6.1. Outer similarity

For $r$ large enough, i.e. $r\gg l_{i} (y)$ (where the inner length-scale $l_{i}$ is to be determined), the most natural choice for outer variables is $v=v_{o}=u_{\tau }$ and $l=l_{o}=y$ given that the distance to the wall should somehow determine the size of large eddies and that their characteristic velocity should scale with the skin friction velocity. With these outer variables, (6.4) becomes

(6.5)\begin{align} & \frac{\kappa}{4} \frac{s_{3}(r/y)}{r/y} - \frac{3\kappa y^{2}}{32{\rm \pi} r^{3} y^{+}}\int_{0}^{r} \rho^{2} \frac{{\rm d}^{2}[s_{2}(\rho/y)]}{{{\rm d}y}^{2}} {\rm d}\rho -\frac{3\kappa y^{2}}{4{\rm \pi} r y^{+}} \frac{{\rm d}}{{\rm d} r}[s_{2}(r/y)] \nonumber\\ &\quad \approx{-}1 - \frac{3}{16{\rm \pi} r^{3}}\int_{0}^{r} \rho^{2} s_{12}(\rho/y)\,{\rm d}\rho. \end{align}

In the limit $y^{+}\gg 1$, viscous diffusion (the second and third terms on the left-hand side) tends to $0$ as $1/y^{+}$ compared with the other terms. This equation therefore suggests outer asymptotic expansions in integer powers of ${1}/{y^{+}}$, which means that the outer similarity functions $s_2$, $s_3$ and $s_{12}$ may be approximated as

(6.6)$$\begin{gather} s_{2}^{o}(r/y, y^{+}) = s_{2}^{o,0} + \frac{1}{y^{+}} s_{2}^{o,1} + \cdots \end{gather}$$

(6.7)$$\begin{gather}s_{3}^{o}(r/y, y^{+}) = s_{3}^{o,0} + \frac{1}{y^{+}} s_{3}^{o,1} + \cdots \end{gather}$$

(6.8)$$\begin{gather}s_{12}^{o}(r/y, y^{+}) = s_{12}^{o,0} + \frac{1}{y^{+}} s_{12}^{o,1} + \cdots \end{gather}$$

with leading orders obeying

(6.9)\begin{equation} \frac{\kappa}{4} \frac{s_{3}^{o,0}(r/y)}{r/y} \approx{-}1 - \frac{3}{16{\rm \pi} r^{3}}\int_{0}^{r} \rho^{2} s_{12}^{o,0}(\rho/y)\,{\rm d}\rho. \end{equation}

The leading order outer scale-by-scale energy balance is therefore a balance between interscale transfer, turbulence dissipation and two-point turbulence production. (Turbulence dissipation appears in this outer balance essentially because the scale-by-scale energy balance that we consider concerns the sphere-averaged second-order structure function which is cumulative with increasing $r$.)

6.2. Inner similarity

For $r$ small enough, i.e. $r\ll l_{o}=y$, we seek inner variables of the form $v_{i}^{2}=v_{o}^{2}({1}/{y^{+}})^{a}=u_{\tau }^{2}({1}/{y^{+}})^{a}$ and $l_{i}=l_{o}({1}/{y^{+}})^{b}=y({1}/{y^{+}})^{b}$, where the exponents $a$, $b$ are positive because inner variables should tend to $0$ relative to outer ones in the limit where the local Reynolds number $y^{+}$ tends to infinity. With such variables, (6.4) becomes

(6.10)\begin{align} & \frac{\kappa}{4} \left(\frac{1}{y^{+}}\right)^{{3a}/{2} -b} \frac{s_{3}(r/l_{i})}{r/l_{i}} -O\left[\left(\frac{1}{y^{+}}\right)^{a+ 3-2b}\right] -\frac{3\kappa}{4{\rm \pi}} \left(\frac{1}{y^{+}}\right)^{a+ 1-2b}\frac{s^{\prime}_{2}(r/l_{i})}{r/l_{i}} \nonumber\\ &\quad \approx{-}1 - \frac{3}{16{\rm \pi} r^{3}}\int_{0}^{r} \rho^{2} \left(\frac{1}{y^{+}}\right)^{a}s_{12}(\rho/l_{i})\,{\rm d}\rho, \end{align}

where $s^{\prime }_{2}(r/l_{i})$ is the derivative of $s_2$ with respect to $r/l_{i}$. In the limit $y^{+}\gg 1$, the two-point turbulence production rate tends to $0$ as $(1/y^{+})^{a}$ compared with the dissipation rate which is represented in this equation by $-1$ on the right-hand side. At inner scales, the leading order scale-by-scale turbulence energy balance must therefore involve interscale energy transfer and viscous diffusion to balance dissipation, which implies $({3a}/{2}) -b=0=a+ 1-2b$ and therefore $a=1/2$ and $b=3/4$. In the limit $y^{+}\to \infty$, i.e. $y^{+}\gg 1$, this equation therefore suggests inner asymptotic expansions in integer powers of $({1}/{y^{+}})^{a} =({1}/{y^{+}})^{1/2}$, which means that the inner similarity functions $s_2$, $s_3$ and $s_{12}$ may be approximated as

(6.11)$$\begin{gather} s_{2}^{i}(r/l_{i}, y^{+}) = s_{2}^{i,0} + \left(\frac{1}{y^{+}}\right)^{1/2} s_{2}^{i,1} + \cdots, \end{gather}$$

(6.12)$$\begin{gather}s_{3}^{i}(r/l_{i}, y^{+}) = s_{3}^{i,0} + \left(\frac{1}{y^{+}}\right)^{1/2} s_{3}^{i,1} + \cdots, \end{gather}$$

(6.13)$$\begin{gather}s_{12}^{i}(r/l_{i}, y^{+}) = s_{12}^{i,0} + \left(\frac{1}{y^{+}}\right)^{1/2} s_{12}^{i,1} + \cdots \end{gather}$$

with leading orders obeying

(6.14)\begin{equation} \frac{\kappa}{4} \frac{s_{3}^{i,0}(r/l_{i})}{r/l_{i}} \approx{-}1 - \frac{3\kappa}{4{\rm \pi}} s_{2}^{{i,0}^{\prime}}(r/l_{i}), \end{equation}

where $s_{2}^{{i,0}^{\prime }}(r/l_{i})$ is the derivative of $s_{2}^{i,0}$ with respect to $r/l_{i}$. The leading order inner scale-by-scale energy balance is therefore a balance between interscale transfer, turbulence dissipation and viscous diffusion.

The values $a=1/2$ and $b=3/4$ that we derived imply that the inner variables are in fact Kolmogorov inner variables, i.e. $v_{i}=u_{\eta }\equiv (\nu \varepsilon ^{v})^{1/4}$ and $l_{i}=\eta \equiv (\nu ^{3}/\varepsilon ^{v})^{1/4}$ (using $\varepsilon ^{v} = u_{\tau }^{3}/(\kappa y)$).

6.3. Intermediate matching

Starting with the second-order structure function $S_2$, matching the leading term $u_{\tau }^{2}s_{2}^{o,0}(r/y)$ of its outer expansion for $r\gg \eta$ with the leading term $u_{\tau }^{2}({1}/{y^{+}})^{1/2} s_{2}^{i,0}(r/\eta )$ of its inner expansion for $r\ll y$ leads to

(6.15)\begin{equation} S_{2}^{0} \sim (\varepsilon^{v} r)^{2/3} \end{equation}

as the overlapping part of the leading order in the intermediate range $\eta \ll r \ll y$.

Similarly,

(6.16)\begin{equation} S_{12}^{0} \sim (\varepsilon^{v} r)^{2/3} \end{equation}

is the overlapping part of the leading order in the intermediate range $\eta \ll r \ll y$ for $S_{12}$.

It may be interesting to note, in passing, the difference compared to turbulence non-homogeneities with negligible turbulence production but non-negligible spatial turbulence transport, such as in certain turbulent wake regions where Chen & Vassilicos (Reference Chen and Vassilicos2022) have shown that a second-order structure function scales as $\sim K (r/L)^{2/3}$, where $K$ is the one-point kinetic energy, $L$ is an integral length scale and turbulence dissipation does not scale as $K^{3/2}/L$. Note that the $K^{3/2}/L$ scaling is effectively the scaling assumed here for $\varepsilon ^{v}$ because, in the range $\delta _{\nu }\ll y \ll \delta$ considered here, the turbulent kinetic energy scales as $u_{\tau }^{2}$ plus logarithmic corrections in $y$ (see Townsend Reference Townsend1976; Dallas et al. Reference Dallas, Vassilicos and Hewitt2009) which we neglect, and because there are integral length scales in FD TCF which are proportional to $y$, see Apostolidis et al. (Reference Apostolidis, Laval and Vassilicos2022). The types of non-homogeneity considered by Chen & Vassilicos (Reference Chen and Vassilicos2022) are opposite to those considered here where spatial turbulence transport is negligible but turbulence production is not.

To obtain the leading order of $S_{3}$, and therefore of the interscale transfer rate $\varPi ^{v}$ via (4.2), we use (6.9) and (6.14). From the leading order outer balance, (6.9) follows

(6.17)\begin{equation} S_{3}^{o,0} \approx{-}\varepsilon^{v} r (1 - A (r/y)^{2/3}), \end{equation}

where $A$ is a dimensionless constant, and from the leading order inner balance, (6.14) follows

(6.18)\begin{equation} S_{3}^{i,0} \approx{-}\varepsilon^{v} r (1 - B (r/\eta)^{{-}4/3}), \end{equation}

where $B$ is another dimensionless constant. The composite leading order (see Van Dyke Reference Van Dyke1964; Cole Reference Cole1968; Hinch Reference Hinch1991) written directly for the interscale transfer $\varPi ^{v}=S_{3}/r$ is $S_{3}^{o,0}/r$ plus $S_{3}^{i,0}/r$ minus their common part $-\varepsilon ^{v}$, i.e.

(6.19)\begin{equation} \varPi^{v}\approx{-}\varepsilon^{v} (1 - A (r/y)^{2/3} -B(r/\eta)^{{-}4/3}), \end{equation}

where we now omit superscripts for ease of notation.

This last equation has the following two verifiable implications, both of which are relatively easy to verify with the DNS data at our disposal. First, it implies that the value of $r$ where $\varPi ^{v}/\varepsilon ^{v}$ is minimal and closest to the Kolmogorov equilibrium value $-1$ is

(6.20)\begin{equation} r_{min} \sim \sqrt{\delta_{\nu} y} \sim \lambda \end{equation}

based on the definition $\lambda ^{2}\equiv 10\nu K/\varepsilon$ (already used by Dallas et al. (Reference Dallas, Vassilicos and Hewitt2009) in the context of FD TCF), and on $K\sim u_{\tau }^{2}$ and $\varepsilon \sim u_{\tau }^{3}/y$ being good enough approximations in the present context for $\delta _{\nu }\ll y \ll \delta$. Conclusions such as (6.19) and (6.20) have recently been obtained by Zimmerman et al. (Reference Zimmerman, Antonia, Djenidi, Philip and Klewicki2022) for the centreline of FD TCF and central axis of turbulent pipe flow where turbulence production is effectively absent.

Second, (6.19) also implies that the value $(\varPi ^{v}/\varepsilon ^{v})_{min}$ of $\varPi ^{v}/\varepsilon ^{v}$ at $r=r_{min}$ obeys

(6.21)\begin{equation} 1+(\varPi^{v}/\varepsilon^{v})_{min} \sim y^{+^{{-}1/3}}\sim Re_{\lambda}^{{-}2/3}, \end{equation}

where $Re_{\lambda } = \sqrt {K}\lambda /\nu$. Consistent with our averages over spheres in $\boldsymbol {r}$-space, these definitions of $\lambda$ and $Re_{\lambda }$ ignore some anisotropies of FD TCF. It is possible to define different Taylor lengths for different directions so as to take explicit account of anisotropies, which is an approach we have taken in another study (Yuvaraj Reference Yuvaraj2022). It may be noteworthy that the Corrsin length (Sagaut & Cambon Reference Sagaut and Cambon2018) does not appear spontaneously from our analysis, whereas the Kolmogorov and Taylor lengths do. The reason for this absence of the Corrsin length is that it equals $\kappa y$ at the approximation level of our theory in the intermediate layer $\delta _{\nu }\ll y \ll \delta$ and is therefore comparable to the outer bound of the range $r\le 2y$ considered here.

In conclusion, the non-homogeneous but statistically stationary case of FD TCF in the intermediate layer $\delta _{\nu }\ll y \ll \delta$ is such that Kolmogorov equilibrium is achieved asymptotically around $\lambda$ and therefore not quite in an inertial range given that $\lambda$ depends on viscosity, and that there is a systematic departure from equilibrium when moving away from $\lambda$, both towards $L$ and towards $\eta$, see (6.19). (Note, however, that the non-zero deviation from Kolmogorov equilibrium as Reynolds number tends to infinity for a fixed small value of $r/y$ or for a fixed large value of $r/\eta$ (necessarily smaller than $\lambda /\eta$ in the limit) is small.) This is the same conclusion that the analysis of Lundgren (Reference Lundgren2002) reached for freely decaying, i.e. non-stationary, but statistically homogeneous and isotropic turbulence far from initial conditions. Two-point turbulence production (which increases with $r$ as confirmed in the following section) and its variation with wall-normal distance play a similar role in FD TCF as the rate of decay of the second-order velocity structure function (which increases with $r$ because unsteadiness increases with $r$) and its variation with time.

8.1. Aligned/anti-aligned decomposition

Equation (4.2) shows that a scale-space flux and a cascade from large to small or from small to large scales correspond to a negative or positive $({3}/{4{\rm \pi} })\int \langle ({\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u}}/{r}) \vert \delta \boldsymbol {u}\vert ^{2}\rangle \,{\rm d}\varOmega _{\boldsymbol {r}}$ and contributes a growth or decrease of TKE at scales $r$ and smaller (see Chen & Vassilicos Reference Chen and Vassilicos2022). Local compression, i.e. $\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {r} < 0$, causes local forward cascade and local stretching, i.e. $\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {r} > 0$, causes local inverse cascade. Our observation that $\varPi ^{v}$ is negative at all scales means that local compressions prevail at all scales, but are they mostly caused by aligned or anti-aligned pairs of fluctuating velocities? This question introduces our first decomposition, namely

(8.1)\begin{equation} \varPi^{v} = \varPi^{v}_{\rightrightarrows} + \varPi^{v}_{\rightleftarrows}= \frac{3}{4 {\rm \pi}} \int \left\langle \frac{\delta \boldsymbol{u}\boldsymbol{\cdot} \hat{\boldsymbol{r}}}{r}\vert \delta \boldsymbol{u}\vert^2\right\rangle_{\rightrightarrows} \,{\rm d}\varOmega_{r} + \frac{3}{4 {\rm \pi}} \int \left\langle \frac{\delta \boldsymbol{u}\boldsymbol{\cdot} \hat{\boldsymbol{r}}}{r}\vert \delta \boldsymbol{u}\vert^2\right\rangle_{\rightleftarrows}\,{\rm d}\varOmega_{r}, \end{equation}

where $\varPi ^{v}_{\rightrightarrows }$ and $\varPi ^{v}_{\rightleftarrows }$ are respectively equal to the first and second terms on the left-hand side which are calculated using averages $\langle \cdots \rangle _{\rightrightarrows }$ conditional on $\boldsymbol {u}^+ \boldsymbol {\cdot } \boldsymbol {u}^{-}>0$ and averages $\langle \cdots \rangle _{\rightleftarrows }$ conditional on $\boldsymbol {u}^+ \boldsymbol {\cdot }\boldsymbol {u}^{-}<0$.

Compressive and stretching relative motions may not balance in terms of energy transfer, resulting in a non-vanishing $\varPi ^{v}$, but they do balance in terms of mass transfer because of incompressibility which implies $\int \delta \boldsymbol {u} \boldsymbol {\cdot } \hat {r}\,{\rm d}\varOmega _{r} =0$. Hence,

(8.2)\begin{equation} \int \langle \delta \boldsymbol{u}\boldsymbol{\cdot}\hat{\boldsymbol{r}}\rangle_{\rightrightarrows} \,{\rm d}\varOmega_{r} + \int \langle \delta \boldsymbol{u}\boldsymbol{\cdot}\hat{\boldsymbol{r}}\rangle_{\rightleftarrows} \,{\rm d}\varOmega_{r}=0. \end{equation}

In figure 10, we plot both terms on the left-hand side of this equation as functions of $r$ for various wall distances $y$. We also plot $\int \delta \boldsymbol {u} \boldsymbol {\cdot } \hat {r} \,{\rm d}\varOmega _{r}$ for comparison and as a check that it is indeed zero in the DNS irrespective of $r$ and $y$. The first observation is that aligned fluctuation pairs are stretching relative motions on average given the positive sign of $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightrightarrows } \,{\rm d}\varOmega _{r}$. The joint PDFs of figure 11 show that relative motions of aligned fluctuation pairs are stretching as a result of $\delta \boldsymbol {u}$ having a tendency to be directed in the same direction as the separation vector $\boldsymbol {r}$ for pairs of aligned fluctuating velocities. This tendency weakens with increasing $r$ irrespective of wall distance $y$ and, consistently, $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightrightarrows } \,{\rm d}\varOmega _{r}$ tends to $0$ with increasing $r$.

Figure 10.

$\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle {\rm d}\varOmega _{r}$ integrated over the whole sphere in black lines, conditionally integrated over anti-aligned pairs in marine colours, and conditionally integrated over aligned pairs in orange colours. Wall-normal distance is increased from light to dark colours. (a) $Re_\tau =932$, (b) $Re_\tau =2003$. (c) $r/\lambda$ positions of the minima/maxima observed in panel (a) as a function of wall-distance $y^+$ for $Re_\tau =932$, similarly in panel (d) for $Re_\tau =2003$.

Figure 11.

Joint probability distribution functions (JPDFs) of $\delta \boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\hat {r}}/|\delta \boldsymbol {u}|$ and $\boldsymbol {u}^{-}\boldsymbol {u}^{+}/\sqrt {|\boldsymbol {u}^{+}|^2|\boldsymbol {u}^{-}|^2}$. (a) For $Re_\tau =932$ and wall-distance $y^+=257$, four different JPDFs with increasing scale $r/\lambda =0.38,0.57,1.05$ and $2.91$. (b) Similarly for $Re_\tau =2003$ and wall-distance $y^+=456$, the JPDFs correspond to scales $r/\lambda =0.35,0.56,1.10$ and $3.43$. The joint PDFs are normalised with their maximum value. Above each JPDF, we also plot the conditional PDF of $\delta \boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\hat {r}}/|\delta \boldsymbol {u}|$, conditioned on aligned (red lines) and anti-aligned (blue lines) pairs.

The second observation in figure 10 is that anti-aligned fluctuation pairs are compressing relative motions, on average, given the negative sign of $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightleftarrows }\,{\rm d}\varOmega _{r}$. Looking at figure 11, it does not seem possible to explain this behaviour purely in terms of velocity directions. However, the joint PDFs of figure 12 reveal that the range of values over which $\delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}$ fluctuates around zero is much wider for anti-aligned than for aligned fluctuations. This effect has to do with the intensity of the fluctuating velocities, not only their relative directions. This very wide fluctuation range is slightly skewed towards negative values of $\delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}$ for pairs of fluctuating velocities which are anti-aligned, thereby accounting for the compressive average behaviour of anti-aligned pairs ($\boldsymbol {u}^+ \boldsymbol {\cdot } \boldsymbol {u}^{-}<0$). This skewness diminishes with increasing $r$ irrespective of wall distance $y$ and, consistently, $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightleftarrows } \,{\rm d}\varOmega _{r}$ tends to $0$ with increasing $r$. Note, finally, that it is far more likely to find aligned ($\boldsymbol {u}^+ \boldsymbol {\cdot } \boldsymbol {u}^{-}>0$) than anti-aligned ($\boldsymbol {u}^+ \boldsymbol {\cdot } \boldsymbol {u}^{-}<0$) pairs as figure 11 shows.

Figure 12.

Joint probability distribution functions (JPDFs) of $\delta \boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\hat {r}}$ and $\boldsymbol {u}^{-}\boldsymbol {u}^{+}$. (a) For $Re_\tau =932$ and wall-distance $y^+=257$, four different JPDFs with increasing scale $r/\lambda =0.38,0.57,1.05$ and $2.91$. (b) Similarly for $Re_\tau =2003$ and wall-distance $y^+=456$, the JPDFs correspond to scales $r/\lambda =0.35,0.56,1.10$ and $3.43$. The joint PDFs are normalised with their maximum value, while the values of the $x$ and $y$ axis are normalised with their own standard deviations.

The third observation in figure 10 is that $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightleftarrows } \,{\rm d}\varOmega _{r}$ has a minimum at $r=r_m$ near $r_{min}\approx 1.2\lambda$ for all $y$ and that $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightrightarrows } \,{\rm d}\varOmega _{r}$ has a maximum at the same value $r=r_m$ for all $y$. As seen in the previous two sections, $r_{min}$ is the value of $r$ where $\varPi ^{v}/\varepsilon ^{v}$ has its minimum. In figure 10(c,d), we plot the positions $r$ of the maxima and minima in figure 10 vs wall distance for both DNS Reynolds numbers at our disposal. It is quite striking that, for all wall distances and both Reynolds numbers tried, $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightleftarrows }\,{\rm d}\varOmega _{r}$ and $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightrightarrows }\,{\rm d}\varOmega _{r}$ peak at $r=r_m$ close to the value $r=r_{min}$, where $\varPi ^{v}/\varepsilon ^{v}$ peaks and is closest to the equilibrium $-1$ value. Even though $r_m$ drifts slightly from $r_{min}\approx 1.2\lambda$ at relatively high wall-normal distances, the suggestion is that, in the layer $\delta _{\nu }\ll y \ll \delta$ of FD TCF, Kolmogorov-like equilibrium may be achieved at those length scales $r$ where aligned fluctuating velocities are stretching with their difference $\delta \boldsymbol {u}$ maximally or near-maximally aligned with the separation vector $\boldsymbol {r}$ and where anti-aligned fluctuations are maximally or near-maximally skewed towards large negative values of $\delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}$. This is a conclusion that is well beyond the reach of the theory in § 6 but which we might not have been able to reach without it. (We refer to Kolmogorov-like rather than Kolmogorov equilibrium because the scale $r_{min}$ is proportional to the Taylor scale and therefore depends on viscosity.)

It is shown in § 5 that anti-aligned fluctuation pairs enhance the positive two-point turbulence production rate in the layer $\delta _{\nu }\ll y \ll \delta$ of FD TCF: we have now seen that these anti-aligned fluctuation pairs are, on average, compressive and figure 13 shows that $\varPi ^{v}_{\rightleftarrows }$ is consistently negative, indicating forward cascade. Therefore, anti-aligned fluctuations do not only enhance the two-point production rate at all $r$, they also contribute a forward cascade at all $r$ in the layer $\delta _{\nu }\ll y \ll \delta$ of FD TCF. Note, however, that the minimum value of $\varPi ^{v}_{\rightleftarrows }$ is not at $r=r_{min}$, where $\varPi ^{v}/\varepsilon ^{v}$ has its minimum value and is closest to the equilibrium $-1$ value. In fact, the $r$-position of the minimum value of $\varPi ^{v}_{\rightleftarrows }$ does not scale with $\lambda$. The scaling of $r_{min}$ therefore requires taking into account both aligned and anti-aligned fluctuations.

Figure 13.

Decomposition of the interscale transfer term $\varPi ^{v}$ (black lines) into $\varPi ^{v}_{\rightleftarrows }$ (blue lines) and $\varPi ^{v}_{\rightrightarrows }$ (red lines): (a) $Re_\tau =932$; (b) $Re_\tau =2003$. Wall-normal distance is increased from light to dark colours.

Aligned fluctuation pairs impose a loss of energy on scales smaller than $r$ by mean flow interaction with turbulence fluctuations and thereby reduce the one-point effect of sweeps and ejections on the two-point turbulence production rate (see § 5). We have now seen that aligned fluctuation pairs are on average stretching, which would suggest the presence of an average inverse cascade element to the interscale transfer rate $\varPi ^{v}_{\rightrightarrows }$. Figure 13 shows that $\varPi ^{v}_{\rightrightarrows }$ is positive (though only slightly so), and an average inverse cascade by aligned fluctuations is indeed present at scales $r$ larger than approximately 2 to 3 times $\lambda$ for the Reynolds numbers of the DNS data used here. However, figure 13 also shows that $\varPi ^{v}_{\rightrightarrows }$ is negative at smaller scales. Stretching aligned fluctuating motions at scales of the order of the Taylor length and below may dominate over compressive aligned fluctuating motions, on average, but they do not dominate interscale energy transfer at these scales. There is no contradiction with the positive values of $\int \langle \delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}\rangle _{\rightrightarrows }\,{\rm d}\varOmega _{r}$ in figure 10. The different signs of this solid angle integral and the solid angle integral in the definition of $\varPi ^{v}_{\rightrightarrows }$ (see (8.1)) are an effect of small-scale anisotropies over which we are averaging. Future studies of interscale transfers in FD TCFs will need to take these anisotropies into account for a finer description of the physics.

Finally, comparing the plots of $\varPi ^{v}$ in figure 5 with those of $\varPi ^{v}_{\rightrightarrows }$ and $\varPi ^{v}_{\rightleftarrows }$ in figure 13 shows that $\varPi ^{v}_{\rightleftarrows }$ dominates over $\varPi ^{v}_{\rightrightarrows }$ at scales of the order of $\lambda$ and larger, and is mostly responsible for the value of $\varPi ^{v}$. At smaller scales, however, $\varPi ^{v}_{\rightrightarrows }$ becomes equally important and of the same negative sign as $\varPi ^{v}_{\rightleftarrows }$ so that the actual negative value of $\varPi ^{v}$ cannot be accounted for by only one or the other: the interscale turbulence energy transfers of both aligned and anti-aligned fluctuations matter.

8.2. Homogeneous/Inhomogeneous energy transfer decomposition

As already mentioned at the start of § 8.1, the right-hand side $({3}/{4{\rm \pi} })\int \langle ({\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u}}/{r}) \vert \delta \boldsymbol {u}\vert ^{2}\rangle {\rm d}\varOmega _{\boldsymbol {r}}$ of (4.2) shows that local compression, i.e. $\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {r} < 0$, causes local forward cascade, whereas local stretching, i.e. $\delta \boldsymbol {u}\boldsymbol {\cdot } \hat {r} > 0$, causes local inverse cascade (see also § 2 of Chen & Vassilicos Reference Chen and Vassilicos2022). These compressions and stretches may be caused either by turbulence inhomogeneities or by correlated ‘eddy’ motions. In an attempt to formalise this distinction, Alves Portela, Papadakis & Vassilicos (Reference Alves Portela, Papadakis and Vassilicos2020) decomposed the interscale energy transfer rate $\varPi = ({\partial }/{\partial r_i}) ( \delta u_i |\delta \boldsymbol {u}|^2 )$ as follows:

(8.3) \begin{equation} \frac{\partial}{\partial r_i} (\delta u_i |\delta \boldsymbol{u}|^2) = \frac{\partial}{\partial r_i} [\delta u_i(|\boldsymbol{u}^+|^2+|\boldsymbol{u}^-|^2)] -2\frac{\partial}{\partial r_i}(\delta u_i \boldsymbol{u}^- \boldsymbol{\cdot} \boldsymbol{u}^+), \end{equation}

where the first term on the right-hand side can be rigorously recast into a gradient in centroid $\boldsymbol {x}$-space leading to

(8.4) \begin{align} \frac{\partial}{\partial r_i} (\delta u_i |\delta \boldsymbol{u}|^2) &= \frac{1}{2}\frac{\partial}{\partial x_i}[ u_i^+ |\boldsymbol{u}^+|^2 + u_i^- |\boldsymbol{u}^-|^2 - u_i^- |\boldsymbol{u}^+|^2 - u_i^+|\boldsymbol{u}^-|^2] \nonumber\\ &\quad -2\frac{\partial}{\partial r_i}(\delta u_i \boldsymbol{u}^- \boldsymbol{\cdot}\boldsymbol{u}^+). \end{align}

Here, $\varPi _I \equiv \frac {1}{2}({\partial }/{\partial x_i})[ u_i^+ |\boldsymbol {u}^+|^2 + u_i^- |\boldsymbol {u}^-|^2 - u_i^- |\boldsymbol {u}^+|^2 - u_i^+ |\boldsymbol {u}^-|^2 ]$ is interpreted as an inhomogeneity-related interscale turbulent energy transfer rate. In statistically homogeneous turbulence, the average $\langle \varPi _I \rangle$ is indeed zero and the interscale turbulent energy transfer rate is only accountable to $\varPi _{H}\equiv -2({\partial }/{\partial r_i})(\delta u_i \boldsymbol {u}^- \boldsymbol {\cdot } \boldsymbol {u}^+)$ on average.

Integrating $\varPi$, $\varPi _{I}$ and $\varPi _H$ over the sphere of radius $r$ in $\boldsymbol {r}$-space to obtain $\varPi ^v$, $\varPi _{I}^v$ and $\varPi _{H}^{v}$, respectively, and then applying the Gauss divergence theorem, we obtain

(8.5)\begin{align} \varPi^{v} = \varPi_I^{v} + \varPi_H^{v} &= \frac{3}{4{\rm \pi}} \left(\int\left\langle\frac{\delta \boldsymbol{u} \boldsymbol{\cdot} \hat{\boldsymbol{r}}}{r}(|\boldsymbol{u}^+|^2 + |\boldsymbol{u}^-|^2) \right\rangle \,{\rm d} \varOmega_r \right. \nonumber\\ &\quad \left.- 2\int\left\langle\frac{\delta \boldsymbol{u} \boldsymbol{\cdot} \hat{\boldsymbol{r}}}{r}(\boldsymbol{u}^- \boldsymbol{\cdot} \boldsymbol{u}^+) \right\rangle \,{\rm d} \varOmega_r \right). \end{align}

This decomposition is partly related to that in § 8.1 because $\varPi _H^{v}$ is linearly dependent on correlations between $\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {r}$ and $\boldsymbol {u}^- \boldsymbol {\cdot } \boldsymbol {u}^+$, and the sign of $\boldsymbol {u}^- \boldsymbol {\cdot } \boldsymbol {u}^+$ indicates whether velocity fluctuation pairs are aligned or anti-aligned, which is the basis of decomposition (8.1). Whilst it follows immediately from (8.4) that $\varPi _I^{v} =0$ if the term inside the $\boldsymbol {x}$-gradient in that equation is statistically homogeneous, (8.5) shows that $\varPi _I^{v}=0$ if $\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {\boldsymbol {r}}$ and $(|\boldsymbol {u}^+|^2 + |\boldsymbol {u}^-|^2)$ are uncorrelated and if $(|\boldsymbol {u}^+|^2 + |\boldsymbol {u}^-|^2)$ is statistically homogeneous. Of course, this is not the only and necessary way for $\varPi _I^{v}$ to vanish. In particular, there may be cases of non-homogeneity for which $\varPi _I^{v}$ vanishes too, for example, cases where $\varPi _I^{v}$ vanishes but $\varPi _I$ does not.

In figure 14, we plot the terms $\varPi _{I}^{v}$ and $\varPi _{H}^{v}$ in 8.5 normalised by the volume integral of the dissipation. For both Reynolds numbers, we observe that $\varPi _{H}^{v}$ dominates and describes almost perfectly the full interscale transfer $\varPi ^{v}$ for all scales $r \le 2y$ in the intermediate range of the channel ($y$ between multiples of $\delta _{\nu }$ and approximately half $\delta$). The average interscale transfer from large to small scales is nearly fully described by the negative value of $\varPi _{H}^{v}$ and the inhomogeneity-related interscale transfer rate $\varPi _{I}^{v}$ is close to zero. In a different non-homogenous turbulent flow, the turbulent wake of a square prism, Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2020) found a significant contribution of the inhomogeneity-related interscale transfer rate to the total interscale transfer rate. It is therefore not trivial that in FD TCF, $\varPi _{I}^{v}$ is negligible compared to $\varPi _{H}^{v}$ in spite of the statistical non-homogeneity of the FD TCF. However, this is partly an artefact of the integration over spheres in $\boldsymbol {r}$-space which we apply to $\varPi _{I}$ to obtain $\varPi _{I}^{v}$. If we lift this integration and use the DNS data to compute $\varPi _{I} (y, r_{1}, r_{2}, r_{3})$ as a function of $r_{2}/y$ for various values of wall-normal distance $y$ and various values of $r_1$ and $r_3$, we find (figure 15) that $\varPi _{I} (y, r_{1}, r_{2}, r_{3})$ is close to $0$ and negligible in most cases except for ‘attached eddies’, i.e. for values of $r_2$ relatively close to $2y$ (wall blocking implies $r_{2}\le 2y$) where it is positive, thereby potentially reflecting interscale transfer from small to large scales (similarly to Cimarelli et al. Reference Cimarelli, De Angelis, Jimenez and Casciola2016; Cho, Hwang & Choi Reference Cho, Hwang and Choi2018) except for $r_2$ near-equal to $2y$ where it is negative. The non-vanishing inhomogeneity-related interscale transfer of ‘attached eddies’ is averaged out when we integrate $\varPi _{I}$ to obtain $\varPi _{I}^{v}$.

Figure 14.

Interscale transfer rate (blue lines), inhomogeneous part $\varPi _I^{v}$ (red lines) and homogeneous part $\varPi _H^{v}$ (green lines), all integrated over the volume of a sphere and normalised by the dissipation rate integrated over the volume of the sphere as a function of $r/\lambda$. Wall-normal distance is increased from light to dark colours. (a) For $Re_\tau =932$ and (b) for $Re_\tau =2003$.

Figure 15.

$\varPi$ (blue markers), $\varPi _I$ (red lines) and $\varPi _H$ (green lines) normalised with the two-point dissipation rate $\varepsilon$ vs the wall-normal scale $r_2$ divided with $y$. (a) $Re_\tau =932$, from left to right we increase the streamwise scale $r_1$ and from top to bottom the spanwise scale $r_3$. (b) Similarly for $Re_\tau =2003$. Wall-normal distance is increased from light to dark colours.

Returning to $\varPi _{H}^{v}$ and the fact that it has very similar dependencies on $r$ and $y$ as $\varPi ^{v}$, we note, in particular, that $\varPi _{H}^{v}$ has a minimum at the near same $r\approx r_{min}$ where $\varPi ^v$ has a minimum, and even that the minimum value of $\varPi _{H}^{v}$ closely obeys the same relation (6.21) that $\varPi ^{v}_{min}$ obeys (see figure 16). As seen in § 6, the two-point separation scale $r=r_{min}$ demarcates between smaller values of $r$ where $\varPi ^{v}$ is balanced by dissipation and viscous diffusion and larger values of $r$ where $\varPi ^{v}$ is balanced by dissipation and two-point turbulence production. However, the theory of § 6, which is conclusive for $\varPi ^{v}$, has no say on $\varPi _{H}^{v}$ and can therefore not explain our observation that $\varPi _{H}^{v}$ behaves very much like $\varPi ^{v}$. We therefore adopt a different point of view from that in § 6 and look at PDFs of instantaneous (in time) and local (in $(x,z)$ planes) interscale transfer rates ${\rm \pi} ^{v} \equiv ({3}/{4 {\rm \pi}}) \int ({\delta \boldsymbol {u}\boldsymbol {\cdot } \hat {\boldsymbol {r}}}/{r})\vert \delta \boldsymbol {u}\vert ^2\,{\rm d}\varOmega _{r}$, ${\rm \pi} _{H}^{v}\equiv -({3}/{2{\rm \pi} }) \int ({\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {\boldsymbol {r}}}/{r})(\boldsymbol {u}^- \boldsymbol {\cdot } \boldsymbol {u}^+) \,{\rm d} \varOmega _r$ and ${\rm \pi} _{I}^{v} \equiv ({3}/{4{\rm \pi} }) \int ({\delta \boldsymbol {u} \boldsymbol {\cdot } \hat {\boldsymbol {r}}}/{r})(|\boldsymbol {u}^+|^2 + |\boldsymbol {u}^-|^2) \,{\rm d} \varOmega _r$. Clearly, $\varPi ^{v} = \langle {\rm \pi}^{v} \rangle$, $\varPi _{H}^{v} = \langle {\rm \pi}_{H}^{v} \rangle$ and $\varPi _{I}^{v} = \langle {\rm \pi}_{I}^{v} \rangle$.

Figure 16.

(a) Values of $r/\lambda$ where minima of $\varPi ^{v}/\varepsilon ^{v}$ and minima of $\varPi _{H}^{v}/\varepsilon ^{v}$ are observed as functions of wall distance $y^+$. (b) Values of $1+(\varPi ^{v}/\varepsilon ^{v})_{min}$ (in blue) and of $1+(\varPi _{H}^{v}/\varepsilon ^{v})_{min}$ (in green), as functions of $Re_\lambda$. Dashed line shows a scaling of $Re_\lambda ^{-2/3}$. Circle markers for $Re_\tau =932$, square markers for $Re_\tau =2003$.

In figure 17, we plot examples of PDFs of ${\rm \pi} ^{v}$, ${\rm \pi} _{H}^{v}$ and ${\rm \pi} _{I}^{v}$ for a couple of wall distances $y$ within the intermediate range $\delta _{\nu }\ll y\ll \delta$ and for different values of separation scale $r$ to see how these PDFs evolve with varying $r$. As pointed out by Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2020), at $r=0$, we have $\varPi ^{v}= \varPi _{H}^{v}=\varPi _{I}^{v}=0$. As $r$ progressively increases, the PDFs of ${\rm \pi} ^{v}$ and ${\rm \pi} _{H}^{v}$ move to the left towards increasingly negative values, as shown in the insets of figure 17(a,b,e, f). This overall PDF drift is most pronounced at the smaller values of $r$, and causes $\varPi ^{v}$ and $\varPi _{H}^{v}$ to progressively decrease below $0$ as $r$ increases. However, the skewnesses of the PDFs of ${\rm \pi} ^{v}$ and of ${\rm \pi} _{H}^{v}$ grow from negative values close to $-10$ at the smallest separations $r$ to values between $-1$ and even slightly positive as $r$ grows (see figure 18(a,b,e, f)). This evolution of the skewnesses of these two PDFs counteracts their overall drift towards increasingly negative values and acts to bring $\varPi ^{v}$ and $\varPi _{H}^{v}$ back towards zero as $r$ increases. The minima of $\varPi ^{v}$ and $\varPi _{H}^{v}$ occur as a result of these two counteracting tendencies, the overall drift dominating at scales $r$ smaller than $r_{min}$ and causing $\varPi ^{v}$ and $\varPi _{H}^{v}$ to decrease, and the decreasingly skewed PDF dominating at scales larger than $r_{min}$ and causing $\varPi ^{v}$ and $\varPi _{H}^{v}$ to increase.

Figure 17.

Probability density functions (PDFs) of (a,b) ${\rm \pi} ^{v}$, (c,d) ${\rm \pi} _I^{v}$ and (e, f) ${\rm \pi} _H^{v}$ normalised with their respective maximum probability. The values of the terms are normalised with their own standard deviation. The scale $r$ is increased from light to dark colours. (a,c,e) $Re_\tau =932$, (b,d, f) $Re_\tau =2003$. Inset is a zoom of the area close to the maximum probability in lin–lin axes.

Figure 18.

Skewness factor of ${\rm \pi} ^{v}$ in blue colours, ${\rm \pi} _I{^v}$ in red colours and of ${\rm \pi} _H^{v}$ in green colours as a function of $r/\lambda$, for different wall-normal locations. The wall-distance $y$ is increased from light to dark colours. (a) For $Re_\tau =932$ and (b) for $Re_\tau =2003$.

The PDF of the inhomogeneity-related interscale transfer rates ${\rm \pi} _{I}^{v}$ is radically different as far as skewness is concerned (see figure 18). Whilst the PDFs of both ${\rm \pi} ^{v}$ and ${\rm \pi} _{H}^{v}$ are skewed towards forward cascade events at small $r$ and evolve with increasing $r$ towards not being skewed or even being slightly skewed towards inverse cascade events, the PDF of ${\rm \pi} _{I}^{v}$ is highly skewed towards inverse cascade events at small $r$ and evolves very quickly with increasing $r$ towards not being very skewed. It remains only slightly skewed (positively or negatively) for all permissible $r$ larger than approximately $2\lambda$ (the word ‘permissible’ refers to $r\le 2y$). The difference is not only that the PDF of ${\rm \pi} _{I}^{v}$ is oppositely skewed to the PDFs of ${\rm \pi} ^{v}$ and ${\rm \pi} _{H}^{v}$ at small $r$, but the equally if not even more important difference is that, as $r$ increases, the skewness of ${\rm \pi} _{I}^{v}$ evolves much faster towards small absolute values (which it actually reaches at $r\approx 2\lambda$) than the skewnesses of ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^v$ which evolve much more gradually towards values around and larger than $-1$.

However, the PDF of ${\rm \pi} _{I}^{v}$ is similar to the PDFs of ${\rm \pi} ^{v}$ and ${\rm \pi} _{H}^{v}$ in that they all have an overall drift to the left, i.e. towards forward cascading negative values, as the separation scale $r$ increases (see insets of plots in figure 17). In the case of the inhomogeneity-related interscale energy transfer rate, this overall PDF drift towards forward cascade events is counteracted at small separations $r$ by the significant PDF skewness towards inverse cascade events leading to small values of $\varPi _{I}^{v}$. As $r$ increases, the drift slows down, and the skewness quickly drops to small absolute values keeping values of $\varPi _{I}^{v}$ small.

In conclusion, the statistics of the inhomogeneity-related interscale transfer rate ${\rm \pi} _{I}^{v}$ are very different from those of ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$. The PDFs of ${\rm \pi} _{I}^{v}$ are characterised by a skewness towards inverse cascade events at the small scales in particular, whereas the PDFs of both ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$ are characterised by a skewness towards forward cascade events at most scales. These PDFs result in relatively small values of $\varPi _{I}^{v}$ and in very similar dependencies on separation $r$ of $\varPi _{H}^{v}$ and $\varPi ^{v}$. As the separation scale $r$ decreases from large values close to $2y$ towards the Taylor length $\lambda$, the PDFs of both ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$ become increasingly skewed towards forward cascading events and the average values $\varPi _{H}^{v}$ and $\varPi ^{v}$ become increasingly negative. However, as $r$ crosses $\lambda$ and tends towards even smaller separation lengths below $\lambda$, these two PDFs drift towards inverse cascading events in their entirety, thereby bringing the average values of $\varPi _{H}^{v}$ and $\varPi ^{v}$ back towards zero.

These two counteracting effects of drift and skewness remain and are therefore confirmed if we consider only the tails of the PDFs of ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$. In figure 19(a,b), we plot the average values of ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$ over the samples of relatively intense values representing only $1\,\%$ of all samples. The average of ${\rm \pi} _{H}^{v}$ over its relatively intense values depends on $y$ and $r$ very much like $\varPi _{H}^{v}$ but with an order of magnitude higher values (compare with figure 14). However, the average of ${\rm \pi} _{I}^{v}$ over these relatively intense values is disproportionally affected by the PDF's positive skewness, and is therefore positive or close to zero and higher than $\varPi _{I}^{v}$ in figure 14 as the cancelling effect of the drift is overcome. To concentrate on the drift and minimise the effect of the skewness, in figure 19(c,d), we report average values of ${\rm \pi} _{H}^{v}$, ${\rm \pi} _{I}^{v}$ and ${\rm \pi} ^{v}$ calculated on the basis of only the most probable part of the PDFs representing $20\,\%$ of all samples. These average values are an order of magnitude smaller than $\varPi _{H}^{v}$, $\varPi _{I}^{v}$ and $\varPi ^{v}$ in figure 14. They are close to zero at the smallest separations $r$ and continuously decrease in negative values till they more or less stabilise at large enough $r$, reflecting the effect of overall drift of the PDFs towards forward interscale transfers and the fact that this drift stabilises at large enough $r$. Without the skewness effect, which is not as present around the peaks of the PDFs as in their extreme tails, these conditional averages (figure 19c,d) do not significantly return towards $0$ with increasing $r$ and therefore look very different from $\varPi _{H}^{v}$, $\varPi _{I}^{v}$ and $\varPi ^{v}$ in figure 14. The averages $\varPi _{H}^{v}$, $\varPi _{I}^{v}$ and $\varPi ^{v}$ in this latter figure emerge as a weighted sum of the conditional averages in figure 19(a,b) with those in figure 19(c,d).

Figure 19.

(a,b) Averages of most intense events accounting for $1\,\%$ of all events for $\varPi ^{v,e}$ (blue lines), $\varPi _I^{v,e}$ (red lines) and $\varPi _H^{v,e}$ (green lines). (c,d) Averages of most probable events accounting for $20\,\%$ of all events for $\varPi ^{v,p}$ (blue lines), $\varPi _I^{v,p}$ (red lines) and $\varPi _H^{v,p}$ (green lines). (a,c) For $Re_\tau =932$, (b,d) for $Re_\tau =2003$. Wall-normal distance is increased from light to dark colours.

Note, finally, that the skewness dominated $r$-range of the PDFs of ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$ coincides with the $r$-range where $\varPi ^{v}$ is balanced by turbulent dissipation rate and two-point turbulence production. The root cause of this coincidence may be anti-aligned velocity fluctuation pairs because they enhance two-point turbulence production (§ 5) while also being the seat of a significant skewness towards compressive, i.e. forward cascading, relative motions (§ 8.1). The drift of the PDFs of ${\rm \pi} _{H}^{v}$ and ${\rm \pi} ^{v}$ towards inverse cascades is in fact, a recentering of the PDFs so that their peak values move towards zero and is mostly present in the $r$-range where $\varPi ^{v}$ is balanced by turbulent dissipation rate and viscous diffusion (see § 6). At these small scales comparable to $\lambda$ and below, both aligned and anti-aligned fluctuation pairs contribute significantly to $\varPi ^{v}$ (see end of § 8.1) and this may be related to the recentring of the PDFs around zero interscale transfer rate.