses $\unicode[STIX]{x1D657}$ ( $\text{II}_{\boldsymbol{b}}$ , $\text{III}_{\boldsymbol{b}}$ ) and of the corresponding dissipation rates $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ ( $\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ , $\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}$ ), at the wall $(\cdot )_{w}$ and at channel centreline $(\cdot )_{CL}$ , and axisymmetry diagnostic $\text{II}_{\boldsymbol{b}}^{-1}(\text{II}_{\boldsymbol{b}}+3(\text{III}_{\boldsymbol{b}}^{2}/4)^{1/3})$ and $\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}^{-1}(\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}+3(\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}^{2}/4)^{1/3})$ , at the centreline $(\cdot )_{CL}$ , as a function of the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}_{w}}$ (A 3k ), from existing (Moser et al. Reference Moser, Kim and Mansour1999; Hu et al. Reference Hu, Morfey and Sandham2006; Hoyas & Jiménez Reference Hoyas and Jiménez2006, Reference Hoyas and Jiménez2008; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014; Vreman & Kuerten Reference Vreman and Kuerten2014b , Reference Vreman and Kuerten2016; Lee & Moser Reference Lee and Moser2015) DNS computations of turbulent plane channel flow, in the range $Re_{\unicode[STIX]{x1D70F}_{w}}\in [80,5200]$ .

At the wall, the data for both ${\text{II}_{\boldsymbol{b}}}_{w}\stackrel{(2.4c)}{=}{\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}}_{w}$ and ${\text{III}_{\boldsymbol{b}}}_{w}\stackrel{(2.4d)}{=}{\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}}}_{w}$ show consistent behaviour with reasonably small scatter (figure 4) suggesting that the near-wall DNS data are quite robust with respect to the variation of computational accuracy indicators of the simulations. It seems likely that, with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ , the wall invariants reach asymptotic values. On the other hand, as $Re_{\unicode[STIX]{x1D70F}_{w}}$ decreases, both invariants increase sharply (figure 4). Notice that the very-small-box $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 4180$ data of Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014), which were only considered at the wall, are consistent with the large-box data (Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lee & Moser Reference Lee and Moser2015) suggesting that the small-box bias mainly impacts the centreline region, whereas the very-near-wall behaviour is less sensitive to the details of the large-scale outer-flow structures.

At the centreline, the data show considerable scatter (figure 4), especially for the $\unicode[STIX]{x1D657}$ invariants $(-\text{II}_{\boldsymbol{b}},\text{III}_{\boldsymbol{b}})_{CL}$ , in line with the observation on the cumulative impact of $\unicode[STIX]{x1D633}_{ij}$ and $\unicode[STIX]{x1D700}_{ij}$ uncertainties on the calculation of the invariants. Notice first that the scatter is more important for the $\unicode[STIX]{x1D657}$ invariants which characterize (Lee & Reynolds Reference Lee, Reynolds, Durst, Launder, Lumley, Schmidt and Whitelaw1987) the large-scale anisotropy and are therefore more sensitive to box size and observation time, whereas the $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ invariants $(-\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}})_{CL}$ which characterize the small-scale anisotropy are more robust. With respect to box size, notice that high $Re_{\unicode[STIX]{x1D70F}_{w}}$ small-box calculations (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014) overpredict the $\unicode[STIX]{x1D657}$ -invariants at the centreline by a factor 2 (not included in figure 4). This is not completely unexpected, and, in view of the consistent wall invariants obtained in these high- $Re_{\unicode[STIX]{x1D70F}_{w}}$ small-box calculations, it suggests that the very-large-scale structures influence quite substantially turbulence at the centreline and much less in the very-near-wall (sublayer) region. The general trend of the $\unicode[STIX]{x1D657}$ invariants $(-\text{II}_{\boldsymbol{b}},\text{III}_{\boldsymbol{b}})_{CL}$ at the centreline (figure 4), is a strong decrease with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ in the range $Re_{\unicode[STIX]{x1D70F}_{w}}<400$ , followed by a slight increase, probably reaching an asymptotic state with increasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ . Nonetheless there is too much scatter in the data to draw definitive conclusions (figure 4). Regarding the $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ invariants $(-\text{II}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}},\text{III}_{\boldsymbol{b}_{\unicode[STIX]{x1D73A}}})_{CL}$ at the centreline, there appears a clear trend of asymptotic decrease to $\approxeq 0$ as $Re_{\unicode[STIX]{x1D70F}_{w}}$ increases (figure 4).

Observation of $\unicode[STIX]{x1D623}_{ij}$ at the centreline (figure 1), where by symmetry $b_{xy_{CL}}=0$ , shows that $b_{yy_{CL}}\approxeq b_{zz_{CL}}<b_{xx_{CL}}$ (figure 1), at least for $Re_{\unicode[STIX]{x1D70F}_{w}}\geqslant 180$ , implying that the Reynolds-stress tensor at the centreline is axisymmetric rod-like (Simonsen & Krogstad Reference Simonsen and Krogstad2005, figure 4, p. 3). However, the componentality of $\unicode[STIX]{x1D700}_{ij}$ at the centreline is not as obvious (figure 1), especially as the $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}}$ invariants approach zero at the centreline (figure 4), contrary to the $\unicode[STIX]{x1D657}$ invariants which seem to reach finite asymptotic values at the centreline (figure 4). At the rod-like axisymmetric boundary of the realizability triangle (figure 4)  $\text{II}+3(\text{III}^{2}/4)^{1/3}=0$ , so that $\text{II}^{-1}(\text{II}+3(\text{III}^{2}/4)^{1/3})$ is a diagnostic function whose distance from zero denotes departure from rod-like axisymmetric componentality. Contrary to $\unicode[STIX]{x1D657}_{CL}$ , $\unicode[STIX]{x1D657}_{\unicode[STIX]{x1D73A}CL}$ at the centreline seems to become increasingly non-axisymmetric as $Re_{\unicode[STIX]{x1D70F}_{w}}$ increases (figure 4). Finally, the data of Hu et al. (Reference Hu, Morfey and Sandham2006) at very low $Re_{\unicode[STIX]{x1D70F}_{w}}<180$ seem to indicate departure from rod-like axisymmetry for $\unicode[STIX]{x1D657}_{CL}$ (figure 4). At $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 180$ , the data of Kim et al. (Reference Kim, Moin and Moser1987) also indicate a slight departure from rod-like axisymmetry of $\unicode[STIX]{x1D657}_{CL}$ , but all other data (Hoyas & Jiménez Reference Hoyas and Jiménez2006; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lee & Moser Reference Lee and Moser2015; Vreman & Kuerten Reference Vreman and Kuerten2016), including the present calculations, indicate that $\unicode[STIX]{x1D657}_{CL}$ is rod-like axisymmetric at $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 180$ (figure 4). Further DNS at very low $Re_{\unicode[STIX]{x1D70F}_{w}}<180$ are therefore needed to verify the departure from rod-like axisymmetry of $\unicode[STIX]{x1D657}_{CL}$ with decreasing $Re_{\unicode[STIX]{x1D70F}_{w}}$ .

3.1 $\unicode[STIX]{x1D700}_{ij}$ -transport equation

Starting from the fluctuating continuity (Mathieu & Scott Reference Mathieu and Scott2000, (4.6), p. 76)

(3.2a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{\ell }^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}=0 & & \displaystyle\end{eqnarray}$$

and fluctuating momentum (Mathieu & Scott Reference Mathieu and Scott2000, (4.31), p. 85)

(3.2b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}\bar{u}_{\ell }\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}=-\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{\ell }}(u_{i}^{\prime }u_{\ell }^{\prime }-\overline{u_{i}^{\prime }u_{\ell }^{\prime }})-\unicode[STIX]{x1D70C}u_{\ell }^{\prime }\frac{\unicode[STIX]{x2202}\bar{u}_{i}}{\unicode[STIX]{x2202}x_{\ell }}-\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }\unicode[STIX]{x2202}x_{\ell }} & & \displaystyle\end{eqnarray}$$

equations we can work out the transport equations for ${\mathcal{E}}_{ijkm}$ (3.1a ), and by contraction (3.1a ) the transport equation for $\unicode[STIX]{x1D700}_{ij}$ , which reads

(3.3) $$\begin{eqnarray}\displaystyle & & \displaystyle \underbrace{\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{ij}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}\bar{u}_{\ell }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{ij}}{\unicode[STIX]{x2202}x_{\ell }}}_{C_{\unicode[STIX]{x1D700}_{ij}}}=\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{\ell }}\left[\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D700}_{ij}}{\unicode[STIX]{x2202}x_{\ell }}\right]}_{d_{\unicode[STIX]{x1D700}_{ij}}^{(\unicode[STIX]{x1D707})}}+\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{\ell }}\left[-\unicode[STIX]{x1D70C}\left(\overline{u_{\ell }^{\prime }2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}\right)\right]}_{d_{\unicode[STIX]{x1D700}_{ij}}^{(u)}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}_{i\ell }\frac{\unicode[STIX]{x2202}\bar{u}_{j}}{\unicode[STIX]{x2202}x_{\ell }}-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}_{j\ell }\frac{\unicode[STIX]{x2202}\bar{u}_{i}}{\unicode[STIX]{x2202}x_{\ell }}}_{P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}}\underbrace{-\unicode[STIX]{x1D70C}\left(2\unicode[STIX]{x1D708}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}}\right)\left(\frac{\unicode[STIX]{x2202}\bar{u}_{k}}{\unicode[STIX]{x2202}x_{\ell }}+\frac{\unicode[STIX]{x2202}\bar{u}_{\ell }}{\unicode[STIX]{x2202}x_{k}}\right)}_{P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{-\unicode[STIX]{x1D70C}\!\left(2\unicode[STIX]{x1D708}\overline{u_{\ell }^{\prime }\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}\right)\frac{\unicode[STIX]{x2202}^{2}\bar{u}_{j}}{\unicode[STIX]{x2202}x_{\ell }\unicode[STIX]{x2202}x_{k}}-\unicode[STIX]{x1D70C}\!\left(2\unicode[STIX]{x1D708}\overline{u_{\ell }^{\prime }\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}}\right)\frac{\unicode[STIX]{x2202}^{2}\bar{u}_{i}}{\unicode[STIX]{x2202}x_{\ell }\unicode[STIX]{x2202}x_{k}}}_{P_{\unicode[STIX]{x1D700}_{ij}}^{(3)}}\underbrace{-\unicode[STIX]{x1D70C}\!\left[\!2\unicode[STIX]{x1D708}\overline{\frac{\unicode[STIX]{x2202}u_{\ell }^{\prime }}{\unicode[STIX]{x2202}x_{k}}\left(\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}+\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{\ell }}\right)}\right]}_{\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D700}_{ij}}=:P_{\unicode[STIX]{x1D700}_{ij}}^{(4)}}\nonumber\\ \displaystyle & & \displaystyle \quad \underbrace{-2\unicode[STIX]{x1D708}\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}^{2}p^{\prime }}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{k}}}-2\unicode[STIX]{x1D708}\overline{\frac{\unicode[STIX]{x2202}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}}\frac{\unicode[STIX]{x2202}^{2}p^{\prime }}{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{k}}}}_{\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{ij}}}-\underbrace{\unicode[STIX]{x1D70C}\overline{\left(2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{k}\unicode[STIX]{x2202}x_{\ell }}\right)\left(2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{k}\unicode[STIX]{x2202}x_{\ell }}\right)}}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{ij}}}.\end{eqnarray}$$

In (3.3) $C_{\unicode[STIX]{x1D700}_{ij}}$ is the convection of $\unicode[STIX]{x1D700}_{ij}$ by the mean-flow velocity field $\bar{u}_{\ell }$ , $d_{\unicode[STIX]{x1D700}_{ij}}^{(\unicode[STIX]{x1D707})}$ is the diffusion of $\unicode[STIX]{x1D700}_{ij}$ by molecular viscosity (notice that $d_{\unicode[STIX]{x1D700}_{ij}}^{(\unicode[STIX]{x1D707})}\stackrel{(3.3)}{=}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D700}_{ij}$ for $\unicode[STIX]{x1D707}=\text{const.}$ ), $d_{\unicode[STIX]{x1D700}_{ij}}^{(u)}$ is the turbulent diffusion (mixing) of $\unicode[STIX]{x1D700}_{ij}$ by the fluctuating velocity field $u_{\ell }^{\prime }$ , $P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}$ is the production of $\unicode[STIX]{x1D700}_{ij}$ by the action of its components on the mean-flow velocity gradients, $P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}$ is the production of $\unicode[STIX]{x1D700}_{ij}$ by the action of ${\mathcal{E}}_{ijkm}$ (3.1) on the mean-flow velocity gradients, $P_{\unicode[STIX]{x1D700}_{ij}}^{(3)}$ is the production of $\unicode[STIX]{x1D700}_{ij}$ by the mean-flow velocity Hessian, $P_{\unicode[STIX]{x1D700}_{ij}}^{(4)}:=\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D700}_{ij}}$ corresponds to triple correlations of fluctuating velocity gradients whose trace $(\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D700}_{\ell \ell }})/2$ was identified by Mansour et al. (Reference Mansour, Kim and Moin1988) as a term ‘producing’ (gain in the budgets of) $\unicode[STIX]{x1D700}:=\unicode[STIX]{x1D700}_{\ell \ell }/2$ (by extension the denomination $P_{\unicode[STIX]{x1D700}_{ij}}^{(4)}$ is used, although this term does not contain gradients of the mean-flow field), $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{ij}}$ are the terms containing the fluctuating pressure Hessian, and $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{ij}}$ is the destruction of $\unicode[STIX]{x1D700}_{ij}$ by the action of molecular viscosity. Following usual practice (Mansour et al. Reference Mansour, Kim and Moin1988, (1), p. 17) for the incompressible $\unicode[STIX]{x1D633}_{ij}$ equations (1.1) and for the $\unicode[STIX]{x1D700}$ equation (Mansour et al. Reference Mansour, Kim and Moin1988, (23), p. 23), a computable viscous diffusion term $d_{\unicode[STIX]{x1D700}_{ij}}^{(\unicode[STIX]{x1D707})}$ was chosen in (3.3), with a corresponding appropriate definition of the destruction-of-dissipation tensor $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{ij}}$ (3.3), in lieu of alternative splittings based on the viscous-stress tensor (Ben Nasr, Gerolymos & Vallet Reference Ben Nasr, Gerolymos and Vallet2014, (2), p. 189).

Using (3.1) the production of $\unicode[STIX]{x1D700}_{ij}$ by mean-flow velocity gradients reads

(3.4) $$\begin{eqnarray}\displaystyle P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}\stackrel{(3.3,3.1a)}{=}\underbrace{-\unicode[STIX]{x1D70C}{\mathcal{E}}_{i\ell kk}\frac{\unicode[STIX]{x2202}\bar{u}_{j}}{\unicode[STIX]{x2202}x_{\ell }}-\unicode[STIX]{x1D70C}{\mathcal{E}}_{j\ell kk}\frac{\unicode[STIX]{x2202}\bar{u}_{i}}{\unicode[STIX]{x2202}x_{\ell }}}_{P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}}\underbrace{-\unicode[STIX]{x1D70C}{\mathcal{E}}_{ijk\ell }\left(\frac{\unicode[STIX]{x2202}\bar{u}_{k}}{\unicode[STIX]{x2202}x_{\ell }}+\frac{\unicode[STIX]{x2202}\bar{u}_{\ell }}{\unicode[STIX]{x2202}x_{k}}\right)}_{P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}} & & \displaystyle\end{eqnarray}$$

highlighting the importance of considering the generating fourth-order tensor ${\mathcal{E}}_{ijkm}$ (3.1). The separation in two terms, $P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}$ and $P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}$ , was made to distinguish between the computable (from the knowledge of $\unicode[STIX]{x1D700}_{ij}$ and the mean-flow field) term $P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}$ , and $P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}$ which involves components of ${\mathcal{E}}_{ijkm}$ that do not simplify by contraction (3.1a ) to  $\unicode[STIX]{x1D700}_{ij}$ .

3.2 $\unicode[STIX]{x1D700}_{ij}$ -transport budgets

Budgets of the $\unicode[STIX]{x1D700}_{ij}$ -transport equations (3.3), for turbulent plane channel flow, were obtained (figure 5) in an $L_{x}\times L_{y}\times L_{z}=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}\times 2\unicode[STIX]{x1D6FF}\times 4/3\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}$ computational box, using a carefully validated DNS solver (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2010, Reference Gerolymos, Sénéchal and Vallet2013; Gerolymos & Vallet Reference Gerolymos and Vallet2014). The resolution in the homogeneous $xz$ -directions is $\unicode[STIX]{x0394}x^{+}\approxeq 5.7$ and $\unicode[STIX]{x0394}z^{+}\approxeq 1.9$ . The wall-normal size of the grid cells adjacent to the wall was $\unicode[STIX]{x0394}y_{w}^{+}\approxeq 0.22$ . More importantly, the mesh (65 % of the nodes were stretched near the walls geometrically with ratio $r_{j}=1.0427$ , the remaining nodes in the outer region being equidistant) was kept fine in the entire near-wall region, with $N_{y^{+}\leqslant 10}=26$ grid cells between the wall and $y^{+}\approxeq 10$ , and remained fine up to the centreline where the wall-normal cell size was $\unicode[STIX]{x0394}y_{CL}^{+}\approxeq 3.1$ . This spatial resolution, combined with the $O(\unicode[STIX]{x0394}\ell ^{17})$ scheme used (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2009, Reference Gerolymos, Sénéchal and Vallet2010) is quite fine in view of current state-of-the-art DNS at this $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 180$ (Vreman & Kuerten Reference Vreman and Kuerten2014a , Reference Vreman and Kuerten2016). Statistics was acquired at high sampling frequency (at every iteration; $\unicode[STIX]{x0394}t_{s}^{+}=\unicode[STIX]{x0394}t^{+}\approxeq 0.0060$ , albeit for a relatively short observation time $t_{OBS}^{+}\approxeq 777$ ). The close agreement (§ 4) of the present results for the budgets of the diagonal components with the highly resolved computations of Vreman & Kuerten (Reference Vreman and Kuerten2016) further substantiate the validity of the computations. Wall asymptotics of the various terms in (3.3) can be obtained using the Taylor expansions (2.3) in the fluctuating momentum equations (3.2b ). Although a full report of these calculations is outside the scope of this paper, some of the limiting wall values obtained from this procedure are included in the following discussion.

Regarding the streamwise component $\unicode[STIX]{x1D700}_{xx}^{+}$ (figure 5), production $P_{\unicode[STIX]{x1D700}_{xx}}^{+}$ (gain) roughly balances destruction $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xx}}^{+}$ (loss) in the major part of the channel ( $y^{+}\gtrapprox 1$ ), with lesser contributions of diffusion ( $d_{\unicode[STIX]{x1D700}_{xx}}^{(u)+}+d_{\unicode[STIX]{x1D700}_{xx}}^{(\unicode[STIX]{x1D707})+}$ ). Production $P_{\unicode[STIX]{x1D700}_{xx}}^{+}$ peaks at $y^{+}\approxeq 4$ and destruction $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xx}}^{+}$ at $y^{+}\approxeq 5$ . The pressure term $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xx}}^{+}$ is negligible in the budgets of $\unicode[STIX]{x1D700}_{xx}^{+}$ throughout the channel. Very near the wall ( $y^{+}\lessapprox 1$ ; figure 5) production $P_{\unicode[STIX]{x1D700}_{xx}}^{+}\underset{y^{+}\rightarrow 0}{\rightarrow 0}$  (wall-asymptotic expansion; § 2.2) and viscous diffusion $d_{\unicode[STIX]{x1D700}_{xx}}^{(\unicode[STIX]{x1D707})+}$ (gain $\forall y^{+}\lessapprox 1$ ) roughly counters destruction $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xx}}^{+}$ ( $y^{+}\rightarrow 0$ ; figure 5). At the wall, the pressure term $[\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xx}}^{+}]_{w}=8\overline{B_{v}^{\prime }\unicode[STIX]{x2202}_{x}A_{u}^{\prime }}^{+}\neq 0$  (wall-asymptotic expansion; § 2.2). However $|\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xx}}^{+}|_{w}\ll [\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xx}}^{+}]_{w}\approxeq [d_{\unicode[STIX]{x1D700}_{xx}}^{(\unicode[STIX]{x1D707})+}]_{w}$ (figure 5).

Figure 5.

Budgets, in wall units (A 3j ), of the transport equations (3.3) for the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ (1.2a ), from the present DNS computations of turbulent plane channel flow ( $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 182$ , $\bar{M}_{CL}\approxeq 0.35$ ), plotted against the inner-scaled (A 3e–g ) wall distance $y^{+}$ (log scale and linear wall zoom).

The budgets of the spanwise component $\unicode[STIX]{x1D700}_{zz}^{+}$ (figure 5) are also dominated by a balance between production $P_{\unicode[STIX]{x1D700}_{zz}}^{+}$ (gain) and destruction $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{zz}}^{+}$ (loss) in the major part of the channel ( $y^{+}\gtrapprox 5$ ). However, production $P_{\unicode[STIX]{x1D700}_{zz}}^{+}$ peaks at $y^{+}\approxeq 10$ and becomes negligible at $y^{+}\approxeq 3$ as $P_{\unicode[STIX]{x1D700}_{zz}}^{+}\underset{y^{+}\rightarrow 0}{\rightarrow 0}$  (wall-asymptotic expansion; § 2.2). In the near-wall region, again destruction $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{zz}}^{+}$ (loss $\forall y^{+}$ ) is balanced by viscous diffusion $d_{\unicode[STIX]{x1D700}_{zz}}^{(\unicode[STIX]{x1D707})+}$ (gain $\forall y^{+}\lessapprox 6$ ), but this zone extends further away from the wall ( $y^{+}\lessapprox 3$ ; figure 5) compared to the streamwise component ( $y^{+}\lessapprox 1$ ; figure 5). As for the streamwise component, the other mechanisms, $d_{\unicode[STIX]{x1D700}_{zz}}^{(u)+}$ and $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{zz}}^{+}$ , have a very small contribution to the $\unicode[STIX]{x1D700}_{zz}^{+}$ -budget. Another difference between the streamwise and the spanwise components is that $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{zz}}^{+}$ is maximum at the wall decreasing monotonically with $y^{+}$ , contrary to $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xx}}^{+}$ (figure 5).

The major difference in the budgets of the wall-normal component $\unicode[STIX]{x1D700}_{yy}^{+}$ , compared to the other two diagonal components, is the importance of the pressure term $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{yy}}^{+}$ which is the dominant gain mechanism throughout the channel (figure 5), except very near the wall ( $y^{+}\lessapprox 1/2$ ; figure 5) where $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{yy}}^{+}\underset{y^{+}\rightarrow 0}{\rightarrow 0}$  (wall-asymptotic expansion; § 2.2) and, as for $\unicode[STIX]{x1D700}_{xx}^{+}$ and $\unicode[STIX]{x1D700}_{zz}^{+}$ , viscous diffusion $d_{\unicode[STIX]{x1D700}_{yy}}^{(\unicode[STIX]{x1D707})+}$ (gain) balances destruction $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{yy}}^{+}$ ( $y^{+}\lessapprox 1/2$ ; figure 5). Production $P_{\unicode[STIX]{x1D700}_{yy}}^{+}$ (gain) is significant in the buffer region ( $10\lessapprox y^{+}\lessapprox 100$ ; figure 5) although it becomes comparable to $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{yy}}^{+}$ only for $y^{+}\gtrapprox 100$ (figure 5). Turbulent diffusion $d_{\unicode[STIX]{x1D700}_{yy}}^{(u)+}$ is generally weak throughout the channel (figure 5).

The budgets of the shear component $\unicode[STIX]{x1D700}_{xy}^{+}<0\,\forall y^{+}>0$ (Mansour et al. Reference Mansour, Kim and Moin1988, figure 4, p. 19) exhibit a fundamentally different behaviour compared to the diagonal components (figure 5). All of the terms in the $\unicode[STIX]{x1D700}_{xy}^{+}$ -budgets ( $P_{\unicode[STIX]{x1D700}_{xy}}^{+}$ , $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xy}}^{+}$ , $d_{\unicode[STIX]{x1D700}_{xy}}^{(u)+}$ , $d_{\unicode[STIX]{x1D700}_{xy}}^{(\unicode[STIX]{x1D707})+}$ , $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xy}}^{+}$ ) are significant (figure 5). Contrary to the diagonal components, the destruction term $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xy}}^{+}$ contributes as gain to the budgets in the major part of the channel ( $y^{+}\gtrapprox 3$ ; figure 5), becoming an actual destruction mechanism (loss) only in the viscous sublayer ( $y^{+}\lessapprox 3$ ; figure 5). Notice that, at the wall, viscous diffusion of the shear component is substantially weaker than the two other mechanisms present in (3.3), viz. $|\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xy}}^{+}|_{w}\gg |d_{\unicode[STIX]{x1D700}_{xy}}^{(\unicode[STIX]{x1D707})+}|_{w}=|\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xy}}^{+}-\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xy}}^{+}|_{w}\ll |\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xy}}^{+}|_{w}$  (wall-asymptotic expansion; § 2.2; figure 5). In a large part of the channel, $-\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{xy}}^{+}<0\,\forall y^{+}\gtrapprox 3$ (figure 5) is an important gain mechanism, along with production $P_{\unicode[STIX]{x1D700}_{xy}}^{+}<0\,\forall y^{+}>0$ (figure 5). On the other hand, $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xy}}^{+}$ is the main loss mechanism in the major part of the channel (figure 5). Notice the complicated $y^{+}$ -evolution of $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xy}}^{+}$ (figure 5), which is ${<}0$ (gain) at the wall, crossing to ${>}0$ (loss) at $y^{+}\gtrapprox 1/2$ , and presents a plateau ( $3\lessapprox y^{+}\lessapprox 7$ ) followed by a global maximum at $y^{+}\approxeq 15$ , before decreasing monotonically to $0$ at the centreline. In the region $1\lessapprox y^{+}\lessapprox 20$ , both diffusion mechanisms, $d_{\unicode[STIX]{x1D700}_{xy}}^{(\unicode[STIX]{x1D707})+}$ and $d_{\unicode[STIX]{x1D700}_{xy}}^{(u)+}$ , contribute significantly to the $\unicode[STIX]{x1D700}_{xy}$ -budgets (figure 5).

A major difference between the $\unicode[STIX]{x1D633}_{ij}$ -budgets (Mansour et al. Reference Mansour, Kim and Moin1988) and the $\unicode[STIX]{x1D700}_{ij}$ -budgets (figure 5) lies in the production mechanisms $P_{ij}$ (1.1) and $P_{\unicode[STIX]{x1D700}_{ij}}$ (3.3). In the $\unicode[STIX]{x1D633}_{ij}$ -budgets of plane channel flow, only $P_{xx}^{+}\neq 0\neq P_{xy}^{+}$ whereas $P_{yy}^{+}=P_{zz}^{+}=0\,\forall y^{+}$ (Mansour et al. Reference Mansour, Kim and Moin1988, figures 2 and 3, pp. 18–19). On the contrary, all of the components $P_{\unicode[STIX]{x1D700}_{ij}}\neq 0\,\forall y^{+}>0$ in general (figure 5). In relation to the above observation, the main gain mechanism in the $\unicode[STIX]{x1D700}_{zz}^{+}$ -budgets is production $P_{\unicode[STIX]{x1D700}_{zz}}^{+}$ (figure 5) in contrast to the $r_{zz}$ -budgets (Mansour et al. Reference Mansour, Kim and Moin1988, figure 3, p. 19) where, in the absence of production $P_{zz}^{+}=0\,\forall y^{+}$ , the pressure term $\unicode[STIX]{x1D6F1}_{zz}^{+}$ is the main gain mechanism. On the contrary, $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{zz}}^{+}$ has negligible contribution to the $\unicode[STIX]{x1D700}_{zz}$ -budgets (figure 5). It is noteworthy that the importance of the pressure terms $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{yy}}^{+}$ and $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{xy}}^{+}$ in the budgets of the $\unicode[STIX]{x1D700}_{yy}^{+}$ and $\unicode[STIX]{x1D700}_{xy}^{+}$ components (figure 5) are also quite generally observed in the budgets of wall-normal fluxes, including the Reynolds stresses $r_{yy}^{+}$ (Mansour et al. Reference Mansour, Kim and Moin1988, figure 2, p. 18) and $r_{xy}^{+}$ (Mansour et al. Reference Mansour, Kim and Moin1988, figure 4, p. 19), and in the compressible case, the fluxes of thermodynamic quantities such as temperature $\overline{T^{\prime }v^{\prime }}$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figures 14 and 15, pp. 737–738), density $\overline{\unicode[STIX]{x1D70C}^{\prime }v^{\prime }}$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 12, p. 731), pressure $\overline{p^{\prime }v^{\prime }}$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 16, p. 741) and entropy $\overline{s^{\prime }v^{\prime }}$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, figure 13, p. 734).

Figure 6.

Comparison of the various mechanisms of production $P_{\unicode[STIX]{x1D700}_{ij}}=P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(3)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(4)}$ (3.3) of the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ (1.2a ), in wall units (A 3j ), from the present DNS computations of turbulent plane channel flow ( $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 182$ , $\bar{M}_{CL}\approxeq 0.35$ ), plotted against the inner-scaled (A 3e–g ) wall distance $y^{+}$ (log scale and linear wall zoom).

Table 3.

Components of different mechanisms of production $P_{\unicode[STIX]{x1D700}_{ij}}=P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(3)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(4)}$ (3.3) for fully developed turbulent plane channel.

Production $P_{\unicode[STIX]{x1D700}_{ij}}^{+}$ (figure 5) of $\unicode[STIX]{x1D700}_{ij}$ contains four different mechanisms, $P_{\unicode[STIX]{x1D700}_{ij}}^{+}=P_{\unicode[STIX]{x1D700}_{ij}}^{(1)+}+P_{\unicode[STIX]{x1D700}_{ij}}^{(2)+}+P_{\unicode[STIX]{x1D700}_{ij}}^{(3)+}+P_{\unicode[STIX]{x1D700}_{ij}}^{(4)+}$ (3.3), which behave differently for each component across the channel (figure 6). Globally, in plane channel flow, $P_{\unicode[STIX]{x1D700}_{ij}}^{+}$ contributes as a gain mechanism in the $\unicode[STIX]{x1D700}_{ij}$ -budgets (figures 5 and 6), except very near the wall for the wall-normal ( $P_{\unicode[STIX]{x1D700}_{yy}}^{+}\leqslant 0\,\forall y^{+}\lessapprox 3$ is a loss mechanism; figure 5) and spanwise ( $P_{\unicode[STIX]{x1D700}_{zz}}^{+}\leqslant 0\,\forall y^{+}\lessapprox 2$ is a loss mechanism; figure 5) diagonal components. For both the spanwise and wall-normal components, $P_{\unicode[STIX]{x1D700}_{yy}}^{(1)+}=P_{\unicode[STIX]{x1D700}_{zz}}^{(1)+}=P_{\unicode[STIX]{x1D700}_{yy}}^{(3)+}=P_{\unicode[STIX]{x1D700}_{zz}}^{(3)+}=0\,\forall y^{+}$ (table 3), whereas production by the action of ${\mathcal{E}}_{ijkm}^{+}$ -components on the mean velocity gradient, $P_{\unicode[STIX]{x1D700}_{yy}}^{(2)+}$ and $P_{\unicode[STIX]{x1D700}_{zz}}^{(2)+}$ (table 3), is the main gain mechanism in the initial part of the buffer layer ( $y^{+}\in [5,30]$ ; figure 6) and is replaced further away from the wall ( $y^{+}\gtrapprox 30$ ; figure 6) by production by triple correlations of fluctuating velocity gradients, $P_{\unicode[STIX]{x1D700}_{yy}}^{(4)+}$ and $P_{\unicode[STIX]{x1D700}_{zz}}^{(4)+}$ (3.3), as expected from the quasi-homogeneous analysis of Tennekes & Lumley (Reference Tennekes and Lumley1972, pp. 88–92). All of the four production mechanisms (3.3) are active for the streamwise $\unicode[STIX]{x1D700}_{xx}^{+}$ and shear $\unicode[STIX]{x1D700}_{xy}^{+}$ components (figure 6). Regarding the streamwise component $\unicode[STIX]{x1D700}_{xx}^{+}$ , the three mechanisms, $P_{\unicode[STIX]{x1D700}_{xx}}^{(1)+}>0$ , $P_{\unicode[STIX]{x1D700}_{xx}}^{(2)+}>0$ and $P_{\unicode[STIX]{x1D700}_{xx}}^{(4)+}>0\,\forall y^{+}>0$ (figure 6) always contribute as gain to the budgets. On the other hand, production by the mean velocity Hessian, $P_{\unicode[STIX]{x1D700}_{xx}}^{(3)+}<0\,\forall y^{+}\lessapprox 10$ (loss) near the wall (figure 6), switching to $P_{\unicode[STIX]{x1D700}_{xx}}^{(3)+}>0\,\forall y^{+}\gtrapprox 10$ (gain) further away from the wall, is generally weaker than the other three mechanisms. Notice that the production by the triple correlations of the fluctuating velocity gradients $P_{\unicode[STIX]{x1D700}_{xx}}^{(4)+}$ (3.3) is important throughout the channel, even as $y^{+}\rightarrow 0$ (figure 6), becoming the dominant mechanism in the outer part of the flow ( $y^{+}\gtrapprox 15$ ; figure 6). Regarding the shear component $\unicode[STIX]{x1D700}_{xy}^{+}$ , $P_{\unicode[STIX]{x1D700}_{xy}}^{(1)+}=-\unicode[STIX]{x1D700}_{yy}^{+}[d_{y}\bar{u}]^{+}<0\,\forall y^{+}>0$ (figure 6) is the main gain mechanism throughout the channel, whereas $P_{\unicode[STIX]{x1D700}_{xy}}^{(2)+}>0$ and $P_{\unicode[STIX]{x1D700}_{xy}}^{(3)+}>0\,\forall y^{+}>0$ contribute as loss to the budgets (figure 6). Finally, $P_{\unicode[STIX]{x1D700}_{xy}}^{(4)+}<0\,\forall y^{+}\lessapprox 6$ (gain) near the wall (figure 6) switches to $P_{\unicode[STIX]{x1D700}_{xy}}^{(4)+}>0\,\forall y^{+}\gtrapprox 6$ (loss) further away from the wall. Interestingly, $P_{\unicode[STIX]{x1D700}_{xy}}^{(4)+}$ , although comparable to the other three mechanisms $\forall y^{+}$ (figure 6), never becomes the dominant mechanism, even as $y^{+}\rightarrow \unicode[STIX]{x1D6FF}^{+}$ , raising the question of applicability of the quasi-homogeneous order-of-magnitude analysis (Tennekes & Lumley Reference Tennekes and Lumley1972, pp. 88–92) to the shear component (although higher- $Re$ data are required to fully resolve this issue).

In plane channel flow, production by interaction with the mean velocity gradient $P_{\unicode[STIX]{x1D700}_{ij}}^{(1)+}+P_{\unicode[STIX]{x1D700}_{ij}}^{(2)+}$ (3.3) involves $\unicode[STIX]{x1D700}_{xy}^{+}$ in $P_{\unicode[STIX]{x1D700}_{xx}}^{(1)+}$ (table 3), $\unicode[STIX]{x1D700}_{yy}^{+}$ in $P_{\unicode[STIX]{x1D700}_{xy}}^{(1)+}$ (table 3), and five different components of ${\mathcal{E}}_{ijkm}$ (3.1) in $P_{\unicode[STIX]{x1D700}_{ij}}^{(2)+}$ (table 3), which are not involved in the generating relation (3.1a ). Since the mean velocity gradient $d_{y}\bar{u}^{+}>0\,\forall y^{+}\in ]0,\unicode[STIX]{x1D6FF}^{+} [$ (Coles Reference Coles1956), the sign of these components directly determines (table 3) whether the corresponding contribution to $P_{\unicode[STIX]{x1D700}_{ij}}^{+}$ is gain or loss (figures 6 and 7).

Concerning $P_{\unicode[STIX]{x1D700}_{xx}}^{(1)+}$ and $P_{\unicode[STIX]{x1D700}_{xx}}^{(2)+}$ (table 3), the corresponding producing components, $\unicode[STIX]{x1D700}_{xy}^{+}$ and ${\mathcal{E}}_{xxxy}^{+}$ are of comparable magnitude (figure 7), but $\unicode[STIX]{x1D700}_{xy}^{+}$ is active nearer to the wall (peak at $y^{+}\in [4,5]$ ; figure 7) compared to ${\mathcal{E}}_{xxxy}^{+}$ (peak at $y^{+}\approxeq 15$ ; figure 7). Therefore, $P_{\unicode[STIX]{x1D700}_{xx}}^{(1)+}$ peaks at $y^{+}\in [4,5]$ whereas $P_{\unicode[STIX]{x1D700}_{xx}}^{(2)+}$ peaks at $y^{+}\approxeq 9$ (figure 6). Near the wall (wall-asymptotic expansion; § 2.2), $P_{\unicode[STIX]{x1D700}_{xx}}^{(2)+}\underset{y^{+}\rightarrow 0}{{\sim}}O(y^{+2})$ whereas $P_{\unicode[STIX]{x1D700}_{xx}}^{(1)+}\underset{y^{+}\rightarrow 0}{{\sim}}O(y^{+})$ (figure 6). For the wall-normal and spanwise diagonal components $P_{\unicode[STIX]{x1D700}_{yy}}^{(1)+}=P_{\unicode[STIX]{x1D700}_{zz}}^{(1)+}=0$ (table 3), so that only $P_{\unicode[STIX]{x1D700}_{yy}}^{(2)+}$ and $P_{\unicode[STIX]{x1D700}_{zz}}^{(2)+}$ appear in the budgets of $\unicode[STIX]{x1D700}_{yy}^{+}$ and $\unicode[STIX]{x1D700}_{zz}^{+}$ respectively (figure 6), with corresponding producing components ${\mathcal{E}}_{yyxy}^{+}$ and ${\mathcal{E}}_{zzxy}^{+}$ respectively (table 3). These components peak at $y^{+}\approxeq 30$ ( ${\mathcal{E}}_{yyxy}^{+}$ ; figure 7) and $y^{+}\approxeq 25$ ( ${\mathcal{E}}_{zzxy}^{+}$ ; figure 7), the ${\mathcal{E}}_{zzxy}^{+}$ -peak being larger than the ${\mathcal{E}}_{yyxy}^{+}$ -peak (figure 7). The producing component ${\mathcal{E}}_{yyxy}^{+}\lessapprox 0\,\forall y^{+}>0$ (figure 7), so that (table 3)  $P_{\unicode[STIX]{x1D700}_{yy}}^{(2)+}\gtrapprox 0\,\forall y^{+}>0$ (gain; figure 6) with a peak at $y^{+}\approxeq 18$ . On the other hand, ${\mathcal{E}}_{zzxy}^{+}<0\,\forall y^{+}\gtrapprox 2$ (gain; figure 7) changes sign near the wall ( ${\mathcal{E}}_{zzxy}^{+}>0\,\forall y^{+}\lessapprox 2$ ; loss; figure 7). Regarding the shear component $\unicode[STIX]{x1D700}_{xy}^{+}$ , the producing terms $\unicode[STIX]{x1D700}_{yy}^{+}>0\,\forall y^{+}>0$ (figure 7) in $P_{\unicode[STIX]{x1D700}_{xy}}^{(1)+}$ (table 3) and $({\mathcal{E}}_{xyxy}^{+}+{\mathcal{E}}_{xyyx}^{+})\lessapprox 0\,\forall y^{+}>0$ (figure 7) in $P_{\unicode[STIX]{x1D700}_{xy}}^{(2)+}$ (table 3), so that $P_{\unicode[STIX]{x1D700}_{xy}}^{(2)+}\gtrapprox 0\,\forall y^{+}>0$ (loss; figure 6) opposes, but is weaker than, $P_{\unicode[STIX]{x1D700}_{xy}}^{(1)+}\lessapprox 0\,\forall y^{+}>0$ (gain; figure 6).

Figure 7.

Components of $\unicode[STIX]{x1D700}_{ij}$ (1.2a ) and of ${\mathcal{E}}_{ijkm}$ (3.1) that appear in the production by mean velocity gradient $P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}$ and $P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}$ terms (3.3) of the $\unicode[STIX]{x1D700}_{ij}$ -transport budgets expressed for the particular case of fully developed plane channel flow (table 3), in wall units (A 3i ), from the present DNS computations of turbulent plane channel flow ( $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 182$ , $\bar{M}_{CL}\approxeq 0.35$ ), plotted against the inner-scaled (A 3e–g ) wall distance $y^{+}$ (log scale and linear wall zoom).

Figure 8.

Examination of the influence of the Reynolds number $Re_{\unicode[STIX]{x1D70F}_{w}}$ (A 3k ) on various terms ( $P_{\unicode[STIX]{x1D700}_{ij}}$ , $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D700}_{ij}}$ , $\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D700}_{ij}}$ and $d_{\unicode[STIX]{x1D700}_{ij}}^{(u)}$ ) in the budgets of the transport equations (3.3) for the diagonal components of the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ (1.2a ), in wall units (A 3j ), by comparison of the present DNS computations of turbulent plane channel flow ( $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 182$ , $\bar{M}_{CL}\approxeq 0.35$ ) with the incompressible DNS data of Vreman & Kuerten (Reference Vreman and Kuerten2014a ,Reference Vreman and Kuerten b , Reference Vreman and Kuerten2016, $~Re_{\unicode[STIX]{x1D70F}_{w}}\in \{180,590\}$ , $\bar{M}_{CL}=0$ ) plotted against the inner-scaled (A 3e–g ) wall-distance $y^{+}$ (log scale and linear wall zoom).

Figure 9.

Examination of the influence of the Reynolds number $Re_{\unicode[STIX]{x1D70F}_{w}}$ (A 3k ) on the of production $P_{\unicode[STIX]{x1D700}_{ij}}=P_{\unicode[STIX]{x1D700}_{ij}}^{(1)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(2)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(3)}+P_{\unicode[STIX]{x1D700}_{ij}}^{(4)}$ (3.3) of the diagonal components of the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ (1.2a ), in wall units (A 3j ), by comparison of the present DNS computations of turbulent plane channel flow ( $Re_{\unicode[STIX]{x1D70F}_{w}}\approxeq 182$ , $\bar{M}_{CL}\approxeq 0.35$ ) with the incompressible DNS data of Vreman & Kuerten (Reference Vreman and Kuerten2014a ,Reference Vreman and Kuerten b , Reference Vreman and Kuerten2016, $~Re_{\unicode[STIX]{x1D70F}_{w}}\in \{180,590\}$ , $\bar{M}_{CL}=0$ ) plotted against the inner-scaled (A 3e–g ) wall distance $y^{+}$ (log scale and linear wall zoom).