1. Introduction
Kolmogorov's equation (Kolmogorov Reference Kolmogorov1941a) describes the scale dependence of the downscale transfer of turbulent kinetic energy (TKE) in the absence of anisotropy and non-stationarity effects. This equation may be written as
\begin{equation} D_{LLL} = 6\nu\frac{\partial}{\partial r}D_{LL} - \frac{4}{5}\langle\epsilon\rangle r, \end{equation}
where 
$D_{LL}$ and 
$D_{LLL}$ are respectively the second- and third-order longitudinal velocity structure functions, 
$r\equiv |\boldsymbol {r}|$ is the magnitude of the separation vector, 
$\nu$ is the kinematic viscosity and 
$\langle \epsilon \rangle$ is the mean dissipation rate of the TKE. For sufficiently large values of 
$r$, the viscous term becomes negligible and (1.1) reduces to the famous ‘4/5’ law
\begin{equation} D_{LLL} ={-} \tfrac{4}{5}\langle\epsilon\rangle r. \end{equation}An expression equivalent to (1.1) in isotropic flows may be written in terms of the TKE rather than the longitudinal velocity (Antonia et al. Reference Antonia, Ould-Rouis, Anselmet and Zhu1997)
\begin{equation} D_{Lii} = 2\nu\frac{\partial}{\partial r}D_{ii} - \frac{4}{3}\langle\epsilon\rangle r, \end{equation}where repeated lowercase indices indicate summation. The equivalent condition to (1.2) written in terms of the TKE is therefore referred to as the ‘4/3’ law herein
\begin{equation} D_{Lii} ={-} \tfrac{4}{3}\langle\epsilon\rangle r. \end{equation}As pointed out by Antonia et al. (Reference Antonia, Ould-Rouis, Anselmet and Zhu1997), a close analogy exists between (1.3) and Yaglom's equation for passive scalars (Yaglom Reference Yaglom1949), which also yields a ‘4/3’ law. While the two results are analogous, note that ‘the 4/3 law’ herein refers specifically to (1.4).
Kolmogorov (Reference Kolmogorov1941a) argued that the degree of anisotropy, initially introduced through the generation of the largest scales in accordance with the boundary conditions of the flow, should diminish as the energy from these scales ‘cascades’ downscale. By this argument, (1.1) will apply to a range of (small) scales in any turbulent flow provided there is sufficient separation in scale between the dissipative range and that at which the anisotropy is introduced (i.e. provided that Reynolds number is sufficiently large). At still larger Reynolds numbers, it follows that the range where (1.1) is valid will include a subrange where (1.2) (or, equivalently, (1.4)) is approximately valid.
In practice, however, the Reynolds numbers at which one can expect to observe close agreement with (1.2) (over an appreciable range of scales) are likely not achievable in the laboratory. A number of studies (employing various techniques) have estimated that the Taylor microscale Reynolds number 
$R_\lambda$ needed in order to satisfy (1.2) over an appreciable range of scales in grid-generated turbulence (for example) is approximately 
$10^5$ (e.g. see Qian Reference Qian1997; Lindborg Reference Lindborg1999; Qian Reference Qian1999; Zhou et al. Reference Zhou, Antonia, Danaila and Anselmet2000; Lundgren Reference Lundgren2002; Antonia & Burattini Reference Antonia and Burattini2006; Tchoufag, Sagaut & Cambon Reference Tchoufag, Sagaut and Cambon2012). Note that, here, 
$\textit {R}_\lambda \equiv \lambda u_{rms}/\nu$, where 
$u_{rms}$ is the fluctuating velocity root mean square (typically of the streamwise component), and 
$\lambda \equiv ({u_{rms}}^2 /\langle (\partial u/\partial x)^2\rangle )^{1/2}$ is the transverse Taylor microscale (e.g. see Pope Reference Pope2000), where angle brackets denote ensemble or time averaging where appropriate. Such Reynolds numbers are well beyond the reach of laboratory experiments and direct numerical simulations. Indeed, a recent review (Antonia et al. Reference Antonia, Tang, Djenidi and Zhou2019) of published data for 
$D_{LLL}$ has shown that (1.2) has yet to be realised satisfactorily in any real flow.
Lack of direct experimental evidence for the 4/5 law has led to proposed modifications to the original arguments of Kolmogorov (Reference Kolmogorov1941b), including Kolmogorov (Reference Kolmogorov1962), that have different implications for the asymptotic behaviour of structure functions (of all orders), and indeed the fundamental nature of high Reynolds number turbulence. Antonia et al. (Reference Antonia, Djenidi, Danaila and Tang2017) have argued, however, that failure to recognise the importance of the finite Reynolds number (FRN) effect in the emergence of (1.2) has, by and large, resulted in misguided assessments of the first two hypotheses of Kolmogorov (Reference Kolmogorov1941b) (i.e. universal similarity with dissipation scales and the tendency of inertial scale transfer to eliminate anisotropy at progressively smaller scales) as well as his third hypothesis (Kolmogorov Reference Kolmogorov1962) which has been generally associated in the literature with the effect of small-scale intermittency. That is, proposed modifications to (1.1) that would persist in the limit of infinite Reynolds number cannot be justified on the basis of observed departures from (1.1) without additional consideration of the FRN effect.
While direct evidence of the 4/5 law is experimentally out of reach, a number of authors have probed the limiting behaviour of 
$D_{LLL}$ by re-deriving (1.1) with relaxed assumptions, such that terms otherwise removed by invoking isotropy are retained. As the resulting equations also constitute scale-by-scale (SBS) energy balances, the additional terms can be examined individually to elucidate the mechanisms responsible for the departure from the 4/5 or 4/3 law behaviour at Reynolds numbers encountered either in the laboratory or in numerical simulations. These SBS budget equations are typically of the following form:
$$\begin{gather} D_{LLL} = 6\nu\frac{\partial}{\partial r}D_{LL} - \frac{4}{5}\langle\epsilon\rangle r + I_{LL}, \end{gather}$$
$$\begin{gather}D_{Lii} = 2\nu\frac{\partial}{\partial r}D_{ii} - \frac{4}{3}\langle\epsilon\rangle r + I_{ii}, \end{gather}$$
where 
$I_{LL}$ and 
$I_{ii}$ represent the (flow-dependent) combined effects of anisotropy, inhomogeneity and/or non-stationarity (as a function of scale) on the spatial and/or interscale transfer of energy (e.g. Danaila et al. Reference Danaila, Anselmet, Zhou and Antonia1999). As with (1.1), (1.5) describes the energy flux to/from each scale (as represented by 
$r$). Unlike (1.1), however, (1.5) is meant to be valid at all 
$r$ regardless of Reynolds number, and reduces to the TKE budget equation as 
$r\rightarrow \infty$ (with the appropriate formulation of 
$I_{LL}$ for each flow).
The main objective of this paper is to examine the causes for departure from the 4/3 law at the centreline of a channel flow and along the axis of a pipe flow. To achieve this, we first derive a proper expression for 
$I_{ii}$ in (1.6) starting from the Navier–Stokes equations without making any assumptions as to the nature of the anisotropy. We also show that (1.6) correctly reduces to the TKE budget equation as 
$r\rightarrow \infty$ and to the budget equation for 
$\langle (\partial u_i/\partial x_1)^2\rangle$ in the limit as 
$r_1\rightarrow 0$ when the ‘longitudinal’ direction is chosen to be along 
$x_1$. To investigate the behaviour of individual terms in (1.6) (including those within 
$I_{ii}$), we have used data obtained experimentally on (or near) the pipe axis in several different facilities, as well as direct numerical simulations (DNSs) of channel flow associated with del Álamo & Jiménez (Reference del Álamo and Jiménez2003), del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004), Hoyas & Jiménez (Reference Hoyas and Jiménez2006), Graham et al. (Reference Graham2016) and Lee & Moser (Reference Lee and Moser2015). The experimental data are from the hot-wire measurements first reported by Zimmerman et al. (Reference Zimmerman2019), which were collected in the long pipe facility at the Center for International Cooperation in Long Pipe Experiments (CICLoPE). We also revisit the pipe flow data obtained by Antonia & Pearson (Reference Antonia and Pearson2000) in order to reach conclusions based on as uniform a treatment as possible of data collected in different facilities. Finally, using the data and the equations, we suggest an approach to the 
$4/3$ law wherein the role of the viscous and 
$I_{ii}$ terms diminishes over a certain range of scales as the Reynolds number increases.
2. Theoretical considerations
We begin by deriving a SBS energy budget equation for turbulent channel and pipe flows. To do so, we follow the framework of Hill (Reference Hill2001b) (i.e. the ‘Archive’ document associated with Hill Reference Hill2001a) and Hill (Reference Hill2002a). The present SBS budget equation is similar to that originally proposed by Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001) and later extended by Danaila, Anselmet & Zhou (Reference Danaila, Anselmet and Zhou2004) to include the effects of (slight) mean shear, but (as with the later budget equation proposed by Marati, Casciola & Piva Reference Marati, Casciola and Piva2004) does not rely upon assumptions regarding the nature of the anisotropy/inhomogeneity. As a result, the present SBS budget equation is found to achieve a superior balance to that of Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001) across the full range of length scales.
2.1. The SBS budget equation
The Navier–Stokes equations in Cartesian coordinates at two points in space, 
$\boldsymbol {x}$ and 
$\boldsymbol {x^\prime }$, may be written as
$$\begin{gather} \frac{\partial \tilde{u}_i}{\partial t} + \tilde{u}_j\frac{\partial \tilde{u}_i}{\partial x_j} ={-}\frac{\partial \tilde{p}}{\partial x_i} + \nu\frac{\partial ^2 \tilde{u}_i}{\partial x_j\partial x_j}, \end{gather}$$
$$\begin{gather}\frac{\partial \tilde{u}^\prime_i}{\partial t} + \tilde{u}^\prime_j \frac{\partial \tilde{u}^\prime_i}{\partial x^\prime_j} ={-}\frac{\partial \tilde{p}^\prime}{\partial x^\prime_i} + \nu\frac{\partial ^2 \tilde{u}^\prime_i}{\partial x^\prime_j\partial x^\prime_j}, \end{gather}$$
where 
$\tilde {p}$ is the instantaneous pressure divided by the density (which is constant), and 
$\tilde {u}_i$ and 
$\tilde {u}_i^\prime$ are instantaneous velocity vectors respectively evaluated at 
$\boldsymbol {x}$ and 
$\boldsymbol {x^\prime }$ (likewise for the pressure). Later, we will decompose the instantaneous velocity (
$\tilde {u}_i$) into the usual mean (
$U_i$) and fluctuating (
$u_i$) components, i.e. 
$\tilde {u}_i = U_i + u_i$. The two expressions given by (2.1) and (2.2) may be manipulated to yield a transport equation for the second-order structure function tensor. To do so, we first define a velocity increment 
$\tilde {v}_i\equiv \tilde {u}_i-\tilde {u}_i^\prime$ (and likewise 
$v_i=u_i-u_i^\prime$), separation vector 
$r_i \equiv x_i-x_i^\prime$, and location vector 
$X_i\equiv (x_i+x_i^\prime )/2$. We also require that 
$\boldsymbol {x}$ and 
$\boldsymbol {x}^\prime$ have no relative motion (Hill Reference Hill2001a), such that
\begin{equation} \frac{\partial \tilde{u}_i}{\partial x^\prime_j} = \frac{\partial \tilde{u}_i^\prime}{\partial x_j} =\frac{\partial \tilde{p}^\prime}{\partial x_j} = \frac{\partial \tilde{p}}{\partial x^\prime_j} = 0. \end{equation}
The full second- and third-order velocity structure functions may then be written in these terms as 
$\tilde {D}_{ij}(\boldsymbol {X},\boldsymbol {r})\equiv \langle \tilde {v}_i\tilde {v}_j \rangle$ and 
$\tilde {D}_{ijk}(\boldsymbol {X},\boldsymbol {r})\equiv \langle \tilde {v}_i\tilde {v}_j\tilde {v}_k \rangle$. Structure functions without the tilde represent the increment in the fluctuating velocity components at 
$\boldsymbol {x}$ and 
$\boldsymbol {x^\prime }$, e.g. 
$D_{ij}\equiv \langle v_iv_j\rangle$. While 
$\tilde {D}_{ij}$ and 
$\tilde {D}_{ijk}$ are both functions of the three-dimensional separation vector 
$\boldsymbol {r}$, we focus our attention herein on the value of these functions along the 
$r_1$, or longitudinal axis for ease of comparison with experimental results.
The transport equation for 
$\langle \tilde {u}_i\tilde {u}_j\rangle$ may be derived by multiplying both sides of (2.1) by 
$\tilde {u}_j$, adding the result to (2.1) written for 
$\tilde {u}_j$ multiplied by 
$\tilde {u}_i$, and ensemble averaging both sides. Similarly, the transport equation for 
$\tilde {D}_{ij}$ may be derived by applying the same steps to the difference between (2.1) and (2.2) (i.e. the transport equations for 
$\tilde {v}_i$, then 
$\tilde {v}_j$) and multiplying by 
$\tilde {v}_i$ and 
$\tilde {v}_j$ rather than 
$\tilde {u}_i$ and 
$\tilde {u}_j$, i.e.
\begin{equation} \frac{\textrm{D}\tilde{v}_i\tilde{v}_j}{\textrm{D} t} = \tilde{v}_j\left(\frac{\textrm{D}\tilde{u}_i}{\textrm{D} t} - \frac{\textrm{D}\tilde{u}^\prime_i}{\textrm{D} t}\right) + \tilde{v}_j\left(\frac{\textrm{D}\tilde{u}_j}{\textrm{D} t} - \frac{\textrm{D}\tilde{u}^\prime_j}{\textrm{D} t}\right). \end{equation}
Note that the spatial gradients in the material derivatives are with respect to the coordinate implied by the numerator, i.e. 
$\boldsymbol {x}$ for 
$\textrm {D}\tilde {u}_i/\textrm {D} t$ and 
$\boldsymbol {x}^\prime$ for 
$\textrm {D}\tilde {u}_i^\prime /\textrm {D} t$.
After transforming from 
$\boldsymbol {x}$ and 
$\boldsymbol {x^\prime }$ coordinates to 
$\boldsymbol {r}$ and 
$\boldsymbol {X}$ coordinates according to the following:
\begin{equation} \frac{\partial}{\partial r_i} = \frac{1}{2}\left(\frac{\partial}{\partial x_i} -\frac{\partial}{\partial x_i^\prime}\right),\quad \frac{\partial}{\partial X_i} = \left(\frac{\partial}{\partial x_i} + \frac{\partial}{\partial x_i^\prime}\right), \end{equation}
the transport equation for 
$\tilde {D}_{ij}$ is written as follows (e.g. Hill Reference Hill2001a, Reference Hill2002b):
\begin{equation} \frac{\partial}{\partial t} \tilde{D}_{ij} + \frac{\partial}{\partial r_k} \tilde{D}_{ijk} + \frac{\partial}{\partial X_k} \tilde{F}_{ijk} ={-}\tilde{T}_{ij} + 2\nu\left[\left(\frac{\partial^2}{\partial r_k \partial r_k} + \frac{1}{4}\frac{\partial^2}{\partial X_k \partial X_k} \right)\tilde{D}_{ij} - \tilde{E}_{ij}\right]. \end{equation}Equation (2.6) makes use of several additional tensors, defined as follows:
$$\begin{gather} \tilde{F}_{ijk} \equiv \tfrac{1}{2}\left\langle \left(\tilde{u}_k+\tilde{u}_k^\prime\right) \tilde{v}_i\tilde{v}_j \right\rangle, \end{gather}$$
$$\begin{gather}\tilde{E}_{ij} \equiv \left\langle \frac{\partial \tilde{u}_i}{\partial x_k} \frac{\partial \tilde{u}_j}{\partial x_k} + \frac{\partial \tilde{u}_i^\prime}{\partial x_k^\prime} \frac{\partial \tilde{u}_j^\prime}{\partial x_k^\prime}\right\rangle \end{gather}$$
$$\begin{gather}\tilde{T}_{ij} \equiv {\frac{\partial}{\partial X_i}}\left\langle \tilde{v}_j\left(\tilde{p}-\tilde{p}^\prime\right)\right\rangle + {\frac{\partial}{\partial X_j}}\left\langle \tilde{v}_i\left(\tilde{p}-\tilde{p}^\prime\right)\right\rangle - 2\left(\tilde{p}-\tilde{p}^\prime\right)\left(\tilde{s}_{ij} - \tilde{s}_{ij}^\prime\right), \end{gather}$$
where 
$\tilde {s}_{ij}\equiv (\partial \tilde {u}_i/\partial x_j + \partial \tilde {u}_j/\partial x_i)/2$ is the instantaneous rate-of-strain tensor. From left to right, the terms in (2.6) represent the unsteady variation in 
$\tilde {D}_{ij}$, the divergence of 
$\tilde {D}_{ij}$ interscale flux (i.e. net inertial transfer of 
$\tilde {D}_{ij}$ to/from a given 
$r_1$), the divergence of 
$\tilde {D}_{ij}$ spatial flux, pressure transport/diffusion and pressure–strain redistribution of 
$\tilde {D}_{ij}$, viscous interscale and spatial diffusion of 
$\tilde {D}_{ij}$ and viscous dissipation of 
$\tilde {D}_{ij}$. Terms dependent on a gradient with respect to 
$\boldsymbol {X}$ represent the effects of inhomogeneity, and would therefore normally be discarded at this stage in the derivation of (1.1). To simplify the analysis, we consider the budget equation for 
$\tilde {D}_{ii}$, where repeated indices indicate summation. This choice eliminates the pressure–strain redistribution term through continuity, since 
$\tilde {s}_{ii}=0$.
For a fully developed flow, restricting the separation vector to a homogeneous direction, e.g. such that 
$U_i = U_i^\prime$, further simplifies the analysis. In this case, several terms in (2.6) can be rewritten in terms of fluctuating rather than instantaneous quantities
\begin{equation} \frac{\partial}{\partial t} D_{ij} + \frac{\partial}{\partial r_k} \tilde{D}_{ijk} + \frac{\partial}{\partial X_k} \tilde{F}_{ijk} ={-}T_{ij} + 2\nu\left[\left(\frac{\partial^2}{\partial r_k \partial r_k} + \frac{1}{4}\frac{\partial^2}{\partial X_k \partial X_k} \right)D_{ij} - E_{ij}\right], \end{equation}
where 
$E_{ij}$ and 
$T_{ij}$ are defined as in (2.8) and (2.9), but contain only fluctuating components of velocity and pressure. This substitution is justified in more detail in Appendix A. In brief, obtaining (2.10) from (2.6) is analogous to obtaining the TKE budget equation by subtracting the equation for 
$U_iU_j$ from that of 
$\left \langle {\tilde {u}_i\tilde {u}_j}\right \rangle$; note further that the terms in the budgets for 
$U_iU_j$ and 
$U_i^\prime U_j^\prime$ will have equal values as one another when 
$\boldsymbol {r}$ is restricted to a homogeneous direction. Note that contributions from the mean shear present in the wall-normal direction of channel and pipe flows are embedded in the second term of (2.10) as will be shown below.
Typically, the next step taken in the derivation of (1.1) and (1.3) is to write the terms representing interscale divergence and diffusion in terms of their variation with a scalar separation value 
$r\equiv \sqrt {r_kr_k}$ (rather than in terms of their variation in three-dimensional 
$\boldsymbol {r}$ space) by invoking isotropy (e.g. see Danaila et al. Reference Danaila, Anselmet, Zhou and Antonia1999; Hill Reference Hill2001b)
$$\begin{gather} \frac{\partial}{\partial r_k} \tilde{D}_{iik} = \left(\frac{\partial}{\partial r} + \frac{2}{r}\right)D_{Lii}, \end{gather}$$
$$\begin{gather}\frac{\partial^2}{\partial r_k \partial r_k} D_{ii} = \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}\right)D_{ii}. \end{gather}$$
Here, the subscript 
$L$ indicates the longitudinal direction, i.e. the same direction as that assigned to 
$r$. Note that 
$D_{iik}$ and 
$\tilde {D}_{iik}$ are interchangeable in homogeneous isotropic flow (as are 
$D_{ii}$ and 
$\tilde {D}_{ii}$). If the streamwise direction in the channel/pipe is assigned as the longitudinal direction, then 
$r$ and 
$D_{Lii}$ in (2.11) and (2.12) can be replaced by 
$r_1$ and 
$D_{1ii}$, respectively. The expressions in (2.11) and (2.12) are, however, only strictly true for globally isotropic flows or in the infinite Reynolds number limit, and therefore introduce imbalance into the resulting SBS budget commensurate with the imbalance between the right- and left-hand sides of (2.11) and (2.12). To account for this, we introduce FRN (or anisotropic) correction terms, 
$\tilde {A}_{ii}$ and 
$B_{ii}$, defined such that their respective addition to the right-hand sides of (2.11) and (2.12) ensures validity of both equations at all 
$r_1$ by construction (for flows homogeneous in the 
$r_1$ direction)
$$\begin{gather} \frac{\partial}{\partial r_k} \tilde{D}_{iik} = \left(\frac{\partial}{\partial r_1} + \frac{2}{r_1}\right)D_{1ii} + \tilde{A}_{ii}, \end{gather}$$
$$\begin{gather}\frac{\partial^2}{\partial r_k \partial r_k} D_{ii} = \left(\frac{\partial^2}{\partial r_1^2} + \frac{2}{r_1}\frac{\partial}{\partial r_1}\right)D_{ii} + B_{ii}. \end{gather}$$
The left-hand side of (2.13), which represents the inertial interscale energy transfer, is expected to be negative (i.e. energy transfers from larger scales to smaller scales). Thus, negative values of 
$\tilde {A}_{ii}$ indicate that the downscale transfer of energy is larger (i.e. more negative) in the 
$r_2$ and 
$r_3$ directions than one would expect based on the transfer in the 
$r_1$ direction as expressed by (2.11). If (2.11) were instead expressed in terms of 
$r_2$ and 
$D_{2ii}$, for example, then negative 
$\tilde {A}_{ii}$ would indicate larger downscale energy transfer in 
$r_1$ and 
$r_3$ than one would expect based on the transfer along 
$r_2$.
Substitution of (2.13) and (2.14) into (2.10) yields the following, which can be cast as an ordinary differential equation for 
$D_{1ii}(r_1,\boldsymbol {X})$ in 
$r_1$ at fixed 
$\boldsymbol {X}$:
\begin{align} \left(\frac{\partial}{\partial r_1} + \frac{2}{r_1}\right)D_{1ii} &={-} \tilde{A}_{ii} - \frac{\partial}{\partial X_k} \tilde{F}_{iik} -2\frac{\partial}{\partial X_i} \langle v_i(p-p^\prime)\rangle \nonumber\\ &\quad +2\nu\left[\left(\frac{\partial^2}{\partial r_1^2} + \frac{2}{{r_1}}\frac{\partial}{\partial r_1} + \frac{1}{4}\frac{\partial^2}{\partial X_2^2} \right)D_{ii} +B_{ii} - E_{ii}\right]. \end{align}
Note that, while the partial derivatives are retained here, this analysis considers 
$D_{1ii}$ to be only a function of 
$r_1$ when 
$\boldsymbol {X}$ is fixed. Equation (2.15) is functionally equivalent to the exact two-point equations that were first used to investigate the flux of scale energy in channel flows by Marati et al. (Reference Marati, Casciola and Piva2004), and then in more detail by Cimarelli, De Angelis & Casciola (Reference Cimarelli, De Angelis and Casciola2013), Cimarelli et al. (Reference Cimarelli, De Angelis, Schlatter, Brethouwer, Talamelli and Casciola2015, Reference Cimarelli, De Angelis, Jiménez and Casciola2016) and Gatti et al. (Reference Gatti, Chiarini, Cimarelli and Quadrio2020). The main differences between these studies and the present one are that 
$(i)$ the interscale divergence and Laplacian terms are separated into isotropic and anisotropic contributions to facilitate examination of their behaviour with Reynolds number, 
$(ii)$ we integrate (2.15) to obtain a form equivalent to (1.6) and 
$(iii)$ we focus our analysis specifically on the emergence of the 4/3 law. Mizuno (Reference Mizuno2016) and Lee & Moser (Reference Lee and Moser2019) also recently studied energy transport by scale in channel flow, but did so using the Fourier-transformed two-point correlation transport equation. While there is an analogy to the 4/3 (or 4/5) law in the spectral domain (Tchoufag et al. Reference Tchoufag, Sagaut and Cambon2012), here we elect to examine the emergence of the inertial energy cascade according to its classical structure function formulation.
The first-order inhomogeneous linear ordinary differential equation represented by (2.15) can be written in the following generic form:
\begin{equation} f^\prime(r) + \frac{2}{r}f(r) = g(r), \end{equation}which has the solution
\begin{equation} f = r^{{-}2}\int_{0}^{r}y^2g(y)\,{\textrm{d}y}. \end{equation}
An expression for 
$D_{1ii}$ is therefore obtained by substituting each term on the right-hand side of (2.15) into the right-hand side of (2.17) and proceeding with the integration where possible.
At this stage, we assign the coordinate directions 
$x_1$, 
$x_2$ and 
$x_3$ to the streamwise, wall-normal and spanwise directions in the channel flow. Restricting the separation vector to the 
$r_1$ axis in this coordinate system allows us to simplify the viscous term by invoking streamwise homogeneity
\begin{equation} 2\nu E_{ii} = 4\langle \tilde{\epsilon}\rangle, \end{equation}
where 
$\langle \tilde {\epsilon }\rangle$ is the pseudo-dissipation, or homogeneous dissipation (e.g. see Pope Reference Pope2000). Restricting the analysis to the centreline (or anywhere away from the viscous sublayer) allows the approximation 
$\langle \tilde {\epsilon }\rangle = \left \langle {\epsilon }\right \rangle$, such that 
$2\nu E_{ii} = 4\left \langle {\epsilon }\right \rangle$.
Kolmogorov's equation written in terms of the TKE, i.e. (1.3), can be obtained from (2.15) via (2.17) if homogeneity 
$(\partial /\partial X_i = 0)$, isotropy 
$(\tilde {A}_{ii}=B_{ii}=0)$ and (2.18) with 
$\langle \tilde {\epsilon }\rangle = \left \langle {\epsilon }\right \rangle$ are invoked (and with the ‘boundary condition’ that 
$D_{1ii}=0$ at 
$r_1=0$). Similarly, (1.6) can be obtained in the same way but with the inhomogeneous and anisotropic terms retained. Note that in both cases, the viscous term (i.e. the first terms on the right-hand side of (1.3) and (1.6)) may be obtained via integration by parts applied to the two viscous 
$r_1$-derivative terms in (2.15) substituted into (2.17). For the particular case of streamwise/spanwise homogeneous flow (e.g. channel flow), the present analysis reveals the inhomogeneous/anisotropic term to take the following form:
\begin{equation} I_{ii} = \frac{1}{r_1^2}\displaystyle\int_0^{r_1} y^2\left(-\tilde{A}_{ii} + 2\nu B_{ii} - \frac{\partial}{\partial X_2} \tilde{F}_{ii2} - 2\frac{\partial}{\partial X_2} \left\langle v_2\left(p-p^\prime\right)\right\rangle + \frac{\nu}{2}\frac{\partial^2}{\partial X_2^2} D_{ii}\right){\textrm{d}y}, \end{equation}which follows from substituting each of the inhomogeneous/anisotropic terms into (2.17).
The following shorthand notation is used throughout for each (integrated) term on the right-hand side of (2.19):
\begin{equation} \left.\begin{gathered} A\equiv{-}\frac{1}{r_1^2}\displaystyle\int_0^{r_1} y^2\tilde{A}_{ii}\,{\textrm{d}y},\quad B\equiv \frac{2\nu}{r_1^2}\displaystyle\int_0^{r_1} y^2\tilde{B}_{ii}\,{\textrm{d}y},\quad H\equiv{-}\frac{1}{r_1^2}\displaystyle\int_0^{r_1} y^2\frac{\partial}{\partial X_2} \tilde{F}_{ii2}\,{\textrm{d}y}, \\ P_I\equiv{-}\frac{2}{r_1^2}\displaystyle\int_0^{r_1} y^2\frac{\partial}{\partial X_2} \left\langle v_2\left(p-p^\prime\right)\right\rangle {\textrm{d}y},\quad L\equiv \frac{\nu}{2r_1^2}\displaystyle\int_0^{r_1} y^2\frac{\partial^2}{\partial X_2^2} D_{ii}\,{\textrm{d}y}. \end{gathered}\right\} \end{equation} These terms comprise 
$I_{ii}$ in (1.6), i.e.
\begin{equation} D_{1ii} = 2\nu\frac{\partial}{\partial r_1}D_{ii} - \frac{4}{3}\langle\epsilon\rangle r_1 + A + B + H + P_I + L. \end{equation} Repeating the above analysis using cylindrical spatial coordinates changes only the 
$H$ and 
$L$ terms (i.e. the spatial divergence and Laplacian terms). These terms take the following form:
$$\begin{gather} H ={-} \frac{1}{r_1^2}\displaystyle\int_0^{r_1} y^2\frac{1}{Y_2}\frac{\partial}{\partial Y_2} Y_2 \tilde{F}_{ii2}\,{\textrm{d}y}, \end{gather}$$
$$\begin{gather}L = \frac{\nu}{2r_1^2}\displaystyle\int_0^{r_1} y^2\frac{1}{Y_2}\frac{\partial}{\partial Y_2} \left(Y_2\frac{\partial}{\partial Y_2}D_{ii}\right){\textrm{d}y}, \end{gather}$$
where 
$Y_1$, 
$Y_2$ and 
$Y_3$ respectively represent the streamwise, radial and azimuthal coordinates.
The contributions of several terms in (2.21) are examined for Reynolds number dependence in § 4. In this way we aim to elucidate the approach towards the 4/3 law at the centreline of channel and pipe flows.
2.2. Limiting behaviour of SBS budget terms
As noted by Antonia et al. (Reference Antonia, Zhou, Danaila and Anselmet2000), necessary (but not sufficient) conditions of an appropriately formulated SBS budget equation (1.6) are that it reduces to the mean TKE budget equation for the flow as 
$r_1\rightarrow \infty$, and to a transport equation for the mean enstrophy (or energy dissipation rate) as 
$r_1\rightarrow 0$. In the present case, the solution to (2.15) (i.e. (2.21)) should reduce to the transport equation for 
$\langle (\partial u_i/\partial x_1)^2\rangle$ as 
$r_1\rightarrow 0$. The evolution with Reynolds number of several of the terms in (2.19) may be characterised in terms of the evolution of these limits. The correspondence between terms in the SBS budget and those in the TKE and 
$\langle (\partial u_i/\partial x_1)^2\rangle$ budgets is also useful in assigning physical significance to the observed behaviours of several SBS budget terms. The limits are therefore described here; more detailed derivations are given in the appendices.
As described in Appendix B, the large-scale limits (i.e. 
$r_1\rightarrow \infty$) of each term in (2.19) can be written in terms of quantities at the point 
$\boldsymbol {x}$ as follows:
$$\begin{gather} \lim_{r_1\rightarrow \infty} -\frac{1}{r_1^2}\int_{0}^{r_1}y^2 \tilde{A}_{ii}\,{\textrm{d}y} ={-}\frac{1}{3}\frac{\partial}{\partial x_2}\left\langle{u_2u_iu_i}\right\rangle r_1 -\frac{4}{3}\left\langle{u_2u_i}\right\rangle \frac{\partial U_i}{\partial x_2}r_1 \end{gather}$$
$$\begin{gather}\lim_{r_1\rightarrow \infty} \frac{2\nu}{r_1^2}\int_{0}^{r_1}y^2 B_{ii}\,{\textrm{d}y} = \frac{\nu}{3}\frac{\partial^2}{\partial x_2^2}\left\langle{u_iu_i}\right\rangle r_1 \end{gather}$$
$$\begin{gather}\lim_{r_1\rightarrow \infty} -\frac{1}{r_1^2}\int_{0}^{r_1}y^2 \frac{\partial}{\partial X_2}\tilde{F}_{ii2} {\textrm{d}y} ={-}\frac{1}{3} \frac{\partial}{\partial x_2}\left\langle{u_2u_iu_i}\right\rangle r_1 -\frac{2}{3}U_2 \frac{\partial \left\langle{u_iu_i}\right\rangle}{\partial x_2}r_1 \end{gather}$$
$$\begin{gather}\lim_{r_1\rightarrow \infty} -\frac{2}{r_1^2}\int_{0}^{r_1}y^2 \frac{\partial}{\partial X_2} \,{\textrm{d}y} \left\langle{v_2\left(p-p^\prime\right)}\right\rangle ={-}\frac{4}{3}\frac{\partial}{\partial x_2}\left\langle{u_2p}\right\rangle r_1 \end{gather}$$
$$\begin{gather}\lim_{r_1\rightarrow \infty} \frac{\nu}{2r_1^2} \int_{0}^{r_1}y^2\frac{\partial^2}{\partial X_2^2} D_{ii}\,{\textrm{d}y} = \frac{\nu}{3}\frac{\partial^2}{\partial x_2^2}\left\langle{u_iu_i}\right\rangle r_1. \end{gather}$$
Summing the limits from (2.24)–(2.28) reveals that (2.21) reduces in the limit as 
$r_1\rightarrow \infty$ to the mean TKE budget equation for the channel, i.e.
\begin{equation} \frac{1}{2}\frac{\partial}{\partial x_2}\left\langle u_iu_iu_2\right\rangle ={-}\left\langle{u_2u_i}\right\rangle \frac{\partial U_i}{\partial x_2} -\frac{\partial}{\partial x_2}\left\langle u_2p\right\rangle + \frac{\nu}{2}\frac{\partial^2}{\partial x_2^2}\left\langle u_iu_i\right\rangle - \left\langle{\epsilon}\right\rangle. \end{equation}
From left to right, the terms in (2.29) represent turbulent diffusion, production, pressure transport, viscous diffusion and dissipation. In the limit as 
$r_1\rightarrow \infty$, the turbulent diffusion of TKE is divided equally between the 
$A$ and 
$H$ terms (i.e. from (2.24) and (2.26)), while the viscous diffusion is divided equally between the 
$B$ and 
$L$ terms (i.e. from (2.25) and (2.28)). This equipartition is a result of the change of variables from 
$\boldsymbol {x}$ and 
$\boldsymbol {x^\prime }$ to 
$\boldsymbol {r}$ and 
$\boldsymbol {X}$. Note that the production term (which is associated with 
$\tilde {A}$) vanishes at the centreline owing to symmetry, and the term that would otherwise represent mean advection (associated with 
$H$) is zero everywhere in the channel because 
$U_2=0$.
As described in Appendix C, the small-scale limits of the terms in (2.21) can be written as follows (recall that herein all terms are evaluated along the 
$r_1$ axis, such that 
$r_2=r_3=0$):
$$\begin{gather} \lim_{\boldsymbol{r}\rightarrow 0} D_{1ii} = \left\langle{\frac{\partial u_1}{\partial x_1} \frac{\partial u_i}{\partial x_1}\frac{\partial u_i}{\partial x_1}}\right\rangle r_1^3 \end{gather}$$
$$\begin{gather}\lim_{\boldsymbol{r}\rightarrow 0} 2\nu\frac{\partial}{\partial r_1}D_{ii} = 4\nu\left\langle{\frac{\partial u_i}{\partial x_1}\frac{\partial u_i}{\partial x_1}}\right\rangle r_1 + \frac{2\nu}{3}\left\langle\frac{\partial^3 u_i}{\partial x_1^3} \frac{\partial u_i}{\partial x_1}\right\rangle r_1^3 \end{gather}$$
$$\begin{gather}\lim_{\boldsymbol{r}\rightarrow 0} -\frac{1}{r_1^2}\int_0^{r_1} y^2 \tilde{A}_{ii}{\textrm{d}y} = \left[\underbrace{\left\langle{\frac{\partial u_i}{\partial x_1}\frac{\partial u_i}{\partial x_1}\frac{\partial u_1}{\partial x_1}}\right\rangle - \frac{2}{5}\left\langle{\frac{\partial u_i}{\partial x_1}\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_1}}\right\rangle}_{C_A} - \frac{2}{5}\left\langle \frac{\partial u_i}{\partial x_1}\frac{\partial u_j}{\partial x_1} \right\rangle \frac{\partial U_i}{\partial x_j}\right]r_1^3 \end{gather}$$
\begin{align} \lim_{\boldsymbol{r}\rightarrow 0} \frac{2\nu}{r_1^2}\int_{0}^{r_1} y^2 B_{ii}\,{\textrm{d}y} &= \nu\underbrace{\left[\frac{4}{3}\left\langle{\frac{\partial u_i}{\partial x_j} \frac{\partial u_i}{\partial x_j}}\right\rangle - 4\left\langle{\frac{\partial u_i}{\partial x_1} \frac{\partial u_i}{\partial x_1}}\right\rangle\right]}_{C_B} r_1 \nonumber\\ &\quad +\left[\frac{\nu}{5}\left\langle\frac{\partial^3 u_i}{\partial x_1^2\partial x_j} \frac{\partial u_i}{\partial x_j}\right\rangle + \frac{\nu}{5}\left\langle \frac{\partial^3 u_i}{\partial x_1\partial x_j^2}\frac{\partial u_i}{\partial x_1}\right\rangle - \frac{2\nu}{3}\left\langle\frac{\partial^3 u_i}{\partial x_1^3} \frac{\partial u_i}{\partial x_1}\right\rangle\right] r_1^3 \end{align}
$$\begin{gather} \lim_{\boldsymbol{r}\rightarrow 0} -\frac{1}{r_1^2}\int_{0}^{r_1} y^2 \frac{\partial}{\partial X_2}F_{ii2}{\textrm{d}y} = \underbrace{-\frac{1}{5} \frac{\partial}{\partial x_2}\left\langle{u_2\frac{\partial u_i}{\partial x_1} \frac{\partial u_i}{\partial x_1}}\right\rangle}_{C_H}r_1^3 \end{gather}$$
$$\begin{gather}\lim_{\boldsymbol{r}\rightarrow 0} -\frac{2}{r_1^2}\int_{0}^{r_1} y^2 \frac{\partial}{\partial X_2} \left\langle{v_2\left(p-p^\prime\right)}\right\rangle {\textrm{d}y} ={-}\frac{2}{5} \frac{\partial}{\partial x_2}\left\langle{\frac{\partial u_2}{\partial x_1}\frac{\partial p}{\partial x_1}}\right\rangle r_1^3 \end{gather}$$
$$\begin{gather}\lim_{\boldsymbol{r}\rightarrow 0} \frac{\nu}{2r_1^2}\int_{0}^{r_1} y^2 \frac{\partial^2}{\partial X_2^2} D_{ii} {\textrm{d}y} = \frac{\nu}{10}\frac{\partial^2}{\partial x_2^2}\left\langle{\frac{\partial u_i}{\partial x_1}\frac{\partial u_i}{\partial x_1}}\right\rangle r_1^3. \end{gather}$$
Note that these limits are obtained through Taylor series expansions of 
$\tilde {v}_i$ (or 
$v_i$) and 
$p-p^\prime$ (see Appendix C) and thus only apply at values of 
$r_1$ small enough to justify neglecting higher order terms. The limits in (2.32) and (2.34), for example, are obtained by retaining only the linear Taylor series terms. This will be discussed further in § 4.
The Reynolds number dependencies of the coefficient terms labelled 
$C_A$ and 
$C_H$ both play a key role in setting the rate at which the isotropic 4/3 law condition emerges at the centreline. These terms can each be shown to equal zero in a perfectly isotropic flow (for 
$C_A$ see Champagne Reference Champagne1978), and thus are expected to approach zero as the Reynolds number increases. The coefficient 
$C_B$, for example, represents the deviation of the pseudo-dissipation from the isotropic approximation based on 
$\langle (\partial u_1/\partial x_1)^2\rangle$. The coefficients 
$C_A$ and 
$C_H$ are analogous to 
$C_B$, and are particularly relevant to the approach towards (approximate) satisfaction of the 4/3 law, as these terms are shown below to be associated with the primary source of anisotropic/inhomogeneous contributions to the SBS budget in the expected ‘inertial’ range.
Plugging each of the limits from (2.30)–(2.36) into the SBS budget (2.21) reveals that the SBS budget reduces in the limit as 
$r_1\rightarrow 0$ to the budget equation for 
$\left \langle {(\partial u_i/\partial x_1)^2}\right \rangle$ (summed across 
$i$) in a channel flow, which is given by the following:
\begin{align} 2\left\langle\frac{\partial u_i}{\partial x_1}\frac{\partial u_i}{\partial x_j} \frac{\partial u_j}{\partial x_1}\right\rangle &={-}2\left\langle\frac{\partial u_1}{\partial x_1} \frac{\partial u_2}{\partial x_1}\right\rangle \frac{\partial U_1}{\partial x_2} -\frac{\partial}{\partial x_2}\left\langle u_2\left(\frac{\partial u_i}{\partial x_1}\right)^2\right\rangle \nonumber\\ &\quad -2\frac{\partial}{\partial x_2}\left\langle\frac{\partial u_2}{\partial x_1} \frac{\partial p}{\partial x_1}\right\rangle + \nu\frac{\partial^2}{\partial x_2^2} \left\langle \left(\frac{\partial u_i}{\partial x_1}\right)^2\right\rangle -2\nu\left\langle \left(\frac{\partial^2 u_i}{\partial x_1\partial x_j}\right)^2\right\rangle. \end{align}
From left to right, the terms in (2.37) represent turbulent production via vortex stretching, mixed production via mean shear, turbulent diffusion, pressure diffusion, viscous diffusion and viscous dissipation (e.g. see Mansour, Kim & Moin Reference Mansour, Kim and Moin1988; Vreman & Kuerten Reference Vreman and Kuerten2014). Note that the first four terms on the right-hand side of (2.37) vanish under homogeneity, leaving the classical two-term balance between production and dissipation for the locally isotropic/homogeneous case (e.g. see Tennekes & Lumley Reference Tennekes and Lumley1972; Davidson Reference Davidson2004). The first term on the right also vanishes at the channel centreline, which is why it is not included as part of 
$C_A$ here. Evidently, 
$C_H$ represents the turbulent diffusion, while 
$C_A$ (at the centreline) represents the difference between the actual turbulent production via vortex stretching and that which could be inferred from streamwise-only velocity gradients via local isotropy (e.g. see Champagne Reference Champagne1978). Note that it can also be shown that (2.15), to which the SBS budget represents the solution, also reduces to (2.37) in the limit as 
$r_1\rightarrow 0$ and (2.29) in the limit as 
$r_1\rightarrow \infty$ (after dividing out common factors of 
$r_1$ where appropriate).
3. Data
The datasets employed in § 4 are summarised in table 1. These include both experimental pipe flow measurements and channel flow DNSs. Brief descriptions of each dataset are given here; more detailed descriptions can be found in the originating studies.
Table 1. Summary of datasets used in this study. The ‘Study’ acronyms refer (in order) to: Zimmerman et al. (Reference Zimmerman2019), Antonia & Pearson (Reference Antonia and Pearson2000), del Álamo & Jiménez (Reference del Álamo and Jiménez2003), del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004), Hoyas & Jiménez (Reference Hoyas and Jiménez2006), Graham et al. (Reference Graham2016) and Lee & Moser (Reference Lee and Moser2015). Here, ‘
$l^*$’ represents individual hot-wire active sensing length relative to the Kolmogorov length scale. The ‘
$+$’ symbol beside Z19 indicates that an additional case is included herein that was not discussed in that study. Darker shades are used to denote higher Reynolds numbers. 
${\dagger}$ Note that the G16 DNS is time resolved, with 4000 individual snapshots available that span a total of one ‘flow-through time’ 
$tU_{cl}/R$, where 
$U_{cl}$ is the centreline velocity and 
$t$ is the sample time.
The experimental pipe flow cases include multi-sensor hot-wire measurements originally reported by Antonia & Pearson (Reference Antonia and Pearson2000) (hereafter AP00) and Zimmerman et al. (Reference Zimmerman2019) (hereafter Z19
$+$). The AP00 dataset is comprised of three cases with 
$\textit {Re}_\tau \approx 930$, 
${\approx }1900$ and 
${\approx }4000$, where 
$\textit {Re}_\tau \equiv R/(\nu /U_\tau )$ is the friction Reynolds number, 
$R$ is the pipe radius (or channel half-height, where appropriate) and 
$U_\tau \equiv \sqrt {\langle \tau _w\rangle /\rho }$ is a ‘friction’ velocity scale derived from the mean wall shear stress 
$\langle \tau _w \rangle$. These data were measured via a standard 
$\times$-wire array oriented to detect the streamwise and radial velocity components and positioned mainly along the axis of the 
$12.7$cm diameter pipe flow facility at the University of Newcastle. As such, these data do not grant access to any quantities that rely upon 
$X_2$ gradients.
The Z19
$+$ dataset is comprised of four cases (
$\textit {Re}_\tau \approx 5400$, 
${\approx }7700$, 
${\approx }9800$ and 
${\approx }14\,000$) measured via (essentially) four independent 
$\times$-wire sub-arrays: two oriented to measure 
$\tilde {u}_1$ and 
$\tilde {u}_2$, and two oriented to measure 
$\tilde {u}_1$ and 
$\tilde {u}_3$. The experiments were carried out in a pipe of 
$90$ cm diameter and 
$110.9$ m length at the CICLoPE facility at the University of Bologna. These measurements were each collected across a range of radial positions from the buffer layer out to 
$x_2/R\approx 0.93$. For the present purposes, the flow at 
$x_2/R\approx 0.93$ can be considered equivalent to the flow at 
$x_2/R=1$ (this was checked against calculations in the channel flow DNS cases). For reference, the mean velocity at 
$x_2/R\approx 0.93$ is 99.8 % of the centreline velocity (based on the pipe flow DNS of El Khoury et al. Reference El Khoury, Schlatter, Noorani, Fischer, Brethouwer and Johansson2013). The off-axis measurements are also particularly useful for the present purposes, as they can be used to compute 
$X_2$ gradient terms given the known symmetries that apply about the pipe axis. The ‘
$+$’ symbol in Z19
$+$ is in reference to the highest Reynolds number case, which was collected as part of the same measurement campaign and processed in the same way as the other three cases, but was not shown in Zimmerman et al. (Reference Zimmerman2019). The omission of this case in that study was related to the unavailability of matched-
$\textit {Re}$ boundary layer data.
Accurate characterisation of the rate at which the flow approaches the 4/3 law from experimental data is contingent upon accurate estimation of 
$\left \langle {\epsilon }\right \rangle$. In order to maintain as uniform a treatment of all present experimental data as possible, here we use the ‘spectral chart’ method described in Djenidi & Antonia (Reference Djenidi and Antonia2012) based on small-scale Kolmogorov similarity applied to the longitudinal velocity spectrum. Antonia, Djenidi & Danaila (Reference Antonia, Djenidi and Danaila2014) argue that single-curve behaviour of the dissipative range when normalised by dissipative (i.e. Kolmogorov) scales does not require large Reynolds number or isotropy, and demonstrate that similarity breaks down only for 
$R_\lambda$ less than approximately 20. The dissipation rate can therefore be estimated by matching the measured spectrum (over a suitable range) to one in which 
$\left \langle {\epsilon }\right \rangle$ is known with a high degree of confidence. The merits of this approach are demonstrated by Djenidi & Antonia (Reference Djenidi and Antonia2012) for a variety of flow configurations. Although the single-curve behaviour of all spectra should improve with increasing wavenumber according to Kolmogorov's first similarity hypothesis, the corresponding regions of the measured spectrum/structure function are also more susceptible to adverse effects (i.e. a ‘peel-up’ effect) related to finite spatial/temporal resolution and low signal-to-noise ratio (Klewicki & Falco Reference Klewicki and Falco1990). Here, we use two criteria to identify a suitable wavenumber range to apply the ‘spectral chart’ method. First, the ‘known’ longitudinal spectra for the several highest-
$\textit {Re}$ cases must exhibit single-curve behaviour over the range in Kolmogorov scaling. For this purpose we use the channel flow spectra (for several 
$\textit {Re}$) associated with Lee & Moser (Reference Lee and Moser2015). The second criterion is that the experimental longitudinal spectra must not exhibit signs of noise contamination, however slight, within the wavenumber range. Possible noise contamination is identified through inspection of the spectra compensated by wavenumber squared and wavenumber to the fourth power, i.e. 
$k_1^2E_{11}$ and 
$k_1^4E_{11}$. The integrals of these quantities with respect to 
$k_1$ are, respectively, proportional to the variance and the skewness of the longitudinal velocity gradient, and thus both 
$k_1^2E_{11}$ and 
$k_1^4E_{11}$ are expected to approach zero as 
$k_1\rightarrow \infty$ (such that the integrals converge to finite values). We find that the wavenumber range 
$0.1\leq k_1\eta \leq 0.25$ (where 
$\eta$ is the Kolmogorov length scale) satisfies both criteria for the present dataset, and thus 
$\left \langle {\epsilon }\right \rangle$ values for the present experimental cases are chosen such that the Kolmogorov-scaled spectra match those of the LM15 DNS cases in this range. We also find that the values of 
$\left \langle {\epsilon }\right \rangle$ determined in this way differ by no more than 6 % from those based on 
$\langle (\partial u_1/\partial x_1)^2\rangle$ after applying the spectral compensation for spatial resolution described by Wyngaard (Reference Wyngaard1968). This difference is attributable to the fact that 
$\langle (\partial u_1/\partial x_1)^2\rangle$ is the integral of 
$k_1^2E_{11}$, and thus it includes contributions from portions of the spectrum affected by very slight noise contamination.
The channel flow DNS datasets are comprised of those reported by del Álamo & Jiménez (Reference del Álamo and Jiménez2003) (dAJ03), del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) (dA04), Hoyas & Jiménez (Reference Hoyas and Jiménez2006) (HJ06), Graham et al. (Reference Graham2016) (G16) and Lee & Moser (Reference Lee and Moser2015) (LM15). The first three are collectively referred to as the ‘Madrid DNSs’, while the last two are collectively referred to as the ‘Johns Hopkins Turbulence Database (JHTDB) DNSs’ (Perlman et al. Reference Perlman, Burns, Li and Meneveau2007; Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, Chen, Szalay and Eyink2008). Velocity field snapshots of the Madrid DNSs were downloaded through the Madrid turbulence database; in total we use six snapshots of the 
$\textit {Re}_\tau \approx 550$ case of box size 
$8{\rm \pi} \times 4{\rm \pi}$ (streamwise by spanwise) channel half-heights, five snapshots of the 
$\textit {Re}_\tau \approx 934$ case of box size 
$8{\rm \pi} \times 3{\rm \pi}$ and four snapshots of the 
$\textit {Re}_\tau \approx 2003$ case of box size 
$8{\rm \pi} \times 3{\rm \pi}$. Velocity and pressure data were accessed at the centreline of nine snapshots for the 
$\textit {Re}_\tau \approx 1000$ G16 case. The G16 dataset is time resolved, with a total of 4000 snapshots available that collectively represent roughly one channel flow-through time; the nine snapshots were selected to be equally spaced in time over this interval. Velocity and pressure data were also accessed at the centreline of all 11 available snapshots for the LM15 case. These snapshots are each separated by approximately 7/10 of a flow-through time.
4. Results
This section begins by showing how the various terms in the SBS budget conspire to achieve balance at each scale at the centreline of a channel flow using the LM15 
$\textit {Re}_\tau \approx 5200$ dataset. In this case we can compute all terms directly (i.e. without approximation) from the available velocity and pressure fields. We then examine the behaviour, with Reynolds number, of several terms individually for both pipe and channel cases. From this analysis we demonstrate how the emergence of the 4/3 law ‘inertial’ subrange can be characterised via trends observed in the other SBS budget terms.
4.1. SBS budget balance
Figure 1 shows the contribution from each term in the SBS budget for the LM15 channel at 
$\textit {Re}_\tau \approx 5200$ vs separation scaled by the Kolmogorov length, i.e. 
$r_1^*\equiv r_1/\eta$. To better visualise the contributions, each term has been normalised such that
\begin{equation} 1 = \frac{3}{4r_1\left\langle{\epsilon}\right\rangle}\left[{-}D_{1ii} + 2\nu\frac{\partial}{\partial r_1}D_{ii} + A + B + H + P_I + L\right] \end{equation}
i.e. the sum of all terms should equal unity at all values of 
$r_1$ and satisfaction of the 4/3 law is reflected by the first term being equal to unity with all other terms being negligible. The numerical scheme used to compute the terms in (4.1) is detailed in Appendix D. The term associated with the third-order structure function is hereafter referred to as the ‘transfer’ term, or 
$T$, and the term associated with the 
$r_1$ gradient of the second-order structure function is referred to as the ‘viscous’ term, or 
$V$. A ‘hat’ above any of these terms, e.g. 
$\hat {T}$, hereafter denotes normalisation by 
$\frac {4}{3}r_1\left \langle {\epsilon }\right \rangle$, e.g.
\begin{equation} \hat{T} \equiv \frac{-3}{4r_1\left\langle{\epsilon}\right\rangle}D_{1ii}, \quad \hat{V} \equiv 2\nu\frac{3}{4r_1\left\langle{\epsilon}\right\rangle}\frac{\partial}{\partial r_1}D_{ii}, \quad \hat{A} \equiv \frac{3}{4r_1\left\langle{\epsilon}\right\rangle}A, \quad \hat{H} \equiv \frac{3}{4r_1\left\langle{\epsilon}\right\rangle}H. \end{equation}
Note that the divergence anisotropy term 
$\hat {A}$ is positive. As this term is computed from the integral of 
$-\tilde {A}_{ii}$ according to (2.19), 
$\hat {A}>0$ indicates that the inertial downscale transfer of energy in 
$r_2$ and 
$r_3$ is underestimated by the isotropic relation (2.11).
Figure 1. Balance of terms in (4.1) at the centreline of the LM15 channel flow DNS (
$\textit {Re}_\tau \approx 5200$). The superscript 
$*$ denotes normalisation by the Kolmogorov dissipation scales, e.g. 
$r_1^*\equiv r_1/\eta$. Arrows indicate the Reynolds number trend of the corresponding terms, and illustrate the approach of the ‘Transfer’ term to the 4/3 law, i.e. 
$\hat {T}\rightarrow 1$.
The present SBS budget formulation results in overall budget balance to within approximately 2 % across the entire range of separations (even without 
$\hat {L}$, as this term is negligible for all 
$r_1$ at this Reynolds number). In comparison, computation of the SBS budget terms proposed by Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001) for the LM15 case results in imbalance of up to 25 % (even if a pressure term is included). This difference is attributable to the way in which the terms associated with turbulence diffusion are represented: Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001) invoked simplifications that amount (in the present notation) to 
$\hat {A} + \hat {H} \approx 2\hat {H}$ in order to allow determination of all terms via standard 
$\times$ wire measurements on and/or near the channel centreline/pipe axis (as with 
$\hat {H}$), while determination of 
$\hat {A}$ for the present budget requires measurement of the full velocity gradient tensor.
The arrows in figure 1 illustrate the general Reynolds number trend observed in several terms, and make clear the path towards the 4/3 law. The difference between the transfer term 
$\hat {T}$ and unity (i.e. the 4/3 law) is equal to the sum of the contributions from the other terms, and thus 
$\hat {V}$, 
$\hat {A}$, 
$\hat {H}$ and 
$\hat {P}_I$ must become negligible for 
$\hat {T}\approx 1$. The divergence anisotropy 
$\hat {A}$, advective inhomogeneity 
$\hat {H}$, and pressure inhomogeneity 
$\hat {P}_I$ terms move to larger 
$r_1^*$ as Reynolds number increases, while the 
$\hat {V}$ term remains essentially fixed (relative to 
$r_1^*$). The only remaining contribution in this emerging ‘gap’ is that associated with the ‘transfer’ term, and thus it is there that the ‘4/3’ law behaviour will emerge. Note that the limits of all terms as 
$r_1\rightarrow \infty$ reflect the TKE budget at the centreline of the channel, with 
$\hat {A} + \hat {H}$ representing the turbulence diffusion and 
$\hat {P}_I$ representing the pressure transport. The small-scale limits of the terms shown in figure 2 are dominated by the 
${O}(r_1)$ terms in (2.30)–(2.36).
Figure 2. Budget terms for (a) channel centreline DNS cases and (b) experimental pipe axis data vs 
$r_1/R$, where 
$R$ is the channel half-height or pipe radius. Dashed purple lines in (a) represent 
$\hat {P}_I$ inferred from the other terms via (4.1). Red curves in (b) (CICLoPE dataset only) represent 
$\hat {B}+\hat {A}+\hat {P}_I$ inferred from the other terms via (4.1). Curves in (b) associated with the AP00 dataset are terminated with diamond symbols; the remaining curves in (b) correspond to Z19
$+$. Arrows in (a) highlight trends with increasing 
$\textit {Re}$.
The trends suggested by the arrows in figure 1 are shown explicitly in figure 2 for both the channel DNS (a) and pipe experimental (b) cases. Note that several terms in figure 2 have been inferred from (4.1) as they could not be computed directly. These include 
$\hat {P}_I$ for the Madrid DNSs (shown as dashed purple lines), and 
$\hat {P}_I +\hat {A} + \hat {B}$ for the pipe experimental cases. Note further that Taylor's frozen turbulence hypothesis has been invoked to obtain spatial velocity increments and from the time-resolved hot-wire data and ‘ensemble’ averages from time averages. The ‘large-scale’ terms (i.e. 
$\hat {A}$, 
$\hat {H}$, and 
$\hat {P}_I$) exhibit reasonably good agreement for different 
$\textit {Re}_\tau$ at large separations when plotted against 
$r_1/R$. Note that this agreement is equivalent to the trends relative to 
$r_1^*$ suggested by the arrows in figure 1. The scaling of these terms with 
$r_1/R$ is unsurprising, as these terms are related to anisotropy imposed by the boundary conditions of the flow at large scale. There is, however, a noticeable trend in the 
$\hat {H}$ and 
$\hat {A}$ terms in the channel DNS as they approach zero (cf. figure 2a). This is discussed extensively below. Nevertheless, figure 2 illustrates how the increasing scale separation between the viscous and large-scale terms opens a ‘gap’ that is filled by the (growing) transfer term. The migration of viscous and large-scale terms to increasingly disparate scale ranges leaves only the transfer term in between to balance the dissipation in the budget. Note that, while the small-
$r_1$ limits of the 
$\hat {V}$ terms do constitute a measure of the dissipation (cf. (2.31)), this measure is unrelated to the dissipation estimate used to normalise the experimental data herein, as described in § 3. Thus, while the AP00 
$\hat {V}$ curves terminate prematurely (due to their lower ‘effective’ sample rate after spectral filtering, see § 3) and the two higher-
$\textit {Re}$ 
$\hat {V}$ curves from Z19
$+$ approach limits that are slightly lower than expected (likely due to imperfect spatial resolution), this is not expected to have any effect on the intermediate or large scales, from which the primary conclusions of the present study are drawn.
Upon closer inspection, figure 2 also reveals a key difference between the pipe and channel flow cases. In both cases, the maximum present value of the transfer term 
$\hat {T}$ is just below 0.6. The pipe cases, however, reach 
$\textit {Re}_\tau \approx 14\,000$ (or 
$\textit {Re}_\tau \approx 10\,000$ for the second highest case), while the channel flow cases reach just 
$\textit {Re}_\tau \approx 5200$. This difference is far greater than the likely effect of finite measurement resolution; a ‘synthetic’ 
$\times$-wire probe (e.g. see Vukoslavčević & Wallace Reference Vukoslavčević and Wallace2013; Zimmerman, Morrill-Winter & Klewicki Reference Zimmerman, Morrill-Winter and Klewicki2017) evaluated at the centreline of the HJ06 channel DNS predicts the underestimation of the 
$D_{1ii}/r_1$ peak magnitude to be less than 0.008 % for the wire length associated with the Z19 
$\textit {Re}_\tau \approx 5400$ pipe case (i.e. 
$l^*\approx 1.7$). The observed difference in 
$\hat {T}$ magnitude between the two flows therefore suggests that the 4/3 law region emerges more slowly in the pipe than it does in the channel. It is also consistent with the findings of Tang et al. (Reference Tang, Antonia, Djenidi, Danaila and Zhou2017), who evaluated 
$D_{111}$ at 
$r_1=\lambda$ in various flows and found that the pipe values were lower in magnitude than those in the channel over a range of Reynolds numbers spanning 
$R_\lambda \approx 35$–125.
To allow for closer inspection of the difference between the two flows, figure 3 shows the SBS budgets for the pipe and channel flow cases at nominally matched Reynolds numbers 
$\textit {Re}_\tau \approx 5200$ and 
$\approx 5400$ (or 
$\textit {R}_\lambda \approx 200$ and 
$\approx 185$). The channel flow transfer term reaches a substantially higher maximum value than the pipe transfer term. This appears to be entirely related to the divergence anisotropy term 
$\hat {A}$, as the advective inhomogeneity 
$\hat {H}$ and viscous 
$\hat {V}$ terms are virtually indistinguishable between the two cases at values of 
$r_1$ near the peak in 
$\hat {T}$, while the 
$\hat {P}_I$ and 
$\hat {B}$ terms are both negligible. As noted above, the sign of 
$\hat {A}$ indicates that the actual net interscale energy flux (which is also negative, i.e. from large-to-small scales) is of a larger magnitude than is implied by 
$D_{1ii}$ under the assumption of isotropy (cf. (2.11)). Thus, the observation that 
$\hat {A}$ is larger in the pipe than in the channel indicates that the underestimation of the net downscale energy flux based on (2.11) (i.e. based on the transfer along the longitudinal 
$r_1$ direction) is even more severe in the pipe than in the channel at matched Reynolds number. Evidently, there is more downscale transfer of energy in the 
$r_2$ and 
$r_3$ directions (relative to that in the 
$r_1$ direction) at the axis of a pipe flow than at the centreline of a channel. This may be related to the condition of axisymmetry at the pipe axis.
Figure 3. SBS budget terms for 
$\textit {Re}_\tau \approx 5400$ pipe experimental data from Z19
$+$ (solid curves) and for 
$\textit {Re}_\tau \approx 5200$ channel DNS from LM15. Colours correspond to each term as in figure 2.
The data in figure 2 show that with increasing Reynolds number there is a monotonic approach to the 
$4/3$ law. In the following we focus on the 
$\textit {Re}$-dependence of the different terms in the SBS budget equation, which will later allow us to quantify this approach to the 
$4/3$ law.
4.2. The 
$\textit {Re}$-behaviour of terms
As noted above, the Reynolds number behaviour of 
$\hat {T}$ can be inferred through trends in the 
$\hat {V}$, 
$\hat {A}$, 
$\hat {H}$ and 
$\hat {P}_I$ terms (with Reynolds number) as they approach zero. For any 
$r_1$ where 
$\hat {V}$, 
$\hat {A}$, 
$\hat {H}$ and 
$\hat {P}_I$ each fall below some threshold 
$\delta$, the value of the transfer term must be 
$\hat {T}\geq 1-\varDelta$, where 
$\varDelta \equiv N\delta$ and 
$N$ is the number of terms preventing 
$\hat {T} = 1$. In other words, for 
$\delta =0.25$% the ‘4/3’ law will be satisfied to within 1 % for the range of 
$r_1$ where the magnitude of the each of the four terms (i.e. 
$N=4$) is less than 
$\delta$. As such, we next address the scaling behaviours of these terms as they approach zero in order to characterise the rate at which 
$\hat {T}$ approaches unity (i.e. the 4/3 law).
Figure 4 shows the viscous terms from all present datasets on logarithmic axes. This presentation more clearly visualises the behaviour of 
$\hat {V}$ as it approaches zero. The 
$\hat {V}$ curves exhibit a high degree of similarity at small 
$r_1^*$, with the extent of similarity increasing with Reynolds number (in accordance with the first two hypotheses of Kolmogorov Reference Kolmogorov1941b). Note that the observed similarity is at variance with the findings of Morrison, Vallikivi & Smits (Reference Morrison, Vallikivi and Smits2016) at the pipe centreline. In contrast, we find that the second-order structure functions (which are used to compute 
$\hat {V}$) do exhibit Kolmogorov similarity at small 
$r_1$ when using the present ‘spectral chart’ method (Djenidi & Antonia Reference Djenidi and Antonia2012) to determine 
$\left \langle {\epsilon }\right \rangle$ (or even the 
$\langle (\partial u_1/\partial x_1)^2\rangle$ method, to a slightly lesser degree). Indeed, we also find that the 
$D_{11}$ curves implied by the Morrison et al. (Reference Morrison, Vallikivi and Smits2016) velocity spectra, for which they employ a different strategy to determine 
$\left \langle {\epsilon }\right \rangle$ than they do for their 
$D_{11}$ curves, match the present datasets and exhibit reasonable Kolmogorov similarity at small 
$r_1$. That is, if one digitises the spectra from their figure 2 and applies the inverse Fourier transform to compute 
$D^*_{11}$ (i.e. from the autocorrelation), one will observe much better Kolmogorov similarity at small 
$r_1$ than is shown in their figure 3(a) plot of 
$D^*_{11}/{r_1^*}^{2/3}$.
Figure 4. Behaviour with Reynolds number of the viscous term 
$\hat {V}$ in the SBS budget equation at the centreline of pipe and channel flows. Symbols demarcate 
$r^*_{V,\delta }$, i.e. the threshold-crossing position where 
$\hat {V}=\delta$. Colours represent various datasets, with darker shades indicating higher Reynolds numbers: see table 1 for key.
Also plotted in figure 4, as a red dashed line, is the curve associated with the infinite Reynolds number condition for the second-order structure function given by the second similarity hypothesis of Kolmogorov (Reference Kolmogorov1941b), i.e.
\begin{equation} D^*_{ii} = C_{0}{r_1^*}^{2/3},\end{equation}which then gives:
\begin{equation} \hat{V} = \frac{3}{2r_1^*}\frac{\partial}{\partial r_1^*}D_{ii}^* = C_0 {r_1^*}^{{-}4/3}. \end{equation}
Equation (4.3) is also known as the ‘2/3’ law, where the superscript 
$*$ denotes normalisation by the Kolmogorov length/velocity scales. Unlike the 4/3 or 4/5 laws, the coefficient 
$C_0$ is not uniquely determined by Kolmogorov's hypotheses. For the present purpose, this value is chosen based on 
$C_0=\frac {11}{3}C_{L,0}$ coming from the isotropic relation, where 
$D^*_{11} = C_{L,0}{r_1^*}^{2/3}$ is the 2/3 law written for 
$D_{11}$. Here, we use the value 
$C_{L,0}=2.1$, i.e. the upper range of the estimate reported by Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994). Note, however, that the remaining analysis is not sensitive to the exact value of 
$C_{L,0}$. Equation (4.3) may be used to determine the value of 
$r_1^*$ at which 
$\hat {V}\leq \delta$, denoted here as 
$r^*_{V,\delta }$, for all Reynolds numbers. This value is effectively a lower bound for the 4/3 law ‘plateau’ region, or more precisely the lower bound of the region where the transfer term is within 
$(100\times \varDelta )$% of its 4/3 law value. For illustrative purposes in figure (4) and what follows, we choose 
$\delta = 0.0033$. This threshold is shown below to correspond to 
$\varDelta = 0.01$ because 
$\hat {P}_I$ is negligible in the region of interest (and thus 
$N=3$). Although 
$r_{\delta,V}^*$ exhibits a trend with Reynolds number for the data presented herein, the expectation is that this value will eventually saturate as the curves merge with the red dashed line in accordance with (4.3). It will be shown below in the context of plotting 
$r^*_{V,\delta }$ vs 
$R_\lambda$ that the present data support this notion. Saturation of 
$r_{\delta,V}^*$ does not require validity of the 2/3 law, but rather only the validity of Kolmogorov's first similarity hypothesis. The 2/3 law is used here only as a convenient means to estimate 
$r_{\delta,V}^*$ (or any other threshold crossing).
Figure 5 shows the 
$\hat {A}$, 
$\hat {H}$, and 
$\hat {P}_I$ terms in (essentially) the same form as in figure 4, for all present data. Here, however, the curves are plotted vs separation scaled by the pipe radius (or channel half-height) 
$r_1/R$ rather than 
$r_1^*$. This is done to highlight the extent of 
$R$-scaling at large 
$r_1/R$. Note that the ‘
$\hat {A}$’ curves for the experimental cases are actually the same surrogate curves as shown in figure 2(b) and represent 
$\hat {A} + \hat {P}_I +\hat {B}$, which is why they do not exhibit the correct behaviour (i.e. 
$\hat {A}\propto r_1^2$, from (2.32)) in the small 
$r_1$ limit. This figure illustrates that 
$\hat {A}$ and 
$\hat {H}$ are the primary barriers to the emergence of the 4/3 law behaviour (along with 
$\hat {V}$ as discussed above), although 
$\hat {A}$ is the most critical large-scale term. For the remainder of this paper, we therefore use 
$N=3$ to relate 
$\delta$ and 
$\varDelta$, i.e. as 
$\varDelta = 3\delta$.
Figure 5. Behaviour of divergence anisotropy 
$\hat {A}$ (red), advective inhomogeneity 
$\hat {H}$ (green) and pressure inhomogeneity 
$\hat {P}_I$ (purple) terms as they approach zero for all present datasets. Dashed lines suggest a behaviour for the 
$\hat {A}$ and 
$\hat {H}$ terms at sufficiently high Reynolds number, where the terms scale with 
$R$. Grey dashed lines illustrate the projected Reynolds number trend in 
$\hat {H}$ based on trends in the small 
$r_1$ limit for that term as discussed in the main text.
As noted above, the 
$\hat {A}$ and 
$\hat {H}$ terms scale with the largest flow scales (i.e. 
$R$) at large separations. If one chooses a single value for 
$r_1/R$ (for each flow) below which 
$\hat {A}<\delta$ and 
$\hat {H}<\delta$, one only has to know the Reynolds number dependence of 
$R/\eta$ along with the fixed value of 
$r_{V,\delta }^*$ to predict the appearance and growth rate of the region where 
$\hat {T}\geq 1-\varDelta$. We denote this large scale separation value for the given threshold 
$\delta$ as 
$r_{L,\delta }$, chosen such that 
$r_{L,\delta }/R$ is constant. Just as 
$r_{V,\delta }^*$ demarcated a lower 
$r_1$ bound for the approximate (i.e. within 
$\varDelta$) 4/3 law region, here the constant value for 
$r_{L,\delta }/R$ demarcates its upper 
$r_1$ limit (again independent of Reynolds number). Before we can show how these two limits behave with increasing Reynolds number, however, the process by which one can identify a constant value of 
$r_{L,\delta }/R$ for each flow requires explanation.
As noted above in the discussion of figure 2(a), the 
$\hat {A}$ and 
$\hat {H}$ curves exhibit a Reynolds number trend at small separations when plotted against 
$r_1/R$, despite their coalescence at large separations. This trend warrants some discussion, as it is the behaviour of these quantities that determines when and where the 4/3 law behaviour might emerge. The limiting behaviours of the 
$\hat {A}$ and 
$\hat {H}$ terms as 
$r_1\rightarrow 0$ are given by (2.32) and (2.34). These limits, however, only apply at values of 
$r_1$ where velocity increments are well described by linear Taylor series terms alone (e.g. see Appendix C). Empirical evidence below indicates that this region is confined to 
$r_1^*\lesssim 2$. Furthermore, the magnitudes of the 
$\hat {A}$ and 
$\hat {H}$ curves in this region are also shown to decrease monotonically with Reynolds number. Although this implies that the terms do not merge when plotted on log axes vs 
$r_1^*$, this is not at odds with Kolmogorov's first similarity hypothesis, since the trend is for all curves to eventually ‘merge’ at 
$\hat {A}=0$ and 
$\hat {H}=0$ for small 
$r_1$.
The decreasing magnitude and fixed-position (i.e. 
$r_1^*\lesssim 2$) constraints on the ‘linear’ regimes of 
$\hat {A}$ and 
$\hat {H}$ imply that they will appear to shift successively downwards and to the left with increasing Reynolds number when plotted against 
$r_1/R$ as in figure 5, which is precisely what is seen. This is further illustrated by the grey dashed model curves in figure 5, for which the 
$\hat {H}\propto r_1^2$ regions corresponding to higher Reynolds numbers appear at smaller 
$r_1/R$. The construction of these model curves is discussed below in the context of the overall behaviour of the linear regimes with Reynolds number. The migration of the linear regimes to smaller 
$r_1/R$ with increasing Reynolds number implies that for any finite threshold 
$\delta$, there is also a FRN beyond which the linear regimes will not influence the location of the threshold crossing. Given that we observe here a tendency for the 
$\hat {A}$ and 
$\hat {H}$ curves to scale with 
$R$ for ‘large’ 
$r_1$, we posit that this shift of the linear regime to smaller 
$r_1$ will leave 
$\hat {A}$ and 
$\hat {H}$ to scale with 
$R$ in the neighbourhood of these threshold crossings, and thus render 
$r_{L,\delta }/R$ invariant with Reynolds number. It is also clear from figure 5 that the Reynolds numbers associated with the present datasets are not sufficient achieve this 
$\textit {Re}$-invariance when 
$\delta = 0.0033$; each present 
$\hat {A}$ and 
$\hat {H}$ curve crosses the threshold before the onset of the 
$R$-scaling regime. It does appear, however, that the extent of single-curve behaviour vs 
$r_1/R$ grows with increasing Reynolds number, particularly if one examines the 
$\hat {H}$ curves (which are very similar between the two flows, cf. figure 3). This is consistent with the argument above, i.e. that the migration (with 
$\textit {Re}$) of the linear regime to smaller 
$r_1/R$ allows for 
$R$-scaling to be appropriate at successively smaller 
$r_1/R$. It is also consistent with the observations of Boschung et al. (Reference Boschung, Gauding, Hennig, Denker and Pitsch2016), who model the unsteady 
$I_{11}$ and 
$I_{22}$ terms in decaying turbulence via an eddy viscosity closure (cf. figure 9 therein), and Meldi & Vassilicos (Reference Meldi and Vassilicos2021) who present numerical results for decaying turbulence based on an eddy damped quasi-normal Markovian (EDQNM) model.
This raises the question of how to model the growing 
$R$-scaling region between the values of 
$r_1/R$ where we currently observe 
$R$-scaling and those where we expect to observe the linear regime at a given Reynolds number. It will be shown below that there is empirical support for describing the endpoints of the linear regime with a power law fit with a slope of precisely 2/3. Thus, if the scale separation between the end of the linear regime and the onset of 
$R$-scaling is fixed with Reynolds number, then one can also expect the emerging 
$R$-scaling range to be of the form given by
$$\begin{gather} \hat{A}_\infty = C_{\hat{A}_\infty}\left(\frac{r_1}{R}\right)^{2/3}, \end{gather}$$
$$\begin{gather}\hat{H}_\infty = C_{\hat{H}_\infty}\left(\frac{r_1}{R}\right)^{2/3}. \end{gather}$$
This is the basis for the grey dashed model curves in figure 5. Note that the numerical values found by Boschung et al. (Reference Boschung, Gauding, Hennig, Denker and Pitsch2016) to correspond to the power laws for their modelled 
$I_{11}$ and 
$I_{22}$ curves (based on an eddy viscosity closure) correspond to 
$\approx (r_1/L)^{0.6}$ where 
$L$ is the integral length scale. Meldi & Vassilicos (Reference Meldi and Vassilicos2021) also report the slope of precisely 
$2/3$ for their non-stationary term based on an EDQNM closure model. As the method represented by (4.5) and (4.6) ultimately amounts to extrapolation (albeit with a physical basis), § 4.3 addresses the sensitivity of our findings to errors in 
$r_{L,\delta }/R$ of up to 
$\pm$ one half-decade. In the following we show the consistency of (4.5) and (4.6) with the data and the standard turbulence estimates at the channel/pipe centreline.
We now address the separation and magnitude values associated with the linear-regime ‘endpoints’, as this is the basis for the form of (4.5) and (4.6) (which ultimately determines our estimate of 
$r_{L,\delta }/R$). Figure 6 shows the 
$\hat {A}$ and 
$\hat {H}$ curves for the DNS cases compensated by their linear-regime limits. These limits can be obtained by normalising (2.32) and (2.34) by the pre-factor from (4.1) (i.e. using ‘hat’ scaling). For example, 
$A\propto r_1^3$, and therefore 
$\hat {A}\propto r_1^2$, hence 
$\hat {A}{r^*_1}^{-2}$ should go to a constant in the limit as 
$r_1\rightarrow 0$. As noted above, figure 6 suggests that the 
$\hat {A}$ and 
$\hat {H}$ curves begin to deviate from the linear Taylor series approximation for 
$r_1^*\gtrsim 2$. This can also be seen in figure 5 as they shift to smaller 
$r_1/R$ of the linear regimes with increasing 
$\textit {Re}$. Recall here that 
$R/\eta$ increases with Reynolds number. Based on the similarity observed in figure 6, one can predict the values of 
$\hat {A}$ and 
$\hat {H}$ at 
$r_1^*=2$ given a relationship between the Reynolds number and the coefficients 
$C_A$ and 
$C_H$.
Figure 6. Breakdown of linear behaviour for the divergence anisotropy and advective inhomogeneity terms vs 
$r_1^*$ for all DNS cases. Curves diverge from linear Taylor series limits (i.e. those expressed in (2.32) and (2.34)) at 
$r_1^*\approx 2$. Here, 
$C^*_A$ and 
$C^*_H$ are values of 
$C_A$ and 
$C_H$ normalised by the Kolmogorov scales.
The coefficients 
$C_A$ and 
$C_H$ that respectively determine the magnitudes of the 
$\hat {A}$ and 
$\hat {H}$ terms as 
$r_1\rightarrow 0$ are plotted as functions of wall distance 
$x_2$ in figure 7. The coefficients are normalised here by the wall shear stress scales, as indicated by the superscript ‘
$+$’. The empirical evidence strongly suggests 
$C_H^+\propto \textit {Re}_\tau ^{-2}$ at both the pipe axis and channel centreline (albeit with different constants of proportionality) and 
$C_A^+\propto \textit {Re}_\tau ^{-2}$ for at least the channel case. While 
$C_A^+$ cannot be computed from the experimental data, it is expected that the Reynolds number trend is preserved, as the terms both ultimately represent the same physical production mechanism for 
$\left \langle {(\partial u_i/\partial x_1)^2}\right \rangle$ (see (2.32)).
Figure 7. Coefficients from (a) (2.32) and (b) (2.34), respectively representing the small-scale limits of the inhomogeneous transport term and the divergence anisotropy term, at each wall-normal location. Line colours as in table 1. All terms normalised by the wall-friction scales 
$U_\tau$ and 
$\nu$. Dashed lines show a power law slope corresponding to 
$x_2^{-2}$. Filled circular markers highlight the value of each curve at the channel centreline/pipe axis. Darker shades represent higher Reynolds number.
This 
$\textit {Re}_\tau ^{-2}$ scaling is perhaps best understood in the context of classical scaling arguments for the dissipation rate, i.e. 
$(l/{u^\prime }^3)\left \langle {\epsilon }\right \rangle = Const.$, where 
$u^\prime$ here is a turbulent velocity scale and 
$l$ is a ‘dimension defining the scale of the system’ (Taylor Reference Taylor1935). For 
$l\propto R$ and 
$u^\prime \propto U_\tau$, the classical scaling gives 
$\left \langle {\epsilon }\right \rangle ^+ \propto \textit {Re}_\tau ^{-1}$ at the channel centreline/pipe axis. If the length and velocity scales associated with turbulent diffusion are also 
$R$ and 
$U_\tau$ (e.g. see Tennekes & Lumley Reference Tennekes and Lumley1972), respectively, then it follows that the turbulent diffusion of 
$\left \langle {\epsilon }\right \rangle ^+$ (i.e. 
$C_H^+$) scales with 
$\textit {Re}_\tau ^{-2}$, i.e. 
$u_2^+\partial \left \langle {\epsilon }\right \rangle ^+/\partial x_2^+ \sim \left \langle {\epsilon }\right \rangle ^+/R^+ \sim \textit {Re}_\tau ^{-2}$. While this argument is formulated specifically for 
$C_H$, it is not surprising that the two main ‘source’ terms of anisotropy at small 
$r_1$, i.e. 
$C_H$ and 
$C_A$, appear to exhibit the same scaling behaviour as one another.
The 
$\left \langle {\epsilon }\right \rangle ^+\propto \textit {Re}_\tau ^{-1}$ scaling, i.e. 
$(\left \langle {\epsilon }\right \rangle \nu /U_\tau ^4) \propto \textit {Re}_\tau ^{-1}$, can also be used to translate the friction-scaled coefficients 
$C_A^+$ and 
$C_H^+$ to their Kolmogorov-scaled counterparts based on the following relationship between the two:
\begin{equation} C_A^* = C_A^+\left(\frac{U_\tau^4}{\nu \left\langle{\epsilon}\right\rangle}\right)^{3/2},\quad C_H^* = C_H^+\left(\frac{U_\tau^4}{\nu \left\langle{\epsilon}\right\rangle}\right)^{3/2}. \end{equation}
That is, the classical dissipation scaling ultimately yields 
$C^*_A \propto C^*_H \propto \textit {Re}_\tau ^{-1/2}$. From this result, the magnitudes of 
$\hat {A}$ and 
$\hat {H}$ at 
$r_1^*=2$ can then be computed according to
$$\begin{gather} \hat{A}(r_1^*=2) = 3C^\prime_A \textit{Re}_\tau^{{-}1/2}, \end{gather}$$
$$\begin{gather}\hat{H}(r_1^*=2) = 3C^\prime_H \textit{Re}_\tau^{{-}1/2}, \end{gather}$$
where 
$C^\prime _A\equiv C^*_A\textit {Re}_\tau ^{1/2}$ and 
$C^\prime _H\equiv C^*_H\textit {Re}_\tau ^{1/2}$ are 
$\textit {Re}$-invariant for each flow. These constants, for example, are directly proportional to the ‘intercepts’ of the black dashed lines in figure 7.
With the empirically established Reynolds number behaviour of the linear regime endpoints given by (4.8) and (4.9), we now discuss how this relates to the overall behaviour of the 
$\hat {A}$ and 
$\hat {H}$ terms. Figure 8 shows the predicted values of 
$\hat {A}$ and 
$\hat {H}$ at 
$r_1^*=2$ based on (4.8) and (4.9) along with the following relation between 
$R$ and 
$\eta$:
\begin{equation} \frac{R}{\eta} = C_\eta \textit{Re}_\tau^{3/4}. \end{equation}
This, again, is based on the relationship 
$\left \langle {\epsilon }\right \rangle ^+ \propto \textit {Re}_\tau ^{-1}$ at the centreline. Here, we find that 
$C_\eta \approx 1.01$ for the channel and 
$C_\eta \approx 1.14$ for the pipe. Equations (4.8) (and/or (4.9)) along with (4.10) define a set of parametric equations that can be rewritten in terms of 
$r_1/R$ to yield a power law describing the endpoints of the ‘linear’ regimes as they appear in figure 5. The form of this power law is identical to that given by (4.5) and (4.6) with the 2/3 slope. This finding clarifies why this slope is chosen above. This power law is also used to generate the linear portions of the grey dashed line models in figures 5 and 8, which is why their endpoints align with those of the channel DNS. The correspondence between these linear-regime endpoints and the 2/3 power law curves shown above is shown explicitly in figure 8. The ‘extrapolation’ power law curves used to predict 
$r_{L,\delta }/R$ are clearly parallel to the sets of 
$\hat {A}(r_1^*=2)$ and 
$\hat {H}(r_1^*=2)$ points. This supports the notion that the proposed power law curves are representative of the 
$\hat {A}$ and 
$\hat {H}$ curves once they transition into 
$R$-scaling. That is, if the portion of the curves between the end of the ‘linear’ regime and the start of the 
$R$-scaling regime is self-similar across Reynolds numbers, then the endpoints of the ‘linear’ regime can be used as a surrogate for the onset of the 
$R$-scaling regime. Our data in figure 8, although at lower 
$\textit {Re}$, are consistent with this self-similar 
$\textit {Re}$ trend. We find that approximately 1 decade of scales separate the end of the linear regime from the onset of tangency to the proposed power law. In other words, a linear regime that ends one decade to the left of a vertical dashed line in figure 8 corresponds to a 2/3 power law region for the corresponding flow that starts where the vertical and horizontal dashed lines intersect. A linear regime that ends at 
$r_1/R\lesssim 7\times 10^{-6}$ in the channel (or 
$r_1/R\lesssim 3\times 10^{-6}$ in the pipe) therefore corresponds to 
$\hat {A}$ being in the 
$R$-scaling regime at 
$r_{L,\delta }$ (for 
$\delta = 0.0033$). Thus, from (4.10) we can expect 
$r_{L,\delta }/R$ to saturate for 
$\delta =0.0033$ when 
$\textit {Re}_\tau \gtrsim 2\times 10^7$ in the channel and 
$\gtrsim 5\times 10^7$ in the pipe (or 
$R_\lambda \gtrsim 1.2\times 10^4$ and 
$\gtrsim 1.8\times 10^4$).
Figure 8. Comparison between suggested power law extrapolation lines and the endpoints of the ‘linear’ regime. All curves are as plotted in figure 5, but here are shown in grey to focus attention on the correspondence between the extrapolation lines and the ends of the ‘linear’ regimes. Experimental data points shown as square symbols, DNS shown as diamonds. Dashed grey lines illustrate the expected behaviour of the 
$\hat {H}$ curves with increasing Reynolds number, with grey 
$\times$ symbols demarcating the expected endpoints of the linear regime.
4.3. Implications for 4/3 law
The previous section asserts that 
$r^*_{V,\delta }$ and 
$r_{L,\delta }/R$ are constant at sufficiently high Reynolds number, and suggests methods for estimating these constant values for any given 
$\varDelta$. We now combine these methods with (4.10) to predict the Reynolds number at which an approximate 4/3 law ‘plateau’ (to within 
$\varDelta$) will emerge, along with its location and growth rate. This process is shown in figure 9 for 
$\delta = 0.0033$ (i.e. 
$\varDelta = 0.01$). Note that any of the above relations written in terms of 
$\textit {Re}_\tau$ can be rewritten in terms of 
$R_\lambda$ by invoking the following relationship:
\begin{equation} R_\lambda = C_\lambda\textit{Re}_\tau^{1/2}, \end{equation}
where we find 
$C_\lambda \approx 2.69$ at the centreline of the channel and 
$C_\lambda \approx 2.57$ at the axis of the pipe. Note that (4.11) follows from the classical relationship (e.g. see Pope Reference Pope2000) 
$\textit {Re}_L = \frac {1}{2}\sqrt {u_iu_i}L/\nu \propto R_\lambda ^2$ when 
$\sqrt {u_iu_i}\propto U_\tau$ and 
$L\propto R$. The criteria used to identify the upper and lower bounds of the 4/3 ‘plateau’ region can also be written more generally as a function of the Reynolds number and 
$\varDelta$. Now, if we choose the value 
${\hat V} = \varDelta /3$, the following relation derived from (4.4) outputs the lower 
$r_1^*$ bound of the 4/3 ‘plateau’ region in canonical internal flows given an allowable proportional deviation 
$\varDelta = 3\delta$ from the limiting 4/3 value
\begin{equation} r^*_{V,\delta} = \left(\frac{3}{11}\frac{\varDelta}{3}C_{L,0}^{{-}1}\right)^{{-}3/4}. \end{equation}
The same can be done for the upper 
$r_1/R$ bound based on the power laws for 
$\hat {A}$ and 
$\hat {H}$ given by (4.5) and (4.6) and shown in figures 5 and 8. For this purpose, it is more precise to consider 
$r_{L,\delta }$ as the scale at which 
$\hat {A}+\hat {H} = 2\delta$ rather than the threshold-crossing scale for either quantity
\begin{equation} r_{L,\delta}/R = \left(\frac{2\varDelta}{3\left(C_{\hat{A}_\infty}+ C_{\hat{H}_\infty}\right)}\right)^{3/2}. \end{equation}
These two bounds can now be combined via (4.10) to yield expressions for the width (in decades of 
$r_1$) of the 4/3 law ‘plateau’ region (to within 
$\varDelta$) at any given Reynolds number
\begin{align} W &= \frac{3}{4}\log_{10}\left(\textit{Re}_\tau\right) + \left[\frac{3}{2}\log_{10}\left(\frac{2\varDelta}{3\left(C_{\hat{A}_\infty} +C_{\hat{H}_\infty}\right)}\right) \right.\nonumber\\ &\quad \left.\vphantom{\frac{2\varDelta}{3\left(C_{\hat{A}_\infty} +C_{\hat{H}_\infty}\right)}} +\log_{10}\left(C_\eta\right)+\frac{3}{4}\log_{10}\left(\frac{\varDelta}{11C_{L,0}}\right)\right] \end{align}
\begin{align} W &= \frac{3}{2}\log_{10}\left(\textit{R}_\lambda\right) + \left[\frac{3}{2}\log_{10} \left(\frac{2\varDelta}{3C_\lambda \left(C_{\hat{A}_\infty}+C_{\hat{H}_\infty}\right)}\right) \right.\nonumber\\ &\quad \left.\vphantom{\frac{2\varDelta}{3C_\lambda \left(C_{\hat{A}_\infty}+C_{\hat{H}_\infty}\right)}} +\log_{10}\left(C_\eta\right) + \frac{3}{4}\log_{10}\left(\frac{\varDelta}{11C_{L,0}}\right)\right]. \end{align}
Note that negative outputs of (4.14) or (4.15) indicate that 
$\hat {T}<1-\varDelta$ for all 
$r_1$. Equations (4.14) and (4.15) indicate that the width of a ‘plateau’ region, i.e. a scale range satisfying 
$\hat {T}>1-\varDelta$, grows by three quarters of a decade for each decade increase in 
$\textit {Re}_\tau$, and by 1.5 decades for each decade increase in 
$\textit {R}_\lambda$. The terms inside the square brackets are solely functions of 
$\varDelta$ (not Reynolds number), and determine the Reynolds numbers at which 
$W$ becomes positive. The constant values that appear in (4.14) and (4.15) are aggregated in table 2. The 3/2 slope of the growth rate with 
$R_\lambda$ in (4.15) is in agreement with that reported by Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019) for isotropic and grid-generated decaying turbulence, which stems from the correspondence between 
$R$ and the integral length scale 
$L$ and the growth rate of 
$L/\eta$ with 
$R_\lambda$ in an isotropic flow. Antonia et al. (Reference Antonia, Tang, Djenidi and Zhou2019) also predict, based on a model equation for 
$D_{11}$, that 
$D_{111}/r_1$ will be within 
$2.5$% of the 4/5 law (i.e. (1.2)) for two decades of 
$r_1$ in decaying grid-generated turbulence when 
$R_\lambda \approx 8\times 10^4$. In comparison, the present (4.15) indicates that 
$D_{1ii}/r_1$ will be within 2.5 % of the 4/3 law when 
$R_\lambda \approx 3.0\times 10^5$ at the channel centreline and 
$R_\lambda \approx 4.1\times 10^5$ at the pipe axis (or 
$\textit {Re}_\tau \approx 1.3\times 10^{10}$ and 
$\approx 2.6\times 10^{10}$, respectively). This is consistent with the findings of Tang et al. (Reference Tang, Antonia, Djenidi, Danaila and Zhou2017), who evaluated 
$D_{111}$ at 
$r_1=\lambda$ for various flows and found that (between these three flows) the axis of the pipe was the slowest to approach the 4/5 law, followed by the channel centreline, and then grid-generated turbulence.
Figure 9. Estimated behaviour, with Reynolds number, of the upper (black dashed lines) and lower (red dashed line) 
$r_1^*$ bounds of the approximate 4/3 law plateau region (i.e. bounds within which the transfer term will be within 1 % of its ‘4/3’ law value). Grey shaded regions represents the uncertainty associated with a 
$\pm$ half-decade error in 
$r_{L,\delta }/R$ for both the channel and pipe cases. This uncertainty affects only the position of the upper bound line, not its slope.
Table 2. Summary of constants used throughout this study and in (4.14) and (4.15). 
${\dagger}$ Value corresponds to the upper limit of estimates given by Saddoughi & Veeravalli (Reference Saddoughi and Veeravalli1994). Note the isotropic relation 
$C_0 = \frac {11}{3}C_{L,0}$ for evaluating expressions containing 
$C_0$.
From (4.4) along with (4.5) and (4.6) one can compute the implied transfer term 
$\hat {T}$, including the location 
$r_p$ and magnitude 
$\hat {T}_p$ of its peak. Invoking (4.10), the sum of (4.5) and (4.6) can be written
\begin{equation} \hat{A}_\infty + \hat{H}_\infty = \left(\frac{C_{\hat{A}_\infty} + C_{\hat{H}_\infty}}{C_\eta^{2/3}\textit{Re}_\tau^{1/2}}\right){r_1^*}^{2/3}. \end{equation} Since 
$\hat {A} + \hat {H} + \hat {V} + \hat {T} \approx 1$ near 
$\hat {T}_p$, the power law (4.4) for 
$\hat {V}$ and (4.16) imply the following shape for 
$\hat {T}$:
\begin{equation} \hat{T} = 1 - C_0 {r_1^*}^{{-}4/3} - \left(\frac{C_{\hat{A}_\infty} + C_{\hat{H}_\infty}}{C_\eta^{2/3}\textit{Re}_\tau^{1/2}}\right){r_1^*}^{2/3}. \end{equation}
This function is plotted for a range of Reynolds numbers along with several LM15 channel flow cases in figure 10. At lower Reynolds numbers where 
$r_p$ does not fall within the respective power law ranges for 
$\hat {V}$ or 
$\hat {A}+\hat {H}$, the right-hand side of (4.17) instead represents an approximate lower limit for 
$\hat {T}$. The word ‘approximate’ is used here because (4.4) is not strictly an upper bound for 
$\hat {V}$ for all 
$r$, as can be seen in figure 4.
Figure 10. Normalised energy transfer term 
$\hat {T}$ for three highest-
$\textit {Re}$ LM15 channel cases and (4.17) evaluated at matched 
$\textit {Re}$ (with channel flow coefficients) as well as at 
$R_\lambda = 10^3$, 
$10^4$, 
$10^5$ and 
$10^6$.
The dependence of 
$r^*_p$ on the Reynolds number can then be obtained by differentiating (4.17) and solving for 
$r_1$ such that 
$\textrm {d}\hat {T}/\textrm {d} r_1^*=0$
\begin{equation} r_p^* = \left(\frac{2C_\eta^{2/3} C_0}{C_{\hat{A}_\infty} + C_{\hat{H}_\infty}}\right)^{1/2}\textit{Re}_\tau^{1/4}. \end{equation}
The classical relationship between the Taylor microscale and the Kolmogorov length scale is 
$\lambda ^*\propto R_\lambda ^{1/2}$ (e.g. see Tennekes & Lumley Reference Tennekes and Lumley1972). From (4.11) it then follows that 
$r_p\propto \lambda$, a result that has also been reported by a number of other authors employing various techniques (e.g. see Lundgren Reference Lundgren2003; Boschung et al. Reference Boschung, Gauding, Hennig, Denker and Pitsch2016; Meldi & Vassilicos Reference Meldi and Vassilicos2021). This property is also evident in figure 10(b). Evidently, the lowest ebb of combined viscous and large-scale influence on interscale energy flux occurs at a length scale that is in fixed proportion to the (reciprocal of the) mean dispersion of energy about the origin in wavenumber space (Batchelor Reference Batchelor1953).
Finally, the peak magnitude of the transfer term can be computed by inserting (4.18) into (4.17)
\begin{equation} \hat{T}_p = 1 - \frac{3}{2}\left(\frac{2C_0\left(C_{\hat{A}_\infty} + C_{\hat{H}_\infty}\right)^{2}}{C_\eta^{4/3}}\right)^{1/3}\textit{Re}_\tau^{{-}1/3}. \end{equation}
The decay rate 
$1-\hat {T}_p\propto \textit {Re}_\tau ^{-1/3}$, or equivalently 
$1-\hat {T}_p\propto R_\lambda ^{-2/3}$, is also in agreement with the findings of Lundgren (Reference Lundgren2003) and Djenidi & Antonia (Reference Djenidi and Antonia2020) (when the 2/3 law is invoked) for forced isotropic turbulence.
5. Conclusions
We propose a new SBS budget equation for use in flows with homogeneity in at least one direction. This new form differs from that proposed by Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001) through the inclusion of terms associated with (spatial) pressure transport of energy along with the anisotropic portions of the interscale energy divergence and Laplacian of 
$D_{ii}$. The role of mean shear is also made clear here, such that the present analysis can be straightforwardly extended to locations in the channel/pipe away from the centreline. While the new formulation includes terms that are less accessible via physical experiment than those proposed by Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001), it achieves a balance of better than 2 % across all scales in the streamwise direction at the centreline of the 
$\textit {Re}_\tau \approx 5200$ channel flow DNS of Lee & Moser (Reference Lee and Moser2015) (compared with 
${\approx }25$ % for the Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001) budget equation). The large- and small-scale limits of each term in the new SBS budget equation are shown to reduce to those associated with the budget equations for 
$\langle u_iu_i\rangle$ and 
$\langle (\partial u_i/\partial x_1)^2\rangle$, respectively, in accordance with the requirements for SBS equations noted by Antonia et al. (Reference Antonia, Zhou, Danaila and Anselmet2000).
Reynolds number trends in several terms in the new budget equation are discussed. The contribution from the viscous term is shown to vary only very slowly with Reynolds number at fixed 
$r_1^*$, with (4.4) providing an asymptotic limit in the ‘inertial’ subrange. This implies that the lower 
$r_1$ bound of the 4/3 law ‘plateau’ region (to within some proportion 
$\varDelta$) is fixed relative to the Kolmogorov length scale and invariant with Reynolds number (once realised). The term representing the anisotropic component of the interscale energy divergence (i.e. 
$\hat {A}$) is shown to be the primary barrier to emergence of the 4/3 law beyond the viscous subrange at both the channel centreline and the pipe axis, with the term representing inhomogeneous spatial transport of 
$D_{ii}$ (i.e. 
$\hat {H}$) also playing a role. The relative magnitudes of the third-order velocity structure functions in the two flows imply a more significant deviation from isotropy of the interscale energy divergence (i.e. larger 
$\hat {A}$) in the pipe flow. That is, the relative values of 
$D_{1ii}$ between the two flows imply that the downscale transfer of energy in the 
$r_2$ and 
$r_3$ directions is larger than that in the 
$r_1$ direction on the pipe axis than on the channel centreline. The small-scale limiting regimes of 
$\hat {A}$ and 
$\hat {H}$ are shown to decrease in magnitude (i.e. approach zero) and remain confined to 
$r_1^*\lesssim 2$ with increasing Reynolds number. It is argued that the migration of these small-scale regimes to smaller 
$r_1/R$ will leave the remainder of the 
$\hat {A}$ and 
$\hat {H}$ terms to scale with 
$R$, and that the rate of this migration implies a 2/3 power law form (i.e. (4.5) and (4.6)) for 
$\hat {A}$ and 
$\hat {H}$ at sufficiently high Reynolds number. This allows characterisation of the upper 
$r_1$ bound of the 4/3 law ‘plateau’ region (to within some proportion 
$\varDelta$). The behaviours of the upper and lower bounds are then combined in (4.14) and (4.15) to describe the emergence and growth rate of a 4/3 law ‘plateau’ region as a function of Reynolds number and the degree of satisfaction (i.e. 
$\varDelta$). The power laws used to determine these upper and lower bounds are also combined to produce an expression for the transfer term in the vicinity of its peak, from which the peak magnitude and location are also derived (all as functions of Reynolds number). In this way, it is shown that the peak in the energy transfer term occurs at a scale that is in fixed proportion to the Taylor microscale.
Acknowledgements
The authors would like to extend their gratitude to the CICLoPE team at the University of Bologna for granting us access to their pipe flow facility, as well as to the authors of dAJ03, dA04, HJ06, G16 and LM15 for making their channel flow DNS fields freely available online.
Funding
This work was supported by the Australian Research Council.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Reynolds decomposition applied to (2.6)
The viscous terms in (2.6) can be rewritten in terms of 
$\boldsymbol {x}$ and 
$\boldsymbol {x}^\prime$ according to (2.5a,b)
\begin{equation} 2\nu\left[\left(\frac{\partial^2}{\partial r_k \partial r_k} + \frac{1}{4}\frac{\partial^2}{\partial X_k \partial X_k} \right)\tilde{D}_{ij} - \tilde{E}_{ij}\right] = 2\nu\left[\left(\frac{\partial^2}{\partial x_k\partial x_k} + \frac{\partial^2}{\partial x_k^\prime \partial x_k^\prime}\right)\tilde{D}_{ij} - \tilde{E}_{ij}\right]. \end{equation}
Applying the Reynolds decomposition (i.e. 
$\tilde {u}_i = U_i + u_i$) to the Laplacian terms and applying (2.3) (i.e. independence of 
$\tilde {u}_i$ on 
$\boldsymbol {x}^\prime$, etc.) results in (for example)
\begin{equation} \frac{\partial^2}{\partial x_k^2}\left\langle{\left(\tilde{u}_i - \tilde{u}_i^\prime\right) \left(\tilde{u}_j - \tilde{u}_j^\prime\right)}\right\rangle = \frac{\partial^2}{\partial x_k^2}U_iU_j + \frac{\partial^2}{\partial x_k^2}\left\langle{u_iu_j}\right\rangle - U_i^\prime \frac{\partial^2 U_j}{\partial x_k^2} - U_j^\prime \frac{\partial^2 U_i}{\partial x_k^2}. \end{equation}
For the case where 
$U_i = U_i^\prime$ (i.e. a fully developed flow, when the flow is homogeneous in the direction of separation), applying the second-order product rule to the first term on the right-hand side of (A2) then yields
\begin{equation} \frac{\partial^2}{\partial x_k^2}\left\langle{\left(\tilde{u}_i - \tilde{u}_i^\prime\right) \left(\tilde{u}_j - \tilde{u}_j^\prime\right)}\right\rangle = \frac{\partial^2}{\partial x_k^2}\left\langle{u_iu_j}\right\rangle + 2\frac{\partial U_i}{\partial x_k}\frac{\partial U_j}{\partial x_k}.\end{equation}
The last term on the right-hand side of (A3) (and the analogous term from the 
$\boldsymbol {x}^\prime$ Laplacian) then cancels with the mean component of 
$\tilde {E}_{ij}$, and thus
\begin{equation} 2\nu\left[\left(\frac{\partial^2}{\partial r_k \partial r_k} + \frac{1}{4}\frac{\partial^2}{\partial X_k \partial X_k} \right)\tilde{D}_{ij} - \tilde{E}_{ij}\right] = 2\nu\left[\left(\frac{\partial^2}{\partial r_k \partial r_k} + \frac{1}{4}\frac{\partial^2}{\partial X_k \partial X_k} \right)D_{ij} - E_{ij}\right]. \end{equation}
A similar process can then be used to show that 
$\tilde {T}_{ij} = T_{ij}$ and 
$\partial \tilde {D}_{ij}/\partial t = \partial D_{ij}/\partial t$ for the case when the mean flow at 
$\boldsymbol {x}$ is the same as that at 
$\boldsymbol {x}^\prime$.
Appendix B. Derivations of large-scale limits ($r_1\rightarrow \infty$)
The large-scale limits of the terms in (2.19) (for the streamwise homogeneous case) can be determined by invoking (2.3) and (2.5a,b) as well as requiring that all fluctuating quantities at 
$\boldsymbol {x}$ and 
$\boldsymbol {x}^\prime$ eventually become decorrelated at sufficiently large 
$r_1$. When combined with streamwise homogeneity (such that 
$\partial \langle \cdot \rangle /\partial x_1 = 0$), this decorrelation can be expressed as
$$\begin{gather} \lim_{r_1\rightarrow \infty} \frac{\partial \langle \cdot \rangle}{\partial r_1} = 0, \end{gather}$$
$$\begin{gather}\lim_{r_1\rightarrow \infty} \langle{u_iu_j^\prime}\rangle = 0. \end{gather}$$Note that (B2) also applies to pressure–velocity correlations and pressure–pressure correlations.
From (2.13), 
$\tilde {A}_{ii}$ can be written as follows:
\begin{equation} \tilde{A}_{ii} = \frac{\partial}{\partial r_k} \tilde{D}_{kii} - \left(\frac{\partial}{\partial r_1} + \frac{2}{r_1}\right)D_{1ii}. \end{equation}
In the limit as 
$r_1\rightarrow \infty$, the rightmost two terms containing 
$D_{1ii}$ both go to zero by (B1). From (2.5a,b), the definition of 
$\tilde {D}_{kii} = \langle \tilde {v}_k\tilde {v}_i\tilde {v}_i\rangle$ and the definition of 
$\tilde {v}_i\equiv \tilde {u}_i-\tilde {u}_i^\prime$, the limit of 
$\tilde {A}_{ii}$ can then be written as
\begin{equation} \lim_{r_1\rightarrow \infty} \tilde{A}_{ii} = \lim_{r_1\rightarrow \infty} \frac{1}{2}\left(\frac{\partial}{\partial x_k} - \frac{\partial}{\partial x^\prime_k}\right) \left\langle\left(\tilde{u}_k-\tilde{u}_k^\prime\right) \left(\tilde{u}_i-\tilde{u}_i^\prime\right)^2\right\rangle. \end{equation}
Applying the Reynolds decomposition to (B4) produces terms of the following type, for example, which contain products of mean quantities at both 
$\boldsymbol {x}$ and 
$\boldsymbol {x}^\prime$:
$$\begin{gather} \frac{\partial}{\partial x_k}\left(U_k^\prime U_i^\prime U_i\right) = U_k^\prime U_i^\prime \frac{\partial}{\partial x_k}U_i = U_k U_i \frac{\partial}{\partial x_k}U_i, \end{gather}$$
$$\begin{gather}\frac{\partial}{\partial x^\prime_k}\left\langle{u_k u_i U^\prime_i}\right\rangle = \left\langle{u_k u_i}\right\rangle \frac{\partial}{\partial x_k^\prime}U^\prime_i = \left\langle{u_k u_i}\right\rangle \frac{\partial}{\partial x_k}U_i. \end{gather}$$In (B5) and (B6) above, (2.3) is invoked to obtain the middle expression from the first (leftmost), and streamwise homogeneity is invoked to obtain the third from the second.
Applying the Reynolds decomposition to (B4) and expanding produces a large number of terms, many of which cancel out or vanish by (B2), continuity, or 
$\left \langle {u_i}\right \rangle =0$. Collecting these terms and applying streamwise homogeneity in the same way as (B5) and (B6) results in
\begin{equation} \lim_{r_1\rightarrow \infty} \tilde{A}_{ii} = \frac{\partial}{\partial x_k}\left\langle{u_iu_iu_k}\right\rangle + 4\left\langle{u_iu_k}\right\rangle \frac{\partial}{\partial x_k}U_i. \end{equation} The contribution from 
$\tilde {A}_{ii}$ to the SBS budget equation is given by the integral in (2.19), which can be written
\begin{equation} \frac{1}{r_1^2}\int_{0}^{r_1} y^2\tilde{A}_{ii}\,{\textrm{d}y} = \frac{1}{r_1^2} \int_{0}^{L} y^2\tilde{A}_{ii}\,{\textrm{d}y} + \frac{1}{r_1^2} \int_{L}^{r_1} y^2\tilde{A}_{ii}\,{\textrm{d}y}, \end{equation}
where 
$L$ is a large length scale beyond which (B1) and (B2) apply. As originally pointed out by Danaila et al. (Reference Danaila, Anselmet, Zhou and Antonia2001), the integral from 
$0$ to 
$L$ converges to a finite value while the integral from 
$L$ to 
$\infty$ does not (for cases with non-zero turbulence diffusion and/or production), and so
\begin{equation} \lim_{r_1\rightarrow \infty} \frac{1}{r_1^2}\int_{0}^{r_1} y^2\tilde{A}_{ii}\,{\textrm{d}y} = \frac{1}{3}\frac{\partial}{\partial x_k}\langle u_iu_iu_k \rangle r_1 + \frac{4}{3}\langle u_iu_k \rangle\frac{\partial U_i}{\partial x_k} r_1. \end{equation}
The channel centreline limit is then obtained by noting that all mean velocity gradients vanish due to symmetry (i.e. the production term is zero) and 
$x_2$ is the only inhomogeneous direction.
The remaining large-scale limits (2.25)–(2.28) may be computed in a very similar way. From (2.14), 
$B_{ii}$ can be written as follows:
\begin{equation} B_{ii} = \left(\frac{\partial^2}{\partial r_k\partial r_k} - \frac{\partial^2}{\partial r_1^2} - \frac{2}{r}\frac{\partial}{\partial r_1}\right)D_{ii}. \end{equation}
As above, the 
$\partial /\partial r_1$ terms go to zero in the limit as 
$r_1\rightarrow \infty$, as do the 
$\partial /\partial r_3$ terms from spanwise homogeneity, and so from (2.5a,b) we have
\begin{equation} \lim_{r_1\rightarrow \infty} B_{ii} = \frac{1}{4}\left(\frac{\partial^2}{\partial x_2^2} + \frac{\partial^2}{\partial {x_2^\prime}^2} - \frac{\partial^2}{\partial x_2\partial x_2^\prime}\right) \langle (u_i-u_i^\prime )^2\rangle = \frac{1}{2}\frac{\partial^2}{\partial x_2^2} \langle u_i^2 \rangle. \end{equation}
Note that the expression on the far right follows from the same arguments that led to the result in (B7). Integrating 
$2\nu B_{ii}$ in the same manner as (B8) then produces the result given in (2.25). The remaining limits (2.26)–(2.28) can be computed through the same procedures used herein for (2.24) and (2.25).
Appendix C. Derivations of small-scale limits ($r_1\rightarrow 0$)
Each of the limits from (2.32)–(2.36) can be computed from the Taylor series expansions for 
$\tilde {v}_i$ (and 
$v_i$) and 
$p-p^\prime$. The Taylor series for 
$\tilde {v}_i$, for example, can be obtained from those of 
$\tilde {u}_i$ and 
$\tilde {u}_i^\prime$ about 
$\boldsymbol {X}$ (e.g. see Pope Reference Pope2000; Hill Reference Hill2002b)
$$\begin{gather} \tilde{u}_i = u^o_i + \frac{\partial \tilde{u}^o_i}{\partial X_j}\frac{r_j}{2} + \frac{1}{2!}\frac{\partial \tilde{u}^o_i}{\partial X_j}\frac{r_jr_k}{2^2} + \frac{1}{3!}\frac{\partial^3 \tilde{u}^o_i}{\partial X_j\partial X_k\partial X_l} \frac{r_jr_kr_l}{2^3} + \cdots \end{gather}$$
$$\begin{gather}\tilde{u}_i^\prime = \tilde{u}^o_i - \frac{\partial \tilde{u}^o_i}{\partial X_j} \frac{r_j}{2} + \frac{1}{2!}\frac{\partial \tilde{u}^o_i}{\partial X_j}\frac{r_jr_k}{2^2} - \frac{1}{3!}\frac{\partial^3 \tilde{u}^o_i}{\partial X_j\partial X_k\partial X_l} \frac{r_jr_kr_l}{2^3} + \cdots, \end{gather}$$
where 
$\tilde {u}_i^o$ is the velocity evaluated half-way between 
$\tilde {u}_i$ and 
$\tilde {u}_i^\prime$. All even-ordered terms cancel in the corresponding Taylor series for 
$\tilde {v}_i$, i.e.
\begin{equation} \tilde{v}_i = r_j\frac{\partial \tilde{u}^o_i}{\partial X_j} + \frac{r_jr_kr_l}{24} \frac{\partial^3 \tilde{u}^o_i}{\partial X_j\partial X_k\partial X_l} + \cdots . \end{equation}
The Taylor series for 
$v_i$ is the same as that for 
$\tilde {v}_i$ but with 
$u_i$ in place of 
$\tilde {u}_i$. From (C3), the second-order structure function 
$\tilde {D}_{ii}$ can be represented as follows:
\begin{equation} \lim_{\boldsymbol{r}\rightarrow 0} \tilde{v}_i\tilde{v}_i = \left(r_j\frac{\partial \tilde{u}^o_i}{\partial X_j} + \frac{r_jr_kr_l}{24}\frac{\partial^3 \tilde{u}^o_i}{\partial X_j\partial X_k\partial X_l}\right)^2. \end{equation}
The third-order structure function 
$\tilde {D}_{kii}$ can be expressed in a similar way. The third-order terms are retained in (C4) because, as shown below, the viscous terms that result from products of only the linear terms cancel with one another in the limit as 
$r_1\rightarrow 0$. Only a handful of terms from (C4) survive any of the operations 
$\partial /\partial r_1$, 
$\partial ^2/\partial r_k^2$, or 
$\partial ^2/\partial X_k^2$, followed by setting 
$r_2=r_3=0$. These are given in (C5) to facilitate the subsequent derivations of terms involving 
$D_{ii}$
\begin{align} \lim_{\boldsymbol{r}\rightarrow 0} v_iv_i &= r_1^2\left(\frac{\partial u^o_i}{\partial X_1}\right)^2 + r_2^2 \left(\frac{\partial u^o_i}{\partial X_2}\right)^2 + r_3^2\left(\frac{\partial u^o_i}{\partial X_3}\right)^2 + \frac{r_1^4}{12}\frac{\partial^3 u^o_i}{\partial X_1^3}\frac{\partial u^o_i}{\partial X_1} \nonumber\\ &\quad +\frac{r_1^2r_2^2}{4}\left(\frac{\partial^3 u^o_i}{\partial X_1^2\partial X_2} \frac{\partial u^o_i}{\partial X_2} + \frac{\partial^3 u^o_i}{\partial X_1\partial X_2^2} \frac{\partial u^o_i}{\partial X_1}\right) \nonumber\\ &\quad + \frac{r_1^2r_3^2}{4} \left(\frac{\partial^3 u^o_i}{\partial X_1^2\partial X_3}\frac{\partial u^o_i}{\partial X_3} + \frac{\partial^3 u^o_i}{\partial X_1\partial X_3^2}\frac{\partial u^o_i}{\partial X_1}\right) + \cdots. \end{align}
In contrast, it is only necessary for the present purposes to retain the linear Taylor series terms for the non-viscous terms. Again, only a handful of these terms will survive either 
$\partial /\partial r_1$ or 
$\partial /\partial r_k$, followed by setting 
$r_2=r_3=0$. These are given in (C6) to facilitate the subsequent derivations of terms involving 
$\tilde {D}_{iik}$
\begin{align} \lim_{\boldsymbol{r}\rightarrow 0} \tilde{v}_i\tilde{v}_i\tilde{v}_k &= \left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2\frac{\partial \tilde{u}^o_k}{\partial X_1} r_1^3 + \left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2\frac{\partial \tilde{u}^o_k}{\partial X_2}r_1^2r_2 + \left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2\frac{\partial \tilde{u}^o_k}{\partial X_3} r_1^2r_3 \nonumber\\ &\quad +2\frac{\partial \tilde{u}^o_i}{\partial X_1}\frac{\partial \tilde{u}^o_i}{\partial X_2} \frac{\partial \tilde{u}^o_k}{\partial X_1}r_1^2r_2 + 2\frac{\partial \tilde{u}^o_i}{\partial X_1} \frac{\partial \tilde{u}^o_i}{\partial X_3}\frac{\partial \tilde{u}^o_k}{\partial X_1} r_1^2r_3 + \cdots. \end{align} The limit of 
$D_{1ii}$ for 
$r_2=r_3=0$ follows immediately from (C6) (and the substitution of fluctuating velocities in place of instantaneous velocities)
\begin{equation} \lim_{r_1\rightarrow 0} D_{1ii} = \left\langle\left(\frac{\partial u^o_i}{\partial X_1}\right)^2 \frac{\partial u^o_1}{\partial X_1}\right\rangle r_1^3. \end{equation} The viscous term involves differentiation of 
$D_{ii}$ with respect to 
$r_1$, and can be computed from (C5)
\begin{equation} \lim_{r_1\rightarrow 0} 2\nu \frac{\partial}{\partial r_1}D_{ii} = 4\nu \left\langle\left(\frac{\partial u^o_i}{\partial X_1}\right)^2\right\rangle r_1 + \frac{2\nu}{3}\left\langle\frac{\partial^3 u^o_i}{\partial X_1^3} \frac{\partial u^o_i}{\partial X_1}\right\rangle r_1^3. \end{equation} From (C6), the divergence term in (B3) for the case where 
$r_2=r_3=0$ can be written as
\begin{align} \lim_{r_1\rightarrow 0} \frac{\partial }{\partial r_k}\tilde{D}_{iik} &= \left[3\left\langle\left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2 \frac{\partial \tilde{u}^o_1}{\partial X_1}\right\rangle + \left\langle\left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2 \frac{\partial \tilde{u}^o_2}{\partial X_2}\right\rangle + \left\langle\left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2 \frac{\partial \tilde{u}^o_3}{\partial X_3}\right\rangle \right.\nonumber\\ &\quad \left.\vphantom{\left(\frac{\partial \tilde{u}^o_i}{\partial X_1}\right)^2} + 2\left\langle\frac{\partial \tilde{u}^o_i}{\partial X_1}\frac{\partial \tilde{u}^o_i}{\partial X_2} \frac{\partial \tilde{u}^o_2}{\partial X_1}\right\rangle + 2\left\langle\frac{\partial \tilde{u}^o_i}{\partial X_1}\frac{\partial \tilde{u}^o_i}{\partial X_3} \frac{\partial \tilde{u}^o_3}{\partial X_1}\right\rangle\right]r_1^2. \end{align}
For the remaining limits, 
$r_2=r_3=0$ is implied. By invoking continuity, i.e. 
$\partial \tilde {u}^o_i/\partial X_i = 0$, the divergence term can be written simply as
\begin{equation} \lim_{r_1\rightarrow 0} \frac{\partial }{\partial r_k}\tilde{D}_{iik} = 2\left\langle\frac{\partial \tilde{u}^o_i}{\partial X_1}\frac{\partial \tilde{u}^o_i}{\partial X_j} \frac{\partial \tilde{u}^o_j}{\partial X_1}\right\rangle r_1^2. \end{equation}
Finally, applying the Reynolds decomposition and invoking streamwise homogeneity (i.e. 
$\partial U_i/\partial X_1 = 0$) results in
\begin{equation} \lim_{r_1\rightarrow 0} \frac{\partial }{\partial r_k}\tilde{D}_{iik} = 2\left\langle\frac{\partial u^o_i}{\partial X_1}\frac{\partial u^o_i}{\partial X_j} \frac{\partial u^o_j}{\partial X_1}\right\rangle r_1^2 + 2\left\langle\frac{\partial u^o_i}{\partial X_1} \frac{\partial U^o_i}{\partial X_j}\frac{\partial u^o_j}{\partial X_1}\right\rangle r_1^2. \end{equation}
The small-scale limit of 
$-A_{ii}$ can then be written as
\begin{equation} \lim_{r_1\rightarrow 0} -A_{ii} = 5\left\langle\frac{\partial u^o_i}{\partial X_1} \frac{\partial u^o_i}{\partial X_1}\frac{\partial u^o_1}{\partial X_1}\right\rangle r_1^2 - 2\left\langle \frac{\partial u^o_i}{\partial X_1}\frac{\partial u^o_i}{\partial X_j} \frac{\partial u^o_j}{\partial X_1} \right\rangle r_1^2 - 2\left\langle \frac{\partial u^o_i}{\partial X_1}\frac{\partial U^o_i}{\partial X_j} \frac{\partial u^o_j}{\partial X_1} \right\rangle r_1^2. \end{equation}
Plugging this expression into (2.19) and evaluating the integral ultimately results in (C12) being multiplied by 
$r_1/5$, which yields (2.32).
A similar procedure can be used to compute the limit of 
$B_{ii}$ from (B10) and (C5). Equation (2.33) may then be obtained from the resulting expression for 
$B_{ii}$ by noting the following (along with other similar expressions):
\begin{equation} \left\langle \frac{\partial^3 u^o_i}{\partial X_1^2 \partial X_2} \frac{\partial u^o_i}{\partial X_2}\right\rangle + \left\langle \frac{\partial^3 u^o_i}{\partial X_1^2 \partial X_3}\frac{\partial u^o_i}{\partial X_3}\right\rangle = \left\langle \frac{\partial^3 u^o_i}{\partial X_1^2 \partial X_j} \frac{\partial u^o_i}{\partial X_j}\right\rangle - \left\langle \frac{\partial u^o_i}{\partial X_1^3}\frac{\partial u^o_i}{\partial X_1}\right\rangle, \end{equation}
and evaluating the integral in (2.19). The remaining limits (2.34)–(2.36) involve only the first term in the right-hand side of (C5) (or the corresponding linear term for 
$p-p^\prime$), and so are straightforward to evaluate. As noted above, all 
${O}(r_1)$ terms appearing in (2.30)–(2.36) cancel one another when plugged into (2.21) (including the dissipation term). The final step to recover the budget equation (2.37) is then to apply the second-order product rule to two of the third-order terms resulting from the limit of 
$B_{ii}$, i.e.
$$\begin{gather} \left\langle\frac{\partial^3 u^o_i}{\partial X_1\partial X_1 \partial X_j} \frac{\partial u^o_i}{\partial X_j}\right\rangle = \frac{1}{2}\left\langle\frac{\partial^2}{\partial X_1\partial X_1} \left(\frac{\partial u^o_i}{\partial X_j}\frac{\partial u^o_i}{\partial X_j}\right)\right\rangle - \left\langle\frac{\partial^2 u^o_i}{\partial X_1\partial X_j} \frac{\partial^2 u^o_i}{\partial X_1\partial X_j}\right\rangle \end{gather}$$
$$\begin{gather}\left\langle\frac{\partial^3 u^o_i}{\partial X_1\partial X_j \partial X_j} \frac{\partial u^o_i}{\partial X_1}\right\rangle = \frac{1}{2}\left\langle\frac{\partial^2}{\partial X_j\partial X_j} \left(\frac{\partial u^o_i}{\partial X_1}\frac{\partial u^o_i}{\partial X_1}\right)\right\rangle - \left\langle\frac{\partial^2 u^o_i}{\partial X_1\partial X_j} \frac{\partial^2 u^o_i}{\partial X_1\partial X_j}\right\rangle. \end{gather}$$Appendix D. Computation of $r$- and $X$-derivatives
This appendix details the numerical procedure for computation of derivatives with respect to the separation vector 
$\boldsymbol {r}$ and the position vector 
$\boldsymbol {X}$. Relevant points in space are diagrammed in figure 11. Derivatives with respect to the separation vector are computed about the origin via a finite difference scheme applied to vectors whose centrepoints are both at the origin, e.g.
$$\begin{gather} \frac{\partial \tilde{D}_{iik}}{\partial r_1} = \lim_{\delta_1\rightarrow 0} \frac{\langle (\tilde{u}_{i_{E2}}-\tilde{u}_{i_{D1}})^2(\tilde{u}_{k_{E2}}-\tilde{u}_{k_{D1}}) \rangle - \langle (\tilde{u}_{i_{D2}}-\tilde{u}_{i_{E1}})^2(\tilde{u}_{k_{D2}}-\tilde{u}_{k_{E1}}) \rangle}{4\delta_1}, \end{gather}$$
$$\begin{gather}\frac{\partial \tilde{D}_{iij}}{\partial r_2} = \lim_{\delta_1\rightarrow 0} \frac{\langle (\tilde{u}_{i_{A2}}-\tilde{u}_{i_{C1}})^2(\tilde{u}_{k_{A2}}-\tilde{u}_{k_{C1}}) \rangle - \langle (\tilde{u}_{i_{C2}}-\tilde{u}_{i_{A1}})^2(\tilde{u}_{k_{C2}}-\tilde{u}_{k_{A1}}) \rangle}{4\delta_2}. \end{gather}$$
Equations (D1) and (D2) represent first-order central finite differences with respect to components of 
$\boldsymbol {r}$ about the point 
$\boldsymbol {r} = (r_1,0,0) = (\textrm {B2}-\textrm {B1},0,0)$. Here B1, B2, A1, A2, etc. are locations in figure 11 (not equations in the appendices). In actuality, a second-order central finite differencing scheme is employed for the calculations presented in this study. Pinning the centrepoint of the constituent vectors (e.g. A1–C2 and A2–C1) at the origin ensures that spatial heterogeneity does not affect the computed quantities, particularly as 
$r_1\rightarrow 0$. If, for example, the 
$r_2$-derivative in (D2) were instead computed between A2–B1 increments and C2–B1 increments, the result would contain artefacts associated with 
$\partial ^2 U_1/\partial x_2^2$ since 
$\partial U_1/\partial x_2$ would differ between the two vector centrepoints.
Figure 11. Representative schematic of the points used to compute derivatives with respect to 
$\boldsymbol {r}$ and 
$\boldsymbol {X}$. Note that undifferentiated structure functions are computed between the points B2 and B1, e.g. 
$\tilde {D}_{ii} = \langle (\tilde {u}_{i_{B2}} - \tilde {u}_{i_{B1}})^2\rangle$.
Derivatives with respect to 
$\boldsymbol {X}$ are instead computed from vectors that are oriented symmetrically about the origin (with respect to one another), e.g.
$$\begin{gather} \frac{\partial \tilde{F}_{iik}}{\partial X_1} = \lim_{\delta_1\rightarrow 0} \frac{\dfrac{1}{2}\langle (\tilde{u}_{i_{E2}}-\tilde{u}_{i_{E1}})^2(\tilde{u}_{k_{E2}}+\tilde{u}_{k_{E1}}) \rangle - \dfrac{1}{2}\langle (\tilde{u}_{i_{D2}}-\tilde{u}_{i_{D1}})^2(\tilde{u}_{k_{D2}}+\tilde{u}_{k_{D1}}) \rangle}{2\delta_1}, \end{gather}$$
$$\begin{gather}\frac{\partial \tilde{F}_{iik}}{\partial X_2} = \lim_{\delta_1\rightarrow 0} \frac{\dfrac{1}{2}\langle (\tilde{u}_{i_{A2}}-\tilde{u}_{i_{A1}})^2(\tilde{u}_{k_{A2}}+\tilde{u}_{k_{A1}}) \rangle - \dfrac{1}{2}\langle (\tilde{u}_{i_{C2}}-\tilde{u}_{i_{C1}})^2(\tilde{u}_{k_{C2}}+\tilde{u}_{k_{C1}}) \rangle}{2\delta_2}. \end{gather}$$These derivatives therefore reflect spatial inhomogeneity as intended.
Second-order derivatives contained within Laplacian terms are computed via central finite differences in much the same way as the first-order derivatives described above, the only difference being that the (second order) finite differencing scheme employed approximates the second-order derivative.
