2.1 Calculation of the structure functions

We consider a simplified version of the attached-eddy model (e.g. Townsend Reference Townsend1976; Perry & Chong Reference Perry and Chong1982; Perry, Henbest & Chong Reference Perry, Henbest and Chong1986), by adopting a hierarchical random additive process (HRAP) proposed by Meneveau & Marusic (Reference Meneveau and Marusic2013) and Yang et al. (Reference Yang, Meneveau, Marusic and Biferale2016b ). Here, velocity fluctuations at a given point in the flow are modelled as a result of an instantaneous superposition of attached-eddy contributions. For simplicity, we will restrict the discussion to the streamwise velocity component initially, where the fluctuating part is given by the sum

(2.1) $$\begin{eqnarray}u_{x}^{+}(y)=\mathop{\sum }_{l=1}^{N(y)}a_{l}.\end{eqnarray}$$

The random additives $a_{l}$ signify contributions from single, spatially self-similar wall-attached ‘eddy’ structures. The summation is over different hierarchies of eddies, with small $l$ representing the large eddies. Assuming the eddy population density to be inversely proportional to $y$ as originally proposed by Townsend (Reference Townsend1976), the number of summands contributing at $y$ is related to the distance off the wall by

(2.2) $$\begin{eqnarray}N(y)\propto \int _{y}^{\unicode[STIX]{x1D6FF}}\frac{1}{y}\,\text{d}y=\ln (\unicode[STIX]{x1D6FF}/y),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}$ is an outer length scale. A schematic representation of this simple conceptual model is provided in figure 1(a). Following Kolmogorov (Reference Kolmogorov1962), we make the simplifying assumption that the random additives follow a Gaussian distribution with zero mean and variance $\unicode[STIX]{x1D70E}$ denoted by ${\mathcal{N}}(0,\unicode[STIX]{x1D70E}^{2})$ . The mean is zero since we only consider velocity fluctuations. Indeed, the arrival of an attached eddy can be assumed to follow roughly a Poisson process (e.g. Woodcock & Marusic Reference Woodcock and Marusic2015) and the velocity fluctuations are results of the superposition of the eddy-induced velocity fields such that the Gaussian approximation appears justified. In this case, we can restate (2.1) as

(2.3) $$\begin{eqnarray}u_{x}^{+}(y)={\mathcal{N}}\left(0,\mathop{\sum }_{l=1}^{N(y)}a_{l}^{2}\right)={\mathcal{N}}(0,N(y)\langle a^{2}\rangle ),\end{eqnarray}$$

where the second step exploits the self-similarity of the eddies. From this and (2.2), it follows from the properties of the Gaussian distribution that for even-order moments

(2.4) $$\begin{eqnarray}\langle (u_{x}^{+}(y))^{2p}\rangle {>}^{1/p}=\hat{B}_{p}-[(2p-1)!!]^{1/p}\hat{A}_{1}\ln (y/\unicode[STIX]{x1D6FF}),\end{eqnarray}$$

which recovers the result of Meneveau & Marusic (Reference Meneveau and Marusic2013) with the ‘Townsend–Perry’ constant $\hat{A}_{1}$ denoting the slope of the log law for the variance $\langle (u_{x}^{+}(y))^{2}\rangle$ (for which the double factorial $(2-1)!!=1$ ) and $\hat{B}_{p}$ is an additive constant. Note that the hat is used here to differentiate from the actual experimentally obtained constants since (2.1) is only an approximation of the actual velocity field.

Figure 1. (a) Side view sketch of the random additives at different hierarchies I–V with an arbitrarily chosen scaling factor of 1.5; hierarchies that contribute to single-point statistics at point $\unicode[STIX]{x2460}$ are marked blue, those contributing to the difference between $\unicode[STIX]{x2460}$ and $\unicode[STIX]{x2461}$ in red. (b) Top view illustration of the effect of long streamwise structures on the structure functions in different directions. Individual $a$ -eddy contributions at a single hierarchy are represented as boxes and coloured red (blue) if they belong to a high (low) speed structure. The bounds of the large-scale structures (dashed lines) may be interpreted as $b$ -eddies.

Taking the difference between velocity fluctuations at two points cancels out the contributions of common eddies as they are assumed to affect both points equally. Therefore only eddies that affect each point individually remain in this case. We assume that a typical attached eddy has an aspect ratio $R_{x}$ given by its streamwise extent over its height. Then for a given streamwise separation $s_{x}$ only eddies smaller than $y_{d}=s_{x}/R_{x}$ contribute to the difference, i.e. large eddies that are shared between the two points (corresponding to small $l$ ) drop out as illustrated in figure 1(a). As a result

(2.5) $$\begin{eqnarray}u_{x}^{+}(x,y)-u_{x}^{+}(x+s_{x},y)=\unicode[STIX]{x0394}u_{x}^{+}(s_{x})=\mathop{\sum }_{l=N(y_{d})}^{N(y)}a_{l}-\mathop{\sum }_{l=N(y_{d})}^{N(y)}a_{l}^{\prime },\end{eqnarray}$$

where the prime denotes independent eddy contributions at a second point $x+s_{x}$ . The same Gaussian approximation leading to (2.3) yields in this case

(2.6) $$\begin{eqnarray}\unicode[STIX]{x0394}u_{x}^{+}(s_{x})={\mathcal{N}}\left(0,\mathop{\sum }_{l=N(y_{d})}^{N(y)}(a_{l}^{2}+{a_{l}^{\prime }}^{2})\right).\end{eqnarray}$$

Assuming $a_{l}$ and $a_{l}^{\prime }$ , which relate to the same hierarchy at different positions, to be identically distributed based on their self-similarity and again using (2.2), we arrive at

(2.7) $$\begin{eqnarray}\unicode[STIX]{x0394}u_{x}^{+}(s_{x})={\mathcal{N}}\left(0,2[N(y)-N(y_{d})]\langle a^{2}\rangle \right)={\mathcal{N}}(0,2\ln (y_{d}/y)\langle a^{2}\rangle ).\end{eqnarray}$$

We would like to caution the reader that the exact cancelling of ‘shared’ eddies (leading to (2.5)) or the complete independence of eddies at different locations (leading to (2.6)) have to be viewed as first-order approximations and cannot be expected to hold exactly in practice (see also Davidson Reference Davidson2004; Davidson et al. Reference Davidson, Krogstad and Nickels2006a , in this regard). We present in § 2.3 how the framework can be refined to account for streamwise alignment, i.e. incomplete decorrelation, of eddies.

From the result in (2.7), the scaling relation for even-order structure functions is obtained analogous to (2.4) as

(2.8) $$\begin{eqnarray}S_{p}(s_{x};u_{x})=[(2p-1)!!]^{1/p}2\langle a^{2}\rangle \ln (y_{d}/y)=[(2p-1)!!]^{1/p}2\hat{A}_{1}\ln (s_{x}/y)+\hat{E}_{p},\end{eqnarray}$$

where $\hat{E}_{p}$ is a constant. The logarithmic scaling of higher-order structure functions predicted here was recently observed by de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015). Typically, the actual slopes in the semi-logarithmic plots $S_{p}(s_{x};u_{x})$ versus $\ln (s_{x}/y)$ will be smaller than the ones predicted in (2.4) and (2.8) as wall-bounded flows are known to be sub-Gaussian (Jiménez Reference Jiménez1998; Meneveau & Marusic Reference Meneveau and Marusic2013). We will test this for the ECR slopes in § 5.3.

2.2 Extended self-similarity for structure functions

In the following, we will show that the ESS framework relaxes the strong self-similarity assumption necessary to derive (2.7) using an approach similar to Yang et al. (Reference Yang, Meneveau, Marusic and Biferale2016b ). This is useful as in practice self-similarity may be broken by large-scale effects as well as viscous (small-scale) effects such that the scaling according to (2.8) may not be discernible. Without the strong self-similarity assumption we have

(2.9) $$\begin{eqnarray}S_{p}(s_{x};u_{x})=[(2p-1)!!]^{1/p}\left(2\mathop{\sum }_{l=N(y_{d})}^{N(y)}\langle a_{l}^{2}\rangle \right),\end{eqnarray}$$

but now $\langle a_{l}^{2}\rangle \neq \langle a_{k}^{2}\rangle$ if $l\neq k$ . Although no scaling according to (2.8) can be obtained directly in this case, forming the ratio in the spirit of ESS gives

(2.10) $$\begin{eqnarray}\frac{S_{p}(s_{x};u_{x})}{S_{m}(s_{x};u_{x})}=\frac{[(2p-1)!!]^{1/p}\left(2\displaystyle \mathop{\sum }_{l=N(y_{d})}^{N(y)}\langle a_{l}^{2}\rangle \right)}{[(2m-1)!!]^{1/m}\left(2\displaystyle \mathop{\sum }_{l=N(y_{d})}^{N(y)}\langle a_{l}^{2}\rangle \right)}=\frac{[(2p-1)!!]^{1/p}}{[(2m-1)!!]^{1/m}}.\end{eqnarray}$$

This shows that even if the self-similarity of the eddies is broken, ESS establishes a linear relationship between $S_{p}(s_{x};u_{x})$ and $S_{m}(s_{x};u_{x})$ . We note that although (2.10) is trivial under the Gaussian assumption employed here, it is still valuable in demonstrating the functional principle of ESS. In general, $\unicode[STIX]{x0394}u_{x}^{+}(s_{x})$ is non-Gaussian, which e.g. is reflected in the presence of a non-zero additive constant in (1.4), such that this result is non-trivial. Formally, (2.10) holds for all $y_{0}$ and $s_{x}$ . However, at small scales $\unicode[STIX]{x1D702}\ll s_{x}\ll y_{0}$ , where the Kolmogorov scale $\unicode[STIX]{x1D702}$ is the length scale of the smallest, dissipative structures in the flow, the scaling of the inertial subrange will prevail. Unlike the logarithmic relationship (1.3) for the ECR, the ISR scaling has the form of a power law. Hence, instead of  (2.10), the classical ESS in the inertial range (Benzi et al. Reference Benzi, Ciliberto, Tripiccione, Baudet, Massaioli and Succi1993, Reference Benzi, Ciliberto, Baudet and Chavarria1995) manifests itself as the ratio of the logarithms of the structure functions according to

(2.11) $$\begin{eqnarray}\frac{\ln [S_{p}(s_{x};u_{x})]}{\ln [S_{m}(s_{x};u_{x})]}=\text{const.}\end{eqnarray}$$

This serves to highlight the differences between the original ESS hypothesis and the related framework introduced in de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017) and employed here. We will not pursue ISR scaling further in the following.

2.3 Extension to the transversal direction and the spanwise velocity component

It is easy to see that the above arguments can readily be extended to transversal structure functions. To this end, we define the spanwise aspect ratio $R_{z}$ as the ratio of the spanwise extent of the attached eddies to their height. With this definition, we have $y_{d}=s_{z}/R_{z}$ as the height of the largest eddy contributing to the velocity difference between two points separated by $s_{z}$ in the spanwise direction, leading to the same result as (2.8). However, since $R_{x}/R_{z}>1$ , we expect the logarithmic scaling range at smaller separations in the spanwise direction than in the streamwise direction.

It is noteworthy that under the assumptions leading to (2.1), the slope in (2.8) remains unchanged between the streamwise and spanwise directions regardless of $R_{x}/R_{z}$ since the aspect ratio only affects the additive constant. A more realistic prediction can be obtained, however, when experimental evidence for the existence of very long structures (Hutchins & Marusic Reference Hutchins and Marusic2007), whose streamwise extent far exceeds the logarithmic scaling region, is taken into account. Similar to Tomkins & Adrian (Reference Tomkins and Adrian2003), it may be assumed that in this case eddy contributions remain correlated over large streamwise distances as is illustrated in figure 1(b). Hence, the assumption of independence between eddy contributions $a_{l}$ and $a_{l}^{\prime }$ for points separated in $x$ , which was made in deriving (2.8), is no longer readily applicable. Mathematically, we model this effect by considering an additional type of eddy contributions denoted as $b_{l}$ (and referred to as ‘ $b$ eddies’ in the following as opposed to the ‘ $a$ eddies’ considered so far). Doing so is in line with e.g. Perry & Marusic (Reference Perry and Marusic1995), who – albeit for different reasons – considered a total of three eddy types. The streamwise length of the structures ( $O(10\unicode[STIX]{x1D6FF})$ ) is large compared to the upper bound of the ECR scaling ( $\unicode[STIX]{x1D6FF}$ ), such that we assume that $R_{x}^{b}\rightarrow \infty$ (superscript $b$ denotes quantities pertaining to $b$ eddies), while $R_{z}^{b}$ remains finite and all other assumptions for $a$ eddies apply. This implies that $N^{b}(y)=N(y)$ and instead of (2.1) the model velocity is then given to leading order by

(2.12) $$\begin{eqnarray}u_{x}^{+}(y)=\mathop{\sum }_{l=1}^{N(y)}a_{l}+\mathop{\sum }_{l=1}^{N(y)}b_{l}.\end{eqnarray}$$

For structure functions in the streamwise direction, the model prediction (2.8) remains unchanged since $y_{d}^{b}\rightarrow 0$ for $R_{x}^{b}\rightarrow \infty$ , i.e. the $b$ eddies always affect both points and therefore do not contribute to the velocity difference. However, for the spanwise direction using (2.12) leads to

(2.13) $$\begin{eqnarray}\unicode[STIX]{x0394}u_{x}^{+}(s_{z})=\mathop{\sum }_{l=N(y_{d})}^{N(y)}a_{l}-\mathop{\sum }_{l=N(y_{d})}^{N(y)}a_{l}^{\prime }+\mathop{\sum }_{l=N(y_{d})}^{N(y)}b_{l}-\mathop{\sum }_{l=N(y_{d})}^{N(y)}b_{l}^{\prime },\end{eqnarray}$$

from which under the assumption of independent Gaussian random processes and following the same arguments leading to (2.7)–(2.8), the logarithmic scaling according to

(2.14) $$\begin{eqnarray}S_{p}^{T}(s_{z};u_{x})=[(2p-1)!!]^{1/p}2\langle a^{2}+b^{2}\rangle \ln (s_{z}/y)+\hat{E}_{p}\end{eqnarray}$$

is obtained. Comparing this result to the equivalent for $S_{p}(s_{x};u_{x})$ in (2.8), it becomes evident that the logarithmic slope in the spanwise direction is predicted to be greater. Nevertheless, the ESS approach will cancel this effect in a similar vein to (2.10) even if $b_{l}\neq b_{k}$ if $l\neq k$ as is seen from

(2.15) $$\begin{eqnarray}\frac{S_{p}^{T}(s_{z};u_{x})}{S_{m}^{T}(s_{z};u_{x})}=\frac{[(2p-1)!!]^{1/p}2\displaystyle \mathop{\sum }_{l=N(y_{d})}^{N(y)}(\langle a_{l}^{2}\rangle +\langle b_{l}^{2}\rangle )}{[(2m-1)!!]^{1/m}2\displaystyle \mathop{\sum }_{l=N(y_{d})}^{N(y)}(\langle a_{l}^{2}\rangle +\langle b_{l}^{2}\rangle )}=\frac{[(2p-1)!!]^{1/p}}{[(2m-1)!!]^{1/m}},\end{eqnarray}$$

such that the prediction for the relative slopes $D_{p}/D_{m}$ from the attached-eddy model is the same in streamwise and spanwise directions. We will test these implications on the data in § 5.

Moreover, an equivalent ansatz to (2.1) or (2.12) can also be made for the spanwise velocity component $u_{z}$ from which it follows that all of the results of this section are also applicable in this case. Investigations on the spanwise velocity component in wall-bounded flows are generally scarce (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2013; Sillero et al. Reference Sillero, Jiménez and Moser2013; Stevens, Wilczek & Meneveau Reference Stevens, Wilczek and Meneveau2014; Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014) and evidence of an ECR scaling according to (1.3) in this case is missing to date. The data necessary for such an investigation will be available from the simulations employed here. We will therefore include a study of the ECR framework applied to $u_{z}$ in the following. Especially for the TC case, this is of great interest since unlike for the streamwise velocity component, the Taylor rolls directly contribute to $u_{z}$ .

3.1 Numerical dataset for TC flow

The geometry of the TC problem is sketched in figure 2(a). Our results are based on the simulation labelled ‘R3’ in Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016b ) and we refer to the original work for additional details on the numerical set-up. We restrict the discussion to the case of pure inner-cylinder rotation (with angular velocity $\unicode[STIX]{x1D714}_{i}$ ) and a radius ratio $\unicode[STIX]{x1D702}\equiv r_{i}/r_{o}=0.909$ (subscripts ‘ $i$ ’ and ‘ $o$ ’ label quantities related to the inner and outer cylinder, respectively). The Reynolds number based on half the gap width $d=r_{o}-r_{i}$ and the shear stress at the inner cylinder is $Re_{\unicode[STIX]{x1D70F},i}=U_{\unicode[STIX]{x1D70F},i}d/(2\unicode[STIX]{x1D708})=3920$ . In order to be able to afford such a relatively high Reynolds number, the computational domain is kept small by making use of axial as well as an imposed azimuthal symmetry of order 20, that is only a $1/20\text{th}$ segment of the domain is simulated. The simulation employs $N_{\unicode[STIX]{x1D719}}\times N_{r}\times N_{z}=2048\times 1536\times 3072$ grid points resulting in spatial resolutions at mid gap in inner wall units of ${\mathcal{D}}_{\unicode[STIX]{x1D719}}^{+}(r_{i}+r_{o})/2=12.6$ and ${\mathcal{D}}_{z}^{+}=5.1$ , where ${\mathcal{D}}_{\unicode[STIX]{x1D719}}$ and ${\mathcal{D}}_{z}$ are grid spacings in $\unicode[STIX]{x1D719}$ and $z$ , respectively. Due to its limited axial extent, the box only contains one pair of Taylor rolls, which was however shown to be sufficient to represent the relevant flow statistics (Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013; Ostilla-Mónico, Verzicco & Lohse Reference Ostilla-Mónico, Verzicco and Lohse2015). Further, the effect of the number of rolls on the flow statistics has been studied in detail by Ostilla-Mónico, Lohse & Verzicco (Reference Ostilla-Mónico, Lohse and Verzicco2016a ). The rolls are oriented along the azimuthal direction (i.e. they contribute predominantly to the wall-normal and axial velocity components) and we will refer to their axial wavelength as $\unicode[STIX]{x1D706}_{TR}$ . As depicted in figure 2(a), we define the wall-normal coordinate $y$ as the distance from the inner cylinder, while $z$ and $\unicode[STIX]{x1D719}$ denote the spanwise and azimuthal directions, respectively. We will present results for the boundary layer at the inner cylinder only. Therefore, we define $u_{\unicode[STIX]{x1D719}}$ as the velocity deficit $u_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D714}_{i}r_{i}-\widetilde{u}_{\unicode[STIX]{x1D719}}$ with $\widetilde{u}_{\unicode[STIX]{x1D719}}$ denoting the actual azimuthal velocity whose mean profile is sketched in figure 2(a). For convenience, we nevertheless refer to $u_{\unicode[STIX]{x1D719}}$ as the ‘azimuthal’ or ‘streamwise’ velocity component in this paper. Figure 2(a) further includes a sketch of the definitions for structure functions with separations along the streamwise directions. In the case of TC, the distance $s_{\unicode[STIX]{x1D719}}$ is measured along the circumference such that $s_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}(r_{i}+y)$ , where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}$ denotes the difference in $\unicode[STIX]{x1D719}$ between the two points. Apart from the spatial averages over homogeneous directions, our results for TC are averaged over 5 independent snapshots in time, which is sufficient for statistical convergence if not stated otherwise.

Figure 2. Sketches of the geometries of Taylor–Couette (a) and channel flow (b). Also shown are the configurations of structure functions along the streamwise directions.

3.2 Numerical dataset for channel flow and comparison

For comparison to a canonical wall-bounded flow, we also include results based on the channel simulations of Del Alamo et al. (Reference Del Alamo, Jiménez, Zandonade and Moser2004), where 15 independent snapshots are available. The corresponding geometry is shown in figure 2(b). Even though with $Re_{\unicode[STIX]{x1D70F}}=U_{\unicode[STIX]{x1D70F}}h/\unicode[STIX]{x1D708}\approx 1900$ ( $h$ being the half-width), the nominal Reynolds number is somewhat lower in this case compared to the TC data, the flows are indeed quite similar in the proximity of the wall as figure 3 shows. In that figure we plot the mean (figure 3 a) and the streamwise variance (figure 3 b) for both geometries, which display similar behaviour and magnitudes up to $y^{+}\approx 1000$ . Beyond that, the $U_{\unicode[STIX]{x1D719}}^{+}$ profile in TC flattens out in a bulk flow region which has no correspondence in the channel flow. It is important to note that following Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016b ), we employ a $z$ -dependent mean for TC according to $\langle u_{i}^{\prime 2}\rangle ^{+}=\langle (u_{i}^{+}-\langle u_{i}^{+}\rangle _{\unicode[STIX]{x1D719},t})^{2}\rangle _{z}$ , where $\langle \cdot \rangle _{i}$ denotes an average with respect to $i$ . Due to the presence of the Taylor rolls, using the conventional approach of averaging over all directions simultaneously results in a significantly different distribution (cf. the dashed line in figure 3 b). The symbols in figure 3 correspond to the wall-normal positions, $y^{+}=30$ and $y^{+}=90$ , for which results for the structure functions will be reported in § 5.

Figure 3. Mean (a) and variance (b) of the streamwise (for channel flow) and azimuthal (for TC) velocity component; dashed dotted line in (a) is $U^{+}\propto 2.6\ln (y^{+})$ for reference, red solid line in (b): TC variance based on $y$ -dependent mean, dashed line: TC variance based on a conventional mean. Diamonds mark the position for which the structure functions are investigated in § 5.

3.3 Experimental high $Re_{\unicode[STIX]{x1D70F}}$ data from TBL and pipe flows

In order to explore the universality of $E_{p,m}^{\ast }$ across Reynolds numbers and flow types, we employ two sets of high $Re$ measurements. For the TBL, we resort to the hot-wire measurements performed in the Melbourne wind tunnel as first reported in Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009). This dataset encompasses 5 different $Re_{\unicode[STIX]{x1D70F}}$ ranging from 2800 to 19 000 at various wall-normal locations. Additionally, we make use of data recorded in the Princeton Superpipe using the so-called NSTAP probe (Vallikivi et al. Reference Vallikivi, Hultmark, Bailey and Smits2011). These measurements span a range of $2000<Re_{\unicode[STIX]{x1D70F}}<98\,000$ (based on the pipe radius) and are described in more detail in the original publication of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012). As both datasets consist of time resolved point measurements, we make use of Taylor’s hypothesis based on the local mean velocity to transform them into the spatial domain.

For all datasets employed, convergence of the structure functions up to tenth order ( $p=5$ ) was verified by plotting the probability density functions $P(\unicode[STIX]{x0394}u_{i})$ in premultiplied form, i.e. $(\unicode[STIX]{x0394}u_{i})^{p}P$ , at various separation distances for all datasets (not shown). The integral of the plotted quantity then represents the desired structure function and its convergence is judged by ensuring that the tails of $(\unicode[STIX]{x0394}u_{i})^{p}P$ have converged. This procedure has been employed previously by Meneveau & Marusic (Reference Meneveau and Marusic2013), Huisman, Lohse & Sun (Reference Huisman, Lohse and Sun2013a ) and others.

5.1 Streamwise velocity component

We start out by considering structure functions of the streamwise velocity components in the conventional form as a function of $s_{i}/y_{0}$ . Results for the longitudinal direction, which is the most accessible experimentally, are plotted in figure 6(a–c) at increasing orders. For both TC and channel flow data at $y_{0}^{+}=30$ and $y_{0}^{+}=90$ are displayed. After an initial increase, all curves are observed to level off at a constant value. The subsequent drop back to zero is an artefact of the periodic boundary conditions used in the simulations, which implies periodicity also for the structure functions. de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015) were able to determine the logarithmic slopes $D_{p}$ in (1.3) directly based on high $Re_{\unicode[STIX]{x1D70F}}$ data. Comparing their results (included as black lines in figure 6(a–c) to the present data clearly shows that the scaling according to (1.3) does not hold for either of the datasets or wall-normal positions plotted. Apart from not following a straight line, the structure functions also exhibit distinctively different slopes at different wall-normal positions $y_{0}$ . The differences between TC and channel flow in figure 6(a–c) are minimal conforming with the good agreement observed for the $\langle u_{x}^{\prime 2}\rangle$ profile in figure 3(b).

Figure 6. Longitudinal (a–c) and transversal (d–f) structure function of the streamwise (azimuthal) velocity component $u_{x}$ ( $u_{\unicode[STIX]{x1D719}}$ ) for Taylor–Couette (shades of red) and channel (shades of blue). Two different wall-normal positions are shown for structure functions of second (a,d), fourth (b,d) and tenth order (c,f). Pictograms in (a,d) illustrate the geometry of the respective structure function and the black lines represent logarithmic slopes at the respective order for the longitudinal direction from de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015). The legend in (b) applies to all panels.

Similar observations can be made for the transversal structure functions of $u_{\unicode[STIX]{x1D719},x}$ displayed at increasing orders in figure 6(d–f). Generally, the initial increase is steeper for $S_{p}^{T}(s_{z};u_{\unicode[STIX]{x1D719},x})$ and the curves level off at lower normalized distances $s_{z}/y_{0}$ . In the attached-eddy picture, this implies that the eddies have an aspect ratio $R_{x}/R_{z}>1$ . Using the slopes $D_{p}$ determined by de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015) for the longitudinal component (black lines in figure 6 d–f) as reference, it becomes clear that indeed the increase of $S_{p}^{T}(s_{z};u_{\unicode[STIX]{x1D719},x})$ is steeper than that of $S_{p}(s_{x};u_{x})$ at the same $y_{0}$ . This observation is in line with the predictions in § 2 and with findings in the spectral domain by Lee & Moser (Reference Lee and Moser2015) and Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017). However, the striking feature of the transversal structure functions is a pronounced peak at large separations that is uniquely observed in TC flow. This peak occurs at values of $s_{z}$ for which turbulent fluctuations have already become uncorrelated (i.e. the curves have levelled off). Based on this along with the fact that the locations of the peaks correspond to $\unicode[STIX]{x1D706}_{TR}$ as indicated in figure 6(d), the peaks can be related to the existence of high and low speed streaks induced by the Taylor rolls. These velocity differences are substantial. From the fact that the maximum of $S_{1}(s_{x};u_{x})$ is located at about twice the turbulence level, their magnitude can be estimated to be ${\approx}(2\langle u_{\unicode[STIX]{x1D719}}^{\prime 2}\rangle )^{1/2}$ at the investigated wall-normal positions. Also here, the eventual decline to zero is an artefact of the finite box size; further peaks above the turbulence level with a spacing of $\unicode[STIX]{x1D706}_{TR}$ would be expected for domains containing additional pairs of rollers.

Figure 7. Higher-order structure functions plotted versus second order for the streamwise/azimuthal velocity component in longitudinal (a–c) and transversal direction (d–f). Circles of increasing size mark locations where $s_{\unicode[STIX]{x1D719},x}/y_{0}$ is equal to 1, 10 and 100 (a–c) or where $s_{z}/y_{0}$ is 0.2, 1 and 10 (d,e). The triangles in (d–f) mark the position of the Taylor roll peak. All these positions are also marked in figure 6(a,c) for reference. Black lines indicate the respective values of $D_{p}/D_{m}$ determined by de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017) for the longitudinal structure function of $u_{x}$ in TBL flow.

Next, we return to the analysis in the spirit of the ESS method. For this purpose, we present higher-order structure functions in the longitudinal direction with reference to $S_{1}(s_{x};u_{x})$ (in figure 7 a–c) and with reference to $S_{1}^{T}(s_{z};u_{x})$ (in figure 7 d–f), which were displayed in figure 6(a,d), respectively. In order to establish the correspondence to the separation distance, we mark values of $S_{1}(s_{x};u_{x})$ and $S_{1}^{T}(s_{z};u_{x})$ corresponding to $s_{\unicode[STIX]{x1D719},x}/y_{0}=1,10,100$ and $s_{z}/y_{0}=0.2,1,10$ , respectively, with circles of increasing size in figure 7 which are also shown in figure 6(a,d). Additionally, we mark the location of the peak in figure 6(d–f) with a triangle. Altogether, good agreement with the slopes measured in de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017) (indicated by black lines) is observed for the longitudinal direction in figure 7(a,b) at $s_{\unicode[STIX]{x1D719},x}/y_{0}>1$ for both channel and TC. The only small limitation is at the position closer to the wall ( $y_{0}^{+}=30$ ), where due to an initial ‘overshoot’ whose intensity increases with order $p$ , the slope is only recovered at slightly higher values of $r/y_{0}$ . A similar effect is also present in the TBL data as evidenced by the small ‘hump’ at low values of $S_{1}$ for small $y_{0}^{+}$ in figure 4.

Consistent with the above discussion on the aspect ratio of attached eddies, the plots for the transversal direction in figure 7(d–f) attain the respective slopes $D_{p}/D_{1}$ at lower multiples of $y_{0}$ in both flows considered. Besides that, it is remarkable in view of the distinctively different behaviour in figure 6(d–f) how well the Taylor roll peaks align with those slopes even for the tenth-order moment in figure 7(d).

5.2 Spanwise velocity component

While structure functions in both directions of the streamwise component were already considered in de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017) for TBLs, we go beyond their work by investigating the spanwise velocity component in the following. Results for the longitudinal structure functions $S_{p}(s_{z};u_{z})$ , where the separation distance is along the spanwise direction of the flow, are plotted in figure 8(a–c). At a first glance, these plots look very similar to the ones for $u_{\unicode[STIX]{x1D719},x}$ in the transversal directions (figure 6 d–f) with close agreement between channel and TC before once again the Taylor rolls lead to a sharp peak at $s_{z}=\unicode[STIX]{x1D706}_{TR}/2$ . The important difference between the peaks in $S_{p}^{T}(s_{z};u_{\unicode[STIX]{x1D719}})$ and $S_{p}(s_{z};u_{z})$ is that in the latter case, their occurrence is a direct effect of the Taylor rolls, i.e. the rolls contribute directly to $u_{z}$ . This in contrast to the previous case where their action was indirect in transporting low velocity fluid away from the wall and vice versa. A noticeable consequence is that the relative magnitude of the peaks is now inverted such that the peak closer to the wall, where the contribution of the Taylor roll is apparently reduced, is lower for $S_{p}(s_{z};u_{z})$ .

The transversal direction for $u_{z}$ runs along the axis of the Taylor rolls. Hence, their contribution cancels out and the structure functions $S_{p}^{T}(s_{x};u_{z})$ in figure 8(d–f) exhibit no distinct peaks for TC flow. It is however noteworthy that the curves for $S_{p}^{T}(s_{x};u_{z})$ in TC keep increasing even at distances at which their counterparts in channel flow have already saturated. This can be explained by the fact that the Taylor rolls fluctuate in size and position (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco and Lohse2015) which introduces correlations at scales larger than those of the turbulence field. The longest correlation possible is limited by half the box size; thus the fact that the end of the increase of $S_{p}^{T}(s_{x};u_{z})$ coincides with $\unicode[STIX]{x1D706}_{TR}$ is simply a consequence of choosing the domain size equal to $\unicode[STIX]{x1D706}_{TR}$ . It should therefore not be given physical significance beyond demonstrating that modulations of the Taylor rolls with wavelengths up to at least this size are possible.

Figure 8. Same as figure 6 for the spanwise velocity component $u_{z}$ . The legend in (b) applies to all panels. Circles of increasing size mark locations where $s_{z}/y_{0}$ is 0.2, 1 and 10 (a–c) or where $s_{\unicode[STIX]{x1D719},x}/y_{0}$ is equal to 1, 10 and 100 (d,e).

The plots of figure 8 for the spanwise velocity component are repeated in figure 9 in ESS form. For the longitudinal direction (figure 9 a–c), it remains inconclusive whether or not the channel results attain a linear relationship due to the limited scaling range. In the case of TC flow, however, it is obvious that the description by (1.4) does not hold since the slope of the curves at both wall-normal positions is continuously changing. It appears likely that this breakdown is related to the fact that for $S_{p}(s_{z};u_{z})$ the direct contribution of the Taylor rolls does not cancel out. Consequently, the statistics are influenced by structures that are not directly related to the presence of the wall which may be responsible for deviation from the ECR scaling observed in other cases.

In agreement with such a reasoning, it is evident from figure 9(d–f) that $S_{p}^{T}(s_{x};u_{z})$ , for which the $u_{z}$ contribution of the Taylor rolls does cancel out largely, complies with the linear scaling of (1.4) for both channel and TC. Even more so, the slopes in the data for the spanwise velocity component seem consistent with those measured for $u_{x}$ by de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017). Thus, there is evidence that at least in the transversal direction, also the structure function of the spanwise velocity adheres to the same universality as $u_{\unicode[STIX]{x1D719},x}$ .

Figure 9. Same as figure 7 for the spanwise velocity component $u_{z}$ . The legend in (b) applies to all panels.

5.3 Universality of the ratios $D_{p}/D_{1}$

In order to address the question of universality for the slopes in the ESS form in a more structured manner, we plot results for $D_{p}/D_{1}$ in figure 10. In all cases, the data were fitted over a range corresponding to $s_{i}/y_{0}>1$ and at $y_{0}^{+}=90$ where the scaling is generally more robust. The TBL results of de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017) included here are obtained at $Re_{\unicode[STIX]{x1D70F}}=19\,000$ and therefore also serve as a high $Re_{\unicode[STIX]{x1D70F}}$ reference for the present numerical datasets. The most obvious result in this plot is that the actual slopes in all geometries considered significantly deviate from the Gaussian estimates of (2.10) for both the streamwise (figure 10 a) and the spanwise (b) velocity component. This is similar to Meneveau & Marusic (Reference Meneveau and Marusic2013) who observed sub-Gaussian behaviour for higher-order moments of the velocity fluctuations $u^{\prime }$ . Our results indicate that also the distributions of $\unicode[STIX]{x0394}u_{i}$ and $\unicode[STIX]{x1D6E5}^{T}u_{i}$ are sub-Gaussian. Figure 10(a) reinforces the result from figure 7 that for the streamwise velocity the ratios $D_{p}/D_{1}$ agree closely up to tenth order for longitudinal as well as transversal separations. This strongly supports universality of this characteristic across all geometries considered – now also including TC.

Figure 10. Result from fitting the slopes $D_{p}/D_{1}$ in the ESS form for the streamwise (a) and spanwise (b) velocity component in the longitudinal and transversal direction. Fits are computed for a range corresponding to $s_{i}/y>1$ at $y_{0}^{+}=90$ . The legend in (a) applies to both panels, however, only the transversal direction is shown in (b). The boundary layer result is taken from de Silva et al. (Reference de Silva, Krug, Lohse and Marusic2017); grey dots represent the Gaussian prediction from (2.10).

For the spanwise velocity component $u_{z}$ , the scaling for the longitudinal direction remained either inconclusive (channel) or – in violation of (1.4) – a clearly nonlinear relation was found (TC). Therefore, only the results for $S_{p}^{T}(s_{x};u_{z})$ are displayed in figure 10(b). For these results, the agreement with the TBL results for $u_{x}$ is good for both channel and TC flow and it may be concluded that the universality also extends to the spanwise velocity component in at least the transversal direction.

5.4 Universality of $E_{p,m}^{\ast }$ across flow geometries

We now return to the question of how the additive constants $E_{p,m}^{\ast }$ change across flow geometries. For pipe flow, the Superpipe dataset of Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) provides access to Reynolds numbers even exceeding those achieved in TBL experiments. In figure 11(a–c), we compare results from the Superpipe to those of the TBL previously reported in figure 5(d–f). The pipe data are plotted for $y^{+}>10$ at the lowest three $Re_{\unicode[STIX]{x1D70F}}$ ( $Re_{\unicode[STIX]{x1D70F}}\leqslant 5400$ ) but only for $y^{+}>300$ at higher $Re_{\unicode[STIX]{x1D70F}}$ in order to stay clear of spatial resolution issues (cf. Smits et al. Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011). In general, very good agreement is observed between the two geometries even at tenth order with possibly a slight trend of increasing $E_{p,1}^{\ast }$ with increasing $Re_{\unicode[STIX]{x1D70F}}$ in the pipe. It is further noteworthy, that the pipe results for $Re_{\unicode[STIX]{x1D70F}}\leqslant 5400$ ‘peel off’ the TBL data at $\langle u_{x}^{\prime 2}\rangle ^{+}$ -values corresponding to the vicinity of the inner peak, i.e. $\langle u_{x}^{\prime 2}\rangle ^{+}>6$ . A comparable trend is not observed for higher $Re_{\unicode[STIX]{x1D70F}}$ and it therefore appears plausible that the peel off is a low- $Re_{\unicode[STIX]{x1D70F}}$ effect rather than geometry related even though data very close to the inner peak (located at $y^{+}\approx 15$ ) are not available at the highest $Re_{\unicode[STIX]{x1D70F}}$ . The fact that at nominally comparable $Re_{\unicode[STIX]{x1D70F}}$ the low- $Re_{\unicode[STIX]{x1D70F}}$ behaviour is different for the TBL might be related to the difficulty to define consistent outer length scales (and hence consistent values of $Re_{\unicode[STIX]{x1D70F}}$ ) across different flow geometries (Marusic et al. Reference Marusic, Mckeon, Monkewitz, Nagib, Smits and Sreenivasan2010).

Figure 11. (a–c) Results for $E_{p,1}^{\ast }$ from the Superpipe data over a range $2000<Re_{\unicode[STIX]{x1D70F}}<94\,400$ (shades of green) as a function of the streamwise variance $\langle u_{x}^{\prime 2}\rangle ^{+}$ . The legend in (a) also applies to (b,c). (d–f) Additive constant $E_{p,1}^{\ast }$ for channel and TC flow for different directions and velocity components (cf. legend in d); results are shown for $30\leqslant y^{+}\lesssim 500$ . In all panels, the TBL results from figure 5(d–f) serve as a reference and are plotted in shades of grey.

The results for the streamwise velocity component of the channel in figure 11(d–f) (shown as filled blue symbols) display similar behaviour to the pipe at low Reynolds numbers. At lower values of $\langle u_{x}^{\prime 2}\rangle ^{+}$ (further from the wall), the agreement with the TBL results is good but there are significant deviations when approaching the inner peak that start at slightly lower values of $\langle u_{x}^{\prime 2}\rangle ^{+}$ in this case. It is further striking how closely the results for the longitudinal (circles) and transversal (triangles) directions agree.

Values of $E_{p,1}^{\ast }$ corresponding to structure functions of $u_{\unicode[STIX]{x1D719}}$ in TC flow in both longitudinal and transversal directions (red filled symbols in figure 11 d–f) are observed to agree closely with the channel flow results closer to the wall (higher values of $\langle u_{x}^{\prime 2}\rangle ^{+}$ ). In doing so, they also approach the TBL data around $\langle u_{x}^{\prime 2}\rangle ^{+}\approx 4$ (equivalent to $y^{+}\approx 100$ cf. figure 3 b). Unlike for the channel, both directions are seen to deviate upwards from the TBL results in the case of TC flow already at slightly lower values of $\langle u_{x}^{\prime 2}\rangle ^{+}$ . At this point, also the agreement between longitudinal and transversal structure functions ceases. This anisotropy hints that this deviation from the behaviour of the other geometries may be related to the strong large-scale roll modes (Taylor rolls) present in the bulk region of TC flow. It further remains open whether there is indeed a range of universality for $E_{p,1}^{\ast }$ that also encompasses TC flow at higher $Re_{\unicode[STIX]{x1D70F}}$ , where a more pronounced logarithmic region is present.

For completeness, we also report the $E_{p,1}^{\ast }$ values for the spanwise velocity component $u_{z}$ in the transversal direction in figure 11(d–f) for channel and TC flow plotted versus $\langle u_{z}^{\prime 2}\rangle ^{+}$ . However, due to the lack of reference data at high $Re_{\unicode[STIX]{x1D70F}}$ for this case, not much more can be stated than that – similar to the observations for $u_{x,\unicode[STIX]{x1D719}}$ - also the results for $u_{z}$ differ at lower values of $\langle u_{z}^{\prime 2}\rangle ^{+}$ .