2. Deriving $\boldsymbol{V}\!(\boldsymbol{r})$ for homogeneous isotropic turbulence

In the spectral space, the energy constraint of (1.3) underpins the interpretation of $E(k)$ as an energy-density function. Here, we derive $V\!(r)$ for homogeneous isotropic turbulence using a spatial analogue of the energy constraint. (In Appendix A, we derive this expression using Townsend’s approach).

From the sifting property of the Dirac delta function $\delta (r)$ , we can write

(2.1) \begin{align} \frac {1}{2}Q_{ii}(r) &=\int _{0}^{\infty }{\textrm d}r^{\prime }{\left [ \frac {1}{2}Q_{ii}(r^{\prime }) \right ]} \delta (r^{\prime }-r). \end{align}

By introducing the Heaviside function $\Theta (r)$ , the above integral can be expressed as

(2.2) \begin{align} \frac {1}{2}Q_{ii}(r) &=\int _{0}^{\infty }{\textrm d}r^{\prime }{\left [ \frac {1}{2}Q_{ii}(r^{\prime }) \right ]} \frac {\partial }{\partial r^{\prime }}\Theta (r^{\prime }-r),\nonumber \\ &=\int _{0}^{\infty }{\textrm d}r^{\prime }\frac {\partial }{\partial r^{\prime }}{\left [ -\frac {1}{2}Q_{ii}(r^{\prime }) \right ]} \Theta (r^{\prime }-r),\quad (\text{integration by parts})\nonumber \\ &=\int _{r}^{\infty }{\textrm d}r^{\prime }\frac {\partial }{\partial r^{\prime }}{\left [ -\frac {1}{2}Q_{ii}(r^{\prime }) \right ]}, \end{align}

where we have used $Q_{ii}(r\to \infty ) = 0$ . Taking the limit $r\to 0$ and noting $Q_{ii}(0) = \langle {\boldsymbol{u}^{2}}\rangle$ , we have

(2.3) \begin{align} \int _{0}^{\infty } {\textrm d}r^{\prime }{\left [ -\frac {\partial }{\partial r^{\prime }}\frac {1}{2}Q_{ii}(r^{\prime }) \right ]}&= \frac {1}{2}\langle {\boldsymbol{u}^{2}}\rangle , \end{align}

which is a real-space analogue of the spectral-space energy constraint, (1.3). Thus, analogous to the three-dimensional spectral energy-density function, $E(k)$ , we can define a three-dimensional spatial energy-density function

(2.4) \begin{align} V\!(r)&:=-\frac {\partial }{\partial r}\frac {1}{2}Q_{ii}(r), \end{align}

which is the Townsend–Hamba function, $V_{\textit{TH}}(r)$ (1.8).

Some additional remarks may be useful. Although we could have derived (2.4) without (2.2), introducing (2.2) allows us to interpret $Q_{ii}(r)$ in terms of kinetic energy of eddies. Combining (2.2) and (2.4), we get

(2.5) \begin{align} \frac {1}{2}Q_{ii}(r)&=\int _{r}^{\infty } {\textrm d}s\, V\!(s)=\int _{0}^{\infty } {\textrm d}s\, V\!(s)\Theta (s-r). \end{align}

Here, the ideal low-pass filter $\Theta (r)$ ensures that only eddies of size $r$ or greater make contributions to $({1}/{2})Q_{ii}(r)$ , with the implication

(2.6) \begin{align} \frac {1}{2}Q_{ii}(r)= [ \textrm {cumulative kinetic energy held in eddies of size }r\textrm { or greater} ]. \end{align}

This interpretation based on $V\!(r)$ can be contrasted with that based on $E(k)$ . We can express $Q_{ii}(r)$ in terms of $E(k)$ as

(2.7) \begin{align} \frac {1}{2}Q_{ii}(r)&=\int _{0}^{\infty } {\textrm d}k\, E(k)\frac {\sin (kr)}{kr}. \end{align}

Note that, while $\sin (kr)/(kr)$ effectively is a low pass-filter, dominated by contributions from wavenumbers $k\lesssim 1/r$ , it is not an ideal filter and has an infinite extent in $k$ -space. Thus, all Fourier modes contribute to $({1}/{2})Q_{ii}(r)$ . Interpreting the Fourier modes as eddy sizes implies that eddies of all sizes contribute to $({1}/{2})Q_{ii}(r)$ . The above discussion underscores how different definitions of the energy-density function embody distinct physical interpretations; also see Davidson (Reference Davidson2004).

3. Deriving $\boldsymbol{V}\!(\boldsymbol{r})$ for homogeneous anisotropic turbulence

So far, we have focused on homogeneous isotropic turbulence. Next we generalise the analysis of § 2 for homogeneous anisotropic turbulence and derive an expression for the three-dimensional spatial energy-density function, $V\!(\boldsymbol{r})$ .

We begin with extending (2.1) for the general case (without assuming isotropy or homogeneity)

(3.1) \begin{align} \frac {1}{2}Q_{ii}(\boldsymbol{r},\boldsymbol{x})&=\int _{\Omega }d^{3}\boldsymbol{r}^{\prime } {\left [ \frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime },\boldsymbol{x}) \right ]} \delta ^{3}(\boldsymbol{r}^{\prime }-\boldsymbol{r}), \end{align}

where $\Omega$ is any arbitrary flow domain (with the restriction that $\boldsymbol{r}$ is contained within it) and $\delta ^{3}(\boldsymbol{r})$ denotes the Dirac delta function in $\mathbb{R}^{3}$ . Noting that the Dirac delta function satisfies the relation

(3.2) \begin{align} \nabla \boldsymbol{\cdot }{{\left [ \left(\frac {1}{4\pi }\frac {\boldsymbol{r}-\boldsymbol{r}^{\prime }} {{\left ||\boldsymbol{r}-\boldsymbol{r}^{\prime }\right |\!|}^{3}}\right) \right ]}}&=\delta ^{3}(\boldsymbol{r}-\boldsymbol{r}^{\prime }), \end{align}

we write (3.1) as

(3.3) \begin{align} \frac {1}{2}Q_{ii}(\boldsymbol{r},\boldsymbol{x})&=\int _{\Omega }d^{3}\boldsymbol{r}^{\prime }\, \boldsymbol{\nabla }^{\prime }\boldsymbol{\cdot }{\left [ \frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime },\boldsymbol{x}) \frac {1}{4\pi }\frac {\boldsymbol{r}^{\prime }-\boldsymbol{r}}{{\left ||\boldsymbol{r}^{\prime }-\boldsymbol{r}\right |\!|}^{3}} \right ]}\nonumber \\ &\quad -\int _{\Omega }d^{3}\boldsymbol{r}^{\prime }\, \frac {1}{4\pi }\frac {\boldsymbol{r}^{\prime }-\boldsymbol{r}}{{\left ||\boldsymbol{r}^{\prime }-\boldsymbol{r}\right |\!|}^{3}} \boldsymbol{\cdot }\boldsymbol{\nabla }^{\prime } {\left [ \frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime },\boldsymbol{x}) \right ]} .\end{align}

Applying the divergence theorem yields

(3.4) \begin{align} \frac {1}{2}Q_{ii}(\boldsymbol{r},\boldsymbol{x})&=\int _{\partial \Omega }{\textrm d}S\, \frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime },\boldsymbol{x}) \frac {1}{4\pi }\frac {(\boldsymbol{r}^{\prime }-\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{n}^{\prime }}{{\left ||\boldsymbol{r}^{\prime }-\boldsymbol{r}\right |\!|}^{3}}\nonumber \\ &\quad -\int _{\Omega }d^{3}\boldsymbol{r}^{\prime }\, \frac {1}{4\pi }\frac {\boldsymbol{r}^{\prime }-\boldsymbol{r}}{{\left ||\boldsymbol{r}^{\prime }-\boldsymbol{r}\right |\!|}^{3}} \boldsymbol{\cdot }\boldsymbol{\nabla }^{\prime } {\left [ \frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime },\boldsymbol{x}) \right ]}, \end{align}

where $\hat{\boldsymbol{n}}^{\prime }$ is the unit normal to the surface $\partial \Omega$ that bounds the domain $\Omega$ . This brings us to a critical consideration. When the turbulence is homogeneous, the surface integral in (3.4) vanishes, yielding

(3.5) \begin{align} \frac {1}{2}Q_{ii}(\boldsymbol{r})&= \int _{\Omega }d^{3}\boldsymbol{r}^{\prime }\, \frac {1}{4\pi }\frac {\boldsymbol{r}^{\prime }-\boldsymbol{r}}{{\left ||\boldsymbol{r}^{\prime }-\boldsymbol{r}\right |\!|}^{3}} \boldsymbol{\cdot }\boldsymbol{\nabla }^{\prime }{\left [ -\frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime }) \right ]}. \end{align}

This is the generalisation of (2.2) for homogeneous anisotropic turbulence.

Next, taking the limit $\boldsymbol{r}\to \boldsymbol{0}$ , we arrive at the energy constraint

(3.6) \begin{align} \int _{\Omega }d^{3}\boldsymbol{r}^{\prime }\frac {1}{4\pi } \frac {1}{{\left ||\boldsymbol{r}^{\prime }\right |\!|}^{2}}\boldsymbol{\hat r}^{\prime }\boldsymbol{\cdot }\boldsymbol{\nabla }^{\prime } {\left [ -\frac {1}{2}Q_{ii}(\boldsymbol{r}^{\prime }) \right ]} &= \frac {1}{2}\langle {\boldsymbol{u}^{2}}\rangle , \end{align}

where we have used $Q_{ii}(\boldsymbol{0})/2 = \langle {\boldsymbol{u}^{2}} \rangle /2$ and $\boldsymbol{\hat r}^{\prime } := \boldsymbol{r}^{\prime }/{ \|\boldsymbol{r}^{\prime } \|}$ . From (3.6), we can define a three-dimensional spatial energy-density function

(3.7) \begin{align} V\!(\boldsymbol{r})&:=\frac {1}{4\pi } \frac {1}{\|\boldsymbol{r}\|^{2}}\boldsymbol{\hat r}\boldsymbol{\cdot }\boldsymbol{\nabla } {\left [ -\frac {1}{2}Q_{ii}(\boldsymbol{r}) \right ]} = \frac {1}{4\pi \|\boldsymbol{r}\|^{2}}D_{\boldsymbol{r}}{\left [ -\frac {1}{2}Q_{ii}(\boldsymbol{r}) \right ]}, \end{align}

where $D_{\boldsymbol{r}}:=\boldsymbol{\hat r}\boldsymbol{\cdot }\boldsymbol{\nabla }$ is the directional derivative. For the case of homogeneous isotropic turbulence, we can simplify (3.7) by averaging $V\!(\boldsymbol{r})$ over a spherical shell $S(r)$ , , similar to the definition of $E(k)$ (cf. (1.2)). This transforms (3.7) to (2.4), thereby verifying self-consistency of the derivation discussed here.

Can the above analysis be readily generalised for inhomogeneous turbulence? For example, we may expect that (3.7) simply generalises to (cf. (18) in Hamba Reference Hamba2015)

(3.8) \begin{align} V\!(\boldsymbol{x}, \boldsymbol{r})&:= \frac {1}{4\pi \|\boldsymbol{r}\|^{2}}D_{\boldsymbol{r}}{\left [ -\frac {1}{2}Q_{ii}(\boldsymbol{x}, \boldsymbol{r}) \right ]}. \end{align}

We urge caution with this step. As we noted in the discussion of (3.4), the surface integral vanishes for homogeneous turbulence. But that may not hold for inhomogeneous turbulence. Note, however, that for the case of inhomogeneous turbulence where there exists one or more directions of homogeneity (e.g. streamwise and azimuthal directions in a fully developed pipe flow), the surface integral still vanishes when $\boldsymbol{r}$ is restricted to the homogeneous directions.

4. One-dimensional spatial energy-density functions

In the analysis so far, we considered the three-dimensional spatial energy-density function. Here, we turn attention to the corresponding one-dimensional functions. In the spectral space, one-dimensional spectral energy-density functions, typically denoted by $E_{11}(k_1), E_{22}(k_1),\dots$ , are well known (Davidson Reference Davidson2004). They are particularly useful in the analysis of experimental data, wherein measuring the (three-dimensional) $E(k)$ is very challenging. In the same spirit, here we seek to derive one-dimensional spatial energy-density functions.

For a homogeneous anisotropic turbulent flow, consider the direction $\boldsymbol{r}=r_{1}\boldsymbol{\hat e}_{1}$ . The associated longitudinal velocity correlation function can be expressed as (cf. (3.1))

(4.1) \begin{align} \frac {1}{2}Q_{11}(r_{1}\boldsymbol{\hat e}_{1}) &=\int _{0}^{\infty }{\textrm d}\xi {\left [ \frac {1}{2}Q_{11}(\xi \boldsymbol{\hat e}_{1}) \right ]} \delta (\xi -r_{1}). \end{align}

Following the analysis of § 2, we can simplify the above integral as

(4.2) \begin{align} \frac {1}{2}Q_{11}(r_{1}\boldsymbol{\hat e}_{1}) &=\int _{r_{1}}^{\infty }{\textrm d}\xi \frac {\partial }{\partial \xi }{\left [ -\frac {1}{2}Q_{11}(\xi \boldsymbol{\hat e}_{1}) \right ]}, \end{align}

where we have used $Q_{11}(r_{1}\to \infty ) = 0$ . Taking the limit $r_{1}\to 0$ leads to an energy constraint

(4.3) \begin{align} \int _{0}^{\infty }{\textrm d}\xi {\left [ -\frac {\partial }{\partial \xi } \frac {1}{2}Q_{11}(\xi \boldsymbol{\hat e}_{1}) \right ]} &= \frac {1}{2}\langle {u_{1}^{2}} \rangle , \end{align}

where we have used $Q_{11}(\boldsymbol{0})/2 = \langle {u_{1}^{2}} \rangle /2$ . We can now define a (one-dimensional) longitudinal spatial energy-density function

(4.4) \begin{align} V_{11}(r_{1})&:= -\frac {\partial }{\partial r_{1}}\frac {1}{2}Q_{11}(r_{1}\boldsymbol{\hat e}_{1}), \end{align}

where $V_{11}(r_{1}){\textrm d}r_{1}$ is the contribution to the turbulent energy component $({1}/{2})\langle {u_{1}^{2}} \rangle$ from eddies of size in the range $(r_{1},r_{1}+{\textrm d}r_{1})$ .

Following the above procedure, we can define (one-dimensional) transverse spatial energy-density functions

(4.5a) \begin{align} V_{22}(r_{1})&:= -\frac {\partial }{\partial r_{1}}\frac {1}{2}Q_{22}(r_{1}\boldsymbol{\hat e}_{1}), \end{align}

(4.5b) \begin{align} V_{33}(r_{1})&:= -\frac {\partial }{\partial r_{1}}\frac {1}{2}Q_{33}(r_{1}\boldsymbol{\hat e}_{1}). \end{align}

These functions satisfy the corresponding energy constraints

(4.6a) \begin{align} \int _{0}^{\infty }{\textrm d}r_{1} V_{22}(r_{1})&= \frac {1}{2}\langle {u_{2}^{2}} \rangle , \end{align}

(4.6b) \begin{align} \int _{0}^{\infty }{\textrm d}r_{1} V_{33}(r_{1})&= \frac {1}{2}\langle {u_{3}^{2}} \rangle . \end{align}

(In Appendix C, we discuss the relationship between the spectral and spatial variants of the one-dimensional energy-density functions.) Also note that the analysis discussed here can be easily extended to other directions, namely, $\boldsymbol{r}=r_{2}\boldsymbol{\hat e}_{2}$ and $\boldsymbol{r}=r_{3}\boldsymbol{\hat e}_{3}$ . This yields one-dimensional spatial energy-density functions such as $V_{11}(r_{2}), V_{11}(r_{3})$ , etc.

Last, we consider homogeneous isotropic turbulence. In this case, (4.4) and (4.5) transform to

(4.7a) \begin{align} V_{11}(r)&:= -\frac {\partial }{\partial r}\frac {1}{2}Q_{11}(r)=-\frac {u^{2}}{2}\frac {\partial }{\partial r}f(r), \end{align}

(4.7b) \begin{align} V_{22}(r)&:= -\frac {\partial }{\partial r}\frac {1}{2}Q_{22}(r) =-\frac {u^{2}}{2}\frac {\partial }{\partial r}g(r), \end{align}

(4.7c) \begin{align} V_{33}(r)&\;= V_{22}(r), \end{align}

where

(4.8) \begin{align} u^{2}&=\langle {u_{1}^{2}} \rangle =\langle {u_{2}^{2}} \rangle =\langle {u_{3}^{2}} \rangle =\frac {1}{3}\langle {\boldsymbol{u}^{2}} \rangle , \end{align}

and $f(r):=Q_{11}(r)/u^2$ and $g(r):=Q_{22}(r)/u^2$ are the longitudinal and transverse velocity correlation functions, respectively.

Notably, our proposal for $V_{11}(r)$ (4.7a ) can be related with a previous result. The kinematic relationship

(4.9) \begin{align} \langle {{\left [ \Delta \upsilon (r) \right ]}^{2}} \rangle &=2u^{2}{[ 1-f(r) ]}, \end{align}

can be rewritten as

(4.10) \begin{align} \frac {1}{4}\frac {\partial }{\partial r}\langle {{\left [ \Delta \upsilon (r) \right ]}^{2}} \rangle =-\frac {u^{2}}{2}\frac {\partial f}{\partial r}, \end{align}

which, when substituted into (4.7a ), leads to

(4.11) \begin{align} 3 V_{11}(r)=\frac {\partial }{\partial r}{\left [ \frac {3}{4}\langle {{[ }\Delta \upsilon (r)]^{2}} \rangle \right ]}. \end{align}

Note that this is the Townsend–Davidson function, $V_{\textit{TD}}(r)$ (1.6). Now, because $V_{11}(r) \neq V_{22}(r)$ , it follows that $V_{\textit{TD}}(r) = 3 V_{11}(r) \neq V\!(r)$ (where we have used (B1c )). We can now see why, starting from Townsend’s proposal, Davidson (Reference Davidson2004) and Hamba (Reference Hamba2015) arrived at disparate forms of energy-density functions.

5. Characteristic properties of spatial energy-density functions

In § 1, we listed three characteristic properties for a spatial energy-density function. Here, we consider each property in turn. For simplicity, we focus on homogeneous isotropic turbulence.

Interestingly, property (i) that a density function should be non-negative turns out to be the most difficult to satisfy. Indeed, to our knowledge, none of the functional forms proposed to date are guaranteed to be non-negative. The same is true for $V\!(r)$ , $V_{11}(r)$ and $V_{22}(r)$ . The reason can be understood by expressing these functions in terms of $E(k)$ . Starting with the relationship

(5.1) \begin{align} -\frac {1}{2} u^2 f(r)&=\int _{0}^{\infty }{\textrm d}kE(k)G_{1}(kr),\quad G_{1}(x)=\frac {x\cos x - \sin x}{x^{3}}, \end{align}

and using (4.11), (B1a ), and (B1b ), we can write

(5.2a) \begin{align} rV\!(r)&=\int _{0}^{\infty }{\textrm d}kE(k)H(kr), \end{align}

(5.2b) \begin{align} rV_{11}(r)&=\int _{0}^{\infty }{\textrm d}kE(k)H_{11}(kr), \end{align}

(5.2c) \begin{align} rV_{22}(r)&=\int _{0}^{\infty }{\textrm d}kE(k)H_{22}(kr), \end{align}

where

(5.3) \begin{align} \begin{aligned} H(kr)&=\frac {1}{r^{3}}\frac {\partial }{\partial r}{[ r^{4}H_{11}(kr) ]},\\ H_{11}(kr)&=r\frac {\partial }{\partial r}G_{1}(kr),\\ H_{22}(kr)&=\frac {1}{2r^2}\frac {\partial }{\partial r}{[ r^{3}H_{11}(kr) ]}. \end{aligned} \end{align}

Now, since the kernels $H(x)$ , $H_{11}(x)$ and $H_{22}(x)$ all oscillate about the $x$ -axis taking positive and negative values (figure 1), $V\!(r)$ , $V_{11}(r)$ and $V_{22}(r)$ are not guaranteed to be non-negative. Thus, strictly speaking, these functions are not density functions; Davidson (Reference Davidson2004) suggested using the term ‘signature function’ instead. For simplicity, we shall refer to them as density functions.

Figure 1. Shapes of the kernels $H(x)$ , $H_{11}(x)$ and $H_{22}(x)$ .

Property (ii) expresses an energy constraint. Because we have derived $V\!(r)$ and its one-dimensional variants by imposing this constraint (cf. § 2), these functions satisfy the constraint by construction.

For checking property (iii), consider a random distribution of Townsend’s model eddies of a fixed size $\ell _{e}$ in a two-dimensional plane (Townsend Reference Townsend1976). For this turbulent flow, we can write $f(r)=\exp (-r^{2}/\ell _{e}^{2})$ . From (2.4) and (4.7), it follows that

(5.4a) \begin{align} \frac {\ell _e V\!(r)}{u^{2}}&={ (r/\ell _{e}) }{\left [ 5-2{\left [ (r/\ell _{e}) \right ]}^{2} \right ]}\exp \left(-r^{2}/\ell _{e}^{2}\right), \end{align}

(5.4b) \begin{align} \frac {\ell _e V_{11}(r)}{u^{2}}&={(r/\ell _{e}) }\exp \left(-r^{2}/\ell _{e}^{2}\right), \end{align}

(5.4c) \begin{align} \frac {\ell _e V_{22}(r)}{u^{2}}&={ (r/\ell _{e})}{\left [ 2-{\left [ (r/\ell _{e}) \right ]}^{2} \right ]}\exp \left(-r^{2}/\ell _{e}^{2}\right). \end{align}

In figure 2, we plot the energy distribution across eddy sizes from (5.4). In all cases, there is a clear peak around $r \sim \ell _{e}$ . Notably, in this idealised flow, only $V_{11}(r)$ is non-negative for all $r$ .

Figure 2. Distribution of energy across eddy sizes as quantified by $\ell _e V\!(r)/u^2$ , $\ell _e V_{11}(r)/u^2$ and $\ell _e V_{22}(r)/u^2$ for a random distribution of Townsend’s model eddies of fixed size $\ell _{e}$ .

The above discussion of property (iii) points to a crucial advantage of one-dimensional spatial energy-density functions over their spectral counterparts. In the spectral space, this property reads: for a random distribution of eddies of fixed size $\ell _{e}$ , the corresponding $E(k)$ (or any of its one-dimensional variants) manifests a clear peak around $k \sim \ell _{e}^{-1}$ (Davidson Reference Davidson2004). While $E(k)$ satisfies this property, $E_{11}(k)$ and $E_{22}(k)$ do not. Instead of having a peak around $k \sim \ell _{e}^{-1}$ , they attain their maxima at $k = 0$ . (This is due to the phenomenon of aliasing (Davidson Reference Davidson2004).) As a result, it is difficult to interpret $E_{11}(k)$ and $E_{22}(k)$ in terms of eddy sizes, even though they satisfy the energy constraint. By contrast, as we have seen, $V_{11}(r)$ and $V_{22}(r)$ manifest a clear peak around $r \sim \ell _{e}$ (figure 2), underscoring their clear advantage over $E_{11}(k)$ and $E_{22}(k)$ .

6. Analysis of empirical data

In § 4, we noted that one-dimensional spatial energy-density functions can be useful for analysis of empirical data. Here, for illustration, we analyse data from two canonical wall-bounded flows: channel flow and pipe flow. We take $\boldsymbol{\hat e}_1$ along the streamwise direction, $\boldsymbol{\hat e}_2$ along the wall-normal direction and $\boldsymbol{\hat e}_3$ along the spanwise direction (for channel flow) or the azimuthal direction (for pipe flow). The origin of the coordinate system is at the wall (for channel flow) and the centreline (for pipe flow). Similar to most studies of empirical data, we focus on the case where the position $x_2$ is fixed and $\boldsymbol{r}$ is oriented along $\boldsymbol{\hat e}_1$ . Note that the flow is homogeneous along $\boldsymbol{\hat e}_1$ (and $\boldsymbol{\hat e}_3$ ), but not along $\boldsymbol{\hat e}_2$ .

First, we consider the scaling $E_{11}(k_1) \propto k_1^{-1}$ (Perry, Henbest & Chong Reference Perry, Henbest and Chong1986). In the past decades, this scaling has attracted considerable attention and no small measure of debate (see, e.g. Smits, McKeon & Marusic (Reference Smits, McKeon and Marusic2011), Appendix A in Zamalloa et al. (Reference Zamalloa, Ng, Chakraborty and Gioia2014)). The scaling appears over a limited spatial region and at high Reynolds numbers, Re; curiously, however, it disappears at even higher Re (Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013; Yamamoto & Tsuji Reference Yamamoto and Tsuji2018). The range of $k_1$ for this scaling corresponds to the large eddies. In this regime, using Taylor’s frozen-turbulence hypothesis may incur substantial errors. This limits the use of experimental data since most experiments obtain $E_{11}(k_1)$ using Taylor’s hypothesis.

We analyse data from a direct numerical simulation (DNS) of channel flow at friction Reynolds number $Re _\tau \approx 5200$ (Lee & Moser Reference Lee and Moser2015) ( $ Re _\tau := u_\tau \delta /\nu$ , where $u_\tau$ is the friction velocity, $\delta$ the channel half-width and $\nu$ the kinematic viscosity). Lee & Moser (Reference Lee and Moser2015) noted that, over a small span of wall-normal distances ( $90 \leqslant x_2^+ \leqslant 169$ , where $x_2^+ := x_2 u_\tau /\nu$ ), the premultiplied spectrum $k_1 E_{11}(k_1)$ manifests a plateau, signalling $E_{11}(k_1) \propto k_1^{-1}$ (figure 3 a). (Note that Taylor’s hypothesis was not invoked to compute $E_{11}(k_1)$ .) Expressed in the form of $V_{11}(r_1)$ , $E_{11}(k_1) \propto k_1^{-1}$ scaling transforms to $V_{11}(r_1) \propto r_1^{-1}$ . (This transformation follows from dimensional considerations: dimensionally, $[k_1 E_{11}(k_1)] = [r_1 V_{11}(r_1)]$ , and thus, $E_{11}(k_1) \propto k_1^{-1}$ implies $V_{11}(r_1) \propto r_1^{-1}$ .) In figure 3(b), we plot profiles of $r_1 V_{11}(r_1)$ corresponding to the spectra of figure 3(a). (We compute $V_{11}(r_1)$ from $E_{11}(k_1)$ using (C3a).) Unlike $k_1 E_{11}(k_1)$ , $r_1 V_{11}(r_1)$ does not manifest a plateau for most of the spatial region except near $x_2^+ = 169$ . By probing farther from the wall, we find a curious result. For $169 \leqslant x_2^+ \leqslant 191$ , $k_1 E_{11}(k_1)$ does not manifest a plateau but $r_1 V_{11}(r_1)$ does so (figure 4). Thus, $V_{11}(r_1) \propto r_1^{-1}$ scaling prevails over a region that is spatially contiguous to albeit shifted from the $E_{11}(k_1) \propto k_1^{-1}$ scaling region. The results suggest that the spatial domain of the scaling region depends on the diagnostic function employed to evince the scaling.

Figure 3. Testing the $k_1^{-1}$ and $r_1^{-1}$ scalings for DNS of channel flow at $Re _\tau \approx 5200$ (Lee & Moser Reference Lee and Moser2015): (a) normalised premultiplied spectrum $k_{1} E_{11}(k_{1})/u_{\tau }^{2}$ and (b) normalised premultiplied spatial energy-density function $r_{1}V_{11}(r_{1})/u_{\tau }^{2}$ . Each curve corresponds to a fixed value of $x_2^+$ in the range $x_2^+ \in [90, 169]$ .

Figure 4. Testing the $k_1^{-1}$ and $r_1^{-1}$ scalings for DNS of channel flow at $Re_\tau \approx 5200$ (Lee & Moser Reference Lee and Moser2015): (a) normalised premultiplied spectrum $k_{1} E_{11}(k_{1})/u_{\tau }^{2}$ and (b) normalised premultiplied spatial energy-density function $r_{1}V_{11}(r_{1})/u_{\tau }^{2}$ . Each curve corresponds to a fixed value of $x_2^+$ in the range $x_2^+ \in [169, 191]$ . In the nominal $r_1^{-1}$ scaling regime, $0.1\lesssim r_1/\delta \lesssim 0.3$ , the plateau value $r_{1}V_{11}(r_{1})/u_{\tau }^{2} \approx 0.88$ (black line, panel b).

In figure 3(a), we note the bimodal structure of $k_1 E_{11}(k_1)$ , wherein the $k_1^{-1}$ scaling regime is flanked on its sides by two local peaks. These peaks are considered to be signatures of organised motions at large scales, the left peak corresponding to ‘very-large-scale motions’ (VLSMs) and the right peak to ‘large-scale motions’ (LSMs) (Kim & Adrian Reference Kim and Adrian1999). We represent these peaks as $k_1^{\textit{VLSM}}$ and $k_1^{\textit{LSM}}$ , respectively. The attendant eddy sizes are estimated as $\ell _{{VLSM}} = 2\pi /k_1^{\textit{VLSM}}$ and $\ell _{{LSM}} = 2\pi /k_1^{\textit{LSM}}$ , respectively. From figure 3(a), we find $\ell _{{VLSM}} \approx 6 \delta$ and $\ell _{{LSM}} \approx 0.3 \delta$ . That organised motions exist at scales $\gtrsim \!\delta$ , as is the case with VLSMs, is a surprising finding and an active area of inquiry (Smits et al. Reference Smits, McKeon and Marusic2011).

We wish to draw attention to some potential problems in inferring the eddy sizes using $2 \pi /k_1$ . As discussed in § 1, eddies are not waves – consequently, the wavelength of a Fourier mode may not directly correspond to an eddy size. More important, as we have noted in § 5, it is difficult to infer eddy sizes from $E_{11}(k_1)$ . Shifting from the Fourier space to the real space may prove useful. Interestingly, similar to its spectral counterpart, $r_1 V_{11}(r_1)$ also manifests a bimodal structure (figure 3 b). Analogous to the discussion of the peaks of $k_1 E_{11}(k_1)$ , we can ascribe the left peak to signal LSMs and the right peak to signal VLSMs. The attendant eddy sizes can be estimated as the values of $r_1$ corresponding to these peaks. We find $\ell _{{LSM}} \approx 0.03 \delta$ and $\ell _{{VLSM}} \approx 0.8 \delta$ . Notably, now the organised motions inhabit scales $\lesssim \!\delta$ , consistent with the standard conceptual picture of turbulent eddies. We suggest that an extensive study of empirical data on wall-bounded flows along these lines may yield valuable insights into the structure of organised motions at large scales.

Returning to the $k_1^{-1}$ and $r_1^{-1}$ scalings, we now turn attention to components other than $E_{11}(k_1)$ and $V_{11}(r_1)$ . Specifically, we consider $E_{33}(k_1)$ and $V_{33}(r_1)$ for DNS of channel flow at $Re_\tau = 2000$ (Lee & Moser Reference Lee and Moser2015). (We compute $V_{33}(r_1)$ from $E_{33}(k_1)$ using (C3c ).) Over the span $902 \leqslant x_2^+ \leqslant 1198$ , it is difficult to discern a plateau in $k_1 E_{33}(k_1)$ (figure 5 a). By contrast, $r_1 V_{33}(r_1)$ manifests a plateau, signalling $V_{33}(r_1) \propto r_1^{-1}$ (figure 5 b). Unlike the case in figure 3(b), the plateau for the curves corresponding to different $x_2^+$ positions do not collapse when normalised as $r_{1}V_{33}(r_{1})/u_{\tau }^{2}$ . But, when normalised as $r_{1}V_{33}(r_{1})/\langle {u_{3}^{2}} \rangle$ , they collapse (figure 5 d). (The normalised curves for $k_{1} E_{33}(k_{1})/\langle {u_{3}^{2}} \rangle$ also collapse, but unlike its real-space counterpart, there is no plateau; figure 5(c).) Remarkably, the collapse occurs not only in the plateau region, but over the whole range of $r_1$ . In figure 5(b), we can also see one drawback of $V_{33}(r_1)$ – it becomes negative for $r_1 \gt \delta$ . It does, however, satisfy a weaker condition, $\int _0^{r_1} {\textrm d}r_1 V_{33}(r_1) \geqslant 0$ (Davidson Reference Davidson2004).

Figure 5. Testing the $k_1^{-1}$ and $r_1^{-1}$ scalings for DNS of channel flow at Re $_\tau$ = 2000 (Lee & Moser Reference Lee and Moser2015): (a) normalised premultiplied spectrum $k_{1} E_{33}(k_{1})/u_{\tau }^{2}$ ; (b) normalised premultiplied spatial energy-density function $r_{1}V_{33}(r_{1})/u_{\tau }^{2}$ ; (c) normalised premultiplied spectrum $k_{1} E_{33}(k_{1})/\langle {u_{3}^{2}} \rangle$ ; (d) normalised premultiplied spatial energy-density function $r_{1}V_{33}(r_{1})/\langle {u_{3}^{2}} \rangle$ . Each curve corresponds to a fixed value of $x_2^+$ in the range $x_2^+ \in [902, 1198]$ . In the nominal $r_1^{-1}$ scaling regime, $0.07\lesssim r_1/\delta \lesssim 0.17$ , the plateau value $r_{1}V_{33}(r_{1})\langle {u_{3}^{2}} \rangle \approx 0.3$ (black line, panel d).

Figure 6. Testing the $k_1^{-5/3}$ and $r_1^{-1/3}$ scalings at the centreline of a pipe flow from the Princeton superpipe experiment (Bailey et al. Reference Bailey, Hultmark, Schumacher, Yakhot and Smits2009; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013): (a) normalised premultiplied spectrum $(k_{1}\eta )^{5/3} \eta E_{11}(k_{1})/\nu ^{2}$ ; (b) normalised premultiplied spatial energy-density function $(r_1/\eta )^{1/3} \eta ^{3} V_{11}(r_{1})/\nu ^{2}$ . The data correspond to $Re := U D/\nu = 24\, 000,\, 81\, 000,\, 512\, 000$ , where $U$ is the mean flow velocity and $D$ is the pipe diameter.

A few additional remarks on the $V_{33}(r_1) \propto r_1^{-1}$ scaling may be useful. Because the definition of $V_{33}(r_1)$ is a new result, this scaling and the attendant collapse of $r_{1}V_{33}(r_{1})/\langle {u_{3}^{2}} \rangle$ are novel findings. It is worth noting that, to our knowledge, the spectral counterpart of this scaling, $E_{33}(k_1) \propto k_1^{-1}$ , has not been reported, which adds to the novelty of the $V_{33}(r_1) \propto r_1^{-1}$ scaling. Further, we observed this scaling for channel flow at $Re_\tau = 2000$ but not at $Re_\tau = 1000$ or $Re_\tau \approx 5200$ . It would appear that, similar to the $E_{11}(k_1) \propto k_1^{-1}$ scaling, this scaling is also limited to a finite range of $ Re_\tau$ .

Unlike the $k_1^{-1}$ scaling, the classical Kolmogorov inertial-range scaling, $k^{-5/3}$ , and the related phenomenon of small-scale universality manifest systematic trends with increase of Re (Kolmogorov Reference Kolmogorov1941). To illustrate these trends, we turn to experimental measurements of $E_{11}(k_1)$ at the centreline of a pipe from the high-Re Princeton superpipe experiment (Bailey et al. Reference Bailey, Hultmark, Schumacher, Yakhot and Smits2009; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). In figure 6(a), we plot the premultiplied spectrum $k_1^{5/3} E_{11}(k_1)$ . For $Re \gtrsim 81\,000$ , a plateau emerges, signalling $E_{11}(k_1) \propto k_1^{-5/3}$ . With increase in Re, the plateau broadens. Moreover, the spectra normalised in Kolmogorov units collapse onto a universal curve at the small scales (at large $k_1 \eta$ , where $\eta$ is the Kolmogorov length scale), signalling small-scale universality. The collapsed region broadens with increase in Re. (This collapse persists even for low-Re transitional pipe flow (Cerbus et al. Reference Cerbus, Liu, Gioia and Chakraborty2020).) The spatial counterpart of the $E_{11}(k_1) \propto k_1^{-5/3}$ scaling is $V_{11}(r_1) \propto r_1^{-1/3}$ . (This follows from dimensional considerations: using $[k_1 E_{11}(k_1)] = [r_1 V_{11}(r_1)]$ , $E_{11}(k_1) \propto k_1^{-5/3}$ and $k_1 \propto r_1^{-1}$ , we obtain $V_{11}(r_1) \propto r_1^{-1/3}$ .) In figure 6(b), we plot the profiles of $r_1^{1/3} V_{11}(r_1)$ corresponding to the spectra of figure 6(a). Complementary to the spectral results, we note a plateau emerging and broadening with increase in Re. Similarly, we note a collapse onto a universal curve at small scales, with the collapsed region broadening with increase in Re.