2.1. Structure functions at dissipative scales

At the smallest scales (Kolmogorov scales), using a Taylor series expansion with the assumption of homogeneous isotropic turbulence at these scales (Frisch Reference Frisch1995) we expect a scaling behaviour of $\langle {\rm\Delta}u^{2}\rangle$ according to

(2.2) $$\begin{eqnarray}\langle {\rm\Delta}u^{2}\rangle =\frac{{\it\epsilon}}{15{\it\nu}}r^{2},\end{eqnarray}$$

where ${\it\epsilon}$ denotes the dissipation rate. For the presented analysis we limit ourselves to the logarithmic region in wall-normal space of high Reynolds number turbulent boundary layers. Within this region, there is a near balance between the turbulent production ( $\mathscr{P}$ ) and dissipation ( ${\it\epsilon}$ ) rates (Tennekes & Lumley Reference Tennekes and Lumley1972; Townsend Reference Townsend1976, and others). Thus,

(2.3) $$\begin{eqnarray}\mathscr{P}\approx {\it\epsilon}\approx \frac{U_{{\it\tau}}^{3}}{{\it\kappa}z},\end{eqnarray}$$

where ${\it\kappa}$ is von Kármán’s constant. We note that Davidson et al. (Reference Davidson, Nickels and Krogstad2006b ) have explored deviations from this equilibrium assumption at lower $Re$ . In wall-bounded turbulence at wall-normal locations within the logarithmic region (2.2) simplifies to

(2.4) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2}\rangle =\frac{z^{+}}{15{\it\kappa}}\left(\frac{r}{z}\right)^{2},\end{eqnarray}$$

which holds at scales smaller than the Kolmogorov length, $0<r\lesssim {\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ , i.e. $r/z\lesssim z_{+}^{-3/4}{\it\kappa}^{1/4}$ . A Taylor series expansion also leads to integer scaling exponents for higher-order structure functions ( $\langle {\rm\Delta}u^{n}\rangle \sim r^{n}$ ) in the dissipative range (Nelkin Reference Nelkin1990).

2.2. Structure functions in the inertial subrange ${\it\eta}\ll r\ll z$

At these scales we expect scaling behaviour to be representative of local isotropy. Following Anselmet et al. (Reference Anselmet, Gagne, Hopfinger and Antonia1984) the scaling in this inertial subrange can be expressed as

(2.5) $$\begin{eqnarray}\langle {\rm\Delta}u^{n}\rangle =C_{n}({\it\epsilon}r)^{n/3}\left(\frac{r}{\ell }\right)^{{\it\xi}_{n}-n/3},\end{eqnarray}$$

where ${\it\xi}_{n}$ is the anomalous scaling exponent and $\ell$ is a length scale. The Kolmogorov 1941 theory predicts ${\it\xi}_{2}=2/3$ for the second-order structure function as well as ${\it\xi}_{n}=n/3$ for higher-order moments. Data instead show anomalous scaling with ${\it\xi}_{n}<n/3$ for $n>3$ (Anselmet et al. Reference Anselmet, Gagne, Hopfinger and Antonia1984; Sreenivasan & Antonia Reference Sreenivasan and Antonia1997). The trends can be fitted quite well empirically using the log-Poisson model of She & Leveque (Reference She and Leveque1994),

(2.6) $$\begin{eqnarray}{\it\xi}_{n}=\frac{n}{9}+2\left[1-\left(\frac{2}{3}\right)^{n/3}\right],\end{eqnarray}$$

or the $p$ -model of Meneveau & Sreenivasan (Reference Meneveau and Sreenivasan1987),

(2.7) $$\begin{eqnarray}{\it\xi}_{n}=1-\ln _{2}(0.7^{n/3}+0.3^{n/3}).\end{eqnarray}$$

Thus, for even moments $n=(2,4,6,8,10)$ , we obtain from She & Leveque (Reference She and Leveque1994) ${\it\xi}_{n}=(0.6959,1.2797,1.7778,2.2105,2.5934)$ , and ${\it\xi}_{n}=(0.6937,1.2822,1.7859,2.2289,2.6321)$ for the $p$ -model, essentially the same within experimental accuracy. For concordance with recent work on even moments, we shall set $n=2p$ below and only focus on even moments $2p$ . Moreover, for wall-bounded flow, using (2.3) and setting $\ell =z$ (the distance to the wall), in non-dimensional form we obtain

(2.8) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle ^{1/p}=C_{2p}^{1/p}{\it\kappa}^{-2/3}\left(\frac{r}{z}\right)^{{\it\xi}_{2p}/p}=M_{p}\left(\frac{r}{z}\right)^{{\it\xi}_{2p}/p},\end{eqnarray}$$

with $M_{p}=C_{2p}^{1/p}{\it\kappa}^{-2/3}$ . For wall-bounded flows, in the logarithmic layer such scaling is expected to hold in the range ${\it\eta}\ll r\ll z$ , or $z_{+}^{-3/4}{\it\kappa}^{1/4}\ll r/z\ll 1$ .

2.3. Second-order structure functions at scales $z<r\ll {\it\delta}$

Davidson et al. (Reference Davidson, Nickels and Krogstad2006b ) argued that $\langle {\rm\Delta}u^{2}\rangle$ can be considered as the cumulative energy of eddies of size $r$ and less. They postulated that at scales $r>z$ , $\langle {\rm\Delta}u^{2}\rangle$ will be dominated by inertial-scale eddies whose heights are roughly between $z$ and $r$ , with kinetic energy of order $U_{{\it\tau}}^{2}$ . Accordingly, considering $\text{d}\langle {\rm\Delta}u^{2}\rangle /\text{d}r$ as an energy density Davidson et al. (Reference Davidson, Nickels and Krogstad2006b ) proposed that $r\text{d}\langle {\rm\Delta}u^{2}\rangle /\text{d}r$ is a constant proportional to $U_{{\it\tau}}^{2}$ . When integrated, a logarithmic law of the form

(2.9) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2}\rangle =E_{1}+D_{1}\ln \frac{r}{z}\end{eqnarray}$$

is obtained. The coefficients $D_{1}$ and $E_{1}$ were evaluated by Davidson et al. (Reference Davidson, Nickels and Krogstad2006b ) based on modelling assumptions and moderate Reynolds number data.

In a more detailed study, Davidson et al. (Reference Davidson, Krogstad and Nickels2006a ) proposed that $r$ should be normalised by the length scale $U_{{\it\tau}}^{3}/{\it\epsilon}$ rather than $z$ in (2.9). It is worth highlighting that $U_{{\it\tau}}^{3}/{\it\epsilon}$ is equivalent to scaling with $z$ (after substitution for ${\it\epsilon}$ using (2.3)), valid at high Reynolds numbers assuming that equilibrium holds. Therefore, they proposed that (2.9) should be replaced by

(2.10) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2}\rangle =E_{1}^{\ast }-D_{1}\ln \frac{{\it\kappa}\mathscr{P}}{{\it\epsilon}}+D_{1}\ln \frac{r}{z},\end{eqnarray}$$

where $E_{1}^{\ast }$ is a constant of order unity. Based on their modelling, Davidson & Krogstad (Reference Davidson and Krogstad2009) postulated that $E_{1}^{\ast }$ and $D_{1}$ are universal constants in turbulent boundary layers. Indeed, Davidson et al. (Reference Davidson, Krogstad and Nickels2006a ) showed an improvement when using (2.10) (i.e. using $U_{{\it\tau}}^{3}/{\it\epsilon}$ to normalise $r$ rather than $z$ ). This can be attributed to the finite $Re$ range studied. Davidson & Krogstad (Reference Davidson and Krogstad2014) further considered these issues for smooth and rough wall flows for structure functions up to sixth order.

We note that for simplicity in the presentation hereafter, the range $z<r\ll {\it\delta}$ within the inertial region of structure functions will be referred to as the ‘ $\ln (r/z)$ region’, following Davidson and coworkers.

2.4. Second- and higher-order moments of streamwise velocity

As summarised in the introduction, there is growing evidence of logarithmic behaviour in the variance of the streamwise velocity fluctuations (Hultmark Reference Hultmark2012; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013), and a similar behaviour in high-order moments has been observed (Meneveau & Marusic Reference Meneveau and Marusic2013). For arbitrary even-order moments, this can be written according to

(2.11) $$\begin{eqnarray}\langle u_{+}^{2p}\rangle ^{1/p}=B_{p}-A_{p}\ln \frac{z}{{\it\delta}},\end{eqnarray}$$

where for $p=1$ the scaling of the variance involves the ‘Townsend–Perry constant’ $A_{1}\approx 1.25$ . For Gaussian statistics, Meneveau & Marusic (Reference Meneveau and Marusic2013) pointed out that $A_{p}=[(2p-1)!!]^{1/p}A_{1}$ for higher-order even moments. They found that the values of $A_{p}$ for $p>1$ measured from experimental data display sub-Gaussian behaviour, i.e. $A_{p}<[(2p-1)!!]^{1/p}A_{1}$ . Such results have also been recently obtained in large-eddy simulations by Stevens, Wilczek & Meneveau (Reference Stevens, Wilczek and Meneveau2014). Preliminary discussions are also presented for the spanwise and wall-normal velocity components as all velocity components are available from the LES datasets.

It can be noted (Davidson & Krogstad Reference Davidson and Krogstad2009) that (2.11) and (2.9) (evaluated at $r={\it\delta}$ ) display a dependence on height $z$ that is consistent with the expected limiting behaviour when $r\gg {\it\delta}$ , $\langle {\rm\Delta}u_{+}^{2}\rangle =2\langle u_{+}^{2}\rangle$ , with $D_{1}=2A_{1}$ .

2.5. Postulate for higher-order structure functions at scales $z<r\ll {\it\delta}$

Having considered the points summarised in the preceding sections, we now consider higher-order structure functions in the range $z<r\ll {\it\delta}$ (the $\ln (r/z)$ region). If in that range the variance of the velocity increment follows a logarithmic law as in (2.9), then following the same arguments as Meneveau & Marusic (Reference Meneveau and Marusic2013) for higher-order moments of velocity fluctuations, one can postulate the following scaling for higher-order structure functions:

(2.12) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle ^{1/p}=E_{p}+D_{p}\ln \frac{r}{z}.\end{eqnarray}$$

In particular, when $r={\it\delta}$ , one can show (see § 5.1 for more details) that this expression recovers the limiting behaviour consistent with (2.11) for the $z$ dependence of the logarithmic behaviour of high-order moments at $r\gg {\it\delta}$ . Further motivation for expecting a logarithmic behaviour of the high-order structure functions can be provided by model calculations based on the attached-eddy hypothesis. As shown in § 5.2, the spatial structure that arises from a superposition of attached eddies can be used to compute velocity increments, and their moments in that range lead directly to logarithmic scaling.

However, first, in the next sections we present empirical evidence for (2.12) based on the experimental data. The analysis of data is presented in § 4.2. Further derivations and the modelling of higher-order structure functions based on the attached-eddy hypothesis are subsequently presented in § 5.

4.1. Scaling of the second-order structure function and influence of the wall-normal location

It is valuable to consider how the second-order structure function varies with both the wall-normal position and the spatial separation, and an attempt to illustrate this is shown in figure 2. Here, a three-dimensional view of $\langle {\rm\Delta}u_{+}^{2}\rangle$ is presented at all measurement locations across the boundary layer height for the hot-wire dataset from Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) at $Re_{{\it\tau}}=19\,000$ with 50 wall-normal locations. The region highlighted in green on the surface formed from these profiles denotes the inertial scaling region where a $\text{ln}(r/z)$ law is observed. Further, the presented three-dimensional surface provides a visual representation of the relationship between the linear–log region observed in $2\langle u_{+}^{2}\rangle$ (● symbols) and the $\text{ln}(r/z)$ law in $\langle {\rm\Delta}u_{+}^{2}\rangle$ with the same multiplicative constant $A_{1}$ (for $p=1$ ; (2.11)).

Figure 2. Three-dimensional surface generated using profiles of $\langle {\rm\Delta}u_{+}^{2}\rangle$ versus $r/z$ at all wall-normal locations from the hot-wire dataset of Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) at $Re_{{\it\tau}}\approx 19\,000$ . The ● symbols correspond to $2\overline{u^{2}}$ at each wall-normal location. The green shaded region corresponds to where a $\ln (r/z)$ law is expected at wall-normal locations within the logarithmic region, highlighted by the darker contours. The indicated slopes are detailed further in § 6.

Figure 3. (a) A plot of $\langle {\rm\Delta}u_{+}^{2}\rangle$ versus $r/z$ from hot-wire data at $Re_{{\it\tau}}=19\,000$ of Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009). Wall-normal locations: ▵, $z^{+}\approx 400$ ; ▹, 700; ▫, 1000; ◃, 1900. The locations are selected to be within the logarithmic region. (b) A plot of $\langle {\rm\Delta}u_{+}^{2}\rangle$ versus $r/z$ after applying the correction outlined in (2.10) following Davidson et al. (Reference Davidson, Krogstad and Nickels2006a ). The solid lines correspond to formulations based on (2.9) and (2.10) in (a) and (b) respectively.

In the following we will focus primarily on data from within the logarithmic region, which we take to be nominally within the range $3\sqrt{Re_{{\it\tau}}}\lesssim z^{+}\lesssim 0.15Re_{{\it\tau}}$ following recent observations using high Reynolds number datasets (Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). The two thicker, dark grey lines (red online) shown in figure 2 demarcate this region. The results in figure 2 indicate good support for (2.9), where $z$ is used as the normalising length scale. It is worthwhile to consider whether the refined formulation (2.10) of Davidson et al. (Reference Davidson, Krogstad and Nickels2006a ) is required at higher Reynolds numbers. Figure 3 shows a comparison between formulations (2.9) and (2.10) for results within the logarithmic region for the same data as in figure 2. The results show that inclusion of the extra term involving production and dissipation slightly improves the collapse of the data in the inertial range ( $z<r\ll {\it\delta}$ ), although its effect is probably not required at this Reynolds number. This is primarily due to the fact that the turbulent production and dissipation rates are nominally equal in the logarithmic region at sufficiently high Reynolds numbers (Tennekes & Lumley Reference Tennekes and Lumley1972; Townsend Reference Townsend1976). Moreover, accurate estimation of the turbulent production and dissipation rates is a challenge experimentally. Here, we estimate production using $\mathscr{P}\approx 1/{\it\kappa}z^{+}$ , while for the dissipation rate several estimates can be employed and are summarised in table 2 for the hot-wire dataset at $Re_{{\it\tau}}\approx 19\,000$ (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009). All the estimates use the assumption of isotropy (see Pope (Reference Pope2000) for a detailed comparison); however, the third method (column 3) based on the third-order structure function has been reported to have less ambiguity (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997). The estimates of ${\it\epsilon}$ show an uncertainty of ${\approx}\!\pm 20\,\%$ between the methods used, comparable with prior observations by Kunkel & Marusic (Reference Kunkel and Marusic2006) and others. It is also worth highlighting that recent work particularly using numerical databases (where one can accurately compute $\mathscr{P}$ and ${\it\epsilon}$ ) has aimed to answer the validity of the balance between $\mathscr{P}$ and ${\it\epsilon}$ in the inertial region of wall turbulence (Hoyas & Jiménez Reference Hoyas and Jiménez2006; Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lee & Moser Reference Lee and Moser2015, and others). The results have largely shown support for the validity of an approximate balance, thus matching prior experimental observations.

Table 2. Dissipation rates ${\it\epsilon}$ (given in $\text{m}^{2}~\text{s}^{-3}$ ) computed from the hot-wire dataset of Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) at $Re_{{\it\tau}}\approx 19\,000$ , using three different methods.

In figure 4 we consider data in the logarithmic region across the other datasets at varying Reynolds numbers. Results are presented on both log–log and log–linear scales to highlight the scaling described previously in § 2. For the inertial subrange shown in figure 4(a) evidence of a power-law scaling given by (2.8) is prevalent in all datasets with a scaling exponent close to $2/3$ . Similarly, a power-law scaling given by (2.4) is observed at the smallest scales within the dissipative range with a scaling exponent of 2. Further, a $\text{ln}(r/z)$ law in the form given by (2.9) is observed at scales larger than the inertial subrange in figure 4(b). All scaling regimes are indicated by the solid lines, together with their corresponding coefficients. The low Reynolds number dataset (▫ symbols) is included to demonstrate a clear trend with changing Reynolds number, and it emphasises the importance of higher Reynolds number data ( $Re_{{\it\tau}}=O(10^{4}{-}10^{5})$ ) to obtain a sufficiently large scale separation and thus observe all scaling behaviours simultaneously. In future efforts, it will also be valuable to consider whether the scaling behaviour reported extends to the transverse structure function, which has been examined previously by Kurien et al. (Reference Kurien, Lvov, Procaccia and Sreenivasan2000), Jacob et al. (Reference Jacob, Biferale, Iuso and Casciola2004) and others. As a final note, certain studies have reported power-law behaviour for structure functions (Benzi et al. Reference Benzi, Amati, Casciola, Toschi and Piva1999; Toschi et al. Reference Toschi, Amati, Succi, Benzi and Piva1999; Jacob et al. Reference Jacob, Biferale, Iuso and Casciola2004, and others), albeit with particular focus on the smaller scales. Our analysis shows clear evidence of a power law in the range ${\it\eta}\ll r\ll z$ and a $\ln (r/z)$ law in the range $z<r\ll {\it\delta}$ . Some of these differences in interpretation may be attributed to the substantially higher Reynolds numbers examined in this study compared with prior work. This provides a large scale separation (large logarithmic region) enabling us to better discern any scaling behaviour. Moreover, a large proportion of studies that reported power-law behaviour alone employed decompositions primarily due to the lack of any scaling behaviour for the structure function in its classical form at the Reynolds numbers considered. Here, we present the structure function in its classical form, where the reported scaling behaviour is clearly observed.

Figure 4. Plots of $\langle {\rm\Delta}u_{+}^{2}\rangle$ versus $r/z$ on (a) log–log and (b) linear–log scales. The symbols represent different datasets and are defined in table 1. Wall-normal locations: ▫, $z^{+}\approx 200$ ; ♢, $z^{+}\approx 800$ ; ▵, $z^{+}\approx 1.6\times 10^{4}$ ; ○, ●, $z^{+}\approx 800$ . The solid lines correspond to fits (detailed further in § 6) in the ranges ${\it\eta}\ll r\ll z$ and $z<r\ll {\it\delta}$ . The heavy solid lines in (a) at the smallest $r/z$ represent fits to each hot-wire dataset in the dissipative range.

The good agreement between the PIV (denoted by ○, ● symbols) and hot-wire results in figure 4 highlights that the use of Taylor’s frozen turbulence hypothesis to convert the time-series information from the hot wires to spatial information is justified for the data presented here, at least up to $r<{\it\delta}$ . This concurs with previous findings where other statistical information is compared when Taylor’s approximation is invoked, such as in the study of Dennis & Nickels (Reference Dennis and Nickels2008). Other studies that have assessed the accuracy of invoking Taylor’s hypothesis include Del Alamo & Jiménez (Reference Del Alamo and Jiménez2009), Chung & McKeon (Reference Chung and McKeon2010) and Atkinson, Buchmann & Soria (Reference Atkinson, Buchmann and Soria2014).

4.2. Higher-order structure functions

Figure 5. Three-dimensional surface generated using profiles of $\langle {\rm\Delta}u_{+}^{8}\rangle ^{1/4}$ versus $r/z$ from the same dataset as presented in figure 2. The symbols are defined in figure 2. The indicated slopes are detailed further in § 6.

Figure 6. Higher-order even structure functions for the same datasets as presented in figure 4: (a,b)  $2p=4$ ; (c,d)  $2p=6$ ; (e,f)  $2p=8$ ; (g,h)  $2p=10$ . The symbols, solid lines and wall positions are defined in table 1 and figure 4 respectively.

Previously, we described the analogous postulate for higher-order structure functions in (2.12), extending prior work by Meneveau & Marusic (Reference Meneveau and Marusic2013) for higher-order moments of velocity fluctuations. This section considers empirical evidence for (2.12) based on the experimental data.

To begin, we apply the format used in figure 2 for the eighth-order structure function, which is shown in figure 5. It is immediately evident from figure 5 that the behaviour of the eighth-order structure function (after taking the $1/4$ root) is qualitatively comparable with its second-order counterpart. The region highlighted in green on the surface denotes the region where a $\text{ln}(r/z)$ law seems to be prevalent, thus providing empirical support for (2.12). To further validate this observation, figure 6 presents results for the even higher-order moments at a single wall-normal location within the logarithmic region across all the experimental datasets considered, here shown up to $2p=10$ . To accentuate the analogous scaling behaviour observed in the second-order structure function (see figure 4), the solid lines from $2p=4{-}10$ in figure 6 represent power- and log-law fits to the HRNBLWT hot-wire dataset. The corresponding coefficients are also indicated in the figure. The universality of these coefficients for the datasets considered will be explored in § 6. The low $Re$ dataset at higher moments shows a deviation from the high $Re$ datasets. This can be attributed to a much smaller logarithmic layer at low $Re$ . Similar $Re$ dependence was also observed by Meneveau & Marusic (Reference Meneveau and Marusic2013) for higher-order moments of the streamwise velocity fluctuations below $Re_{{\it\tau}}<7000$ . Our results from structure functions of higher orders are consistent with these findings; that is, beyond this threshold, the higher-order moments are independent of $Re$ . Therefore, we omit the low $Re$ dataset in the subsequent analyses, particularly when computing the coefficients associated with the scaling fits described in § 5.

Figure 7 shows the higher-order results for the data shown in figure 3(a) for various wall-normal positions throughout the logarithmic region. The results show good collapse of the higher moments up to $2p=10$ and provide further direct support for (2.12) across the entire extent of the logarithmic region.

Figure 7. Higher-order even structure functions for the same data as presented in figure 3 for various wall-normal positions throughout the logarithmic region for $Re_{{\it\tau}}=19\,000$ : (a,b)  $2p=6$ ; (c,d)  $2p=10$ . The symbols, solid lines and wall positions are defined in table 1 and figure 3 respectively.

5.1. Relations between high-order moments of velocity fluctuations and structure functions

Previously, § 2.4 summarised the results from Meneveau & Marusic (Reference Meneveau and Marusic2013) for the general behaviour of higher-order velocity fluctuations. Following their work, it is fruitful to develop the limit of structure functions at $r\gg {\it\delta}$ , by making use of the binomial theorem:

(5.1) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle =\mathop{\sum }_{k=0}^{2p}\left(\begin{array}{@{}c@{}}2p\\ k\end{array}\right)(-1)^{k}\langle u_{+}(x+r)^{2p-k}u_{+}(x)^{k}\rangle .\end{eqnarray}$$

When $r\gg {\it\delta}$ , we expect $u_{+}(x+r)$ and $u_{+}(x)$ to become statistically independent, and hence $\langle u_{+}(x+r)^{a}u_{+}(x)^{b}\rangle =\langle u_{+}(x)^{a}\rangle \langle u_{+}(x)^{b}\rangle$ . Therefore,

(5.2) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle \rightarrow \mathop{\sum }_{k=0}^{2p}\left(\begin{array}{@{}c@{}}2p\\ k\end{array}\right)(-1)^{k}\langle u_{+}^{2p-k}\rangle \langle u_{+}^{k}\rangle .\end{eqnarray}$$

If we assume that the odd moments all vanish, then this is equivalent to only adding the positive even terms:

(5.3) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle \rightarrow \mathop{\sum }_{k=0}^{p}\frac{(2p)!}{(2k)!(2p-2k)!}\langle u_{+}^{2p-2k}\rangle \langle u_{+}^{2k}\rangle .\end{eqnarray}$$

For $p=1$ , this becomes the familiar $\langle [u_{+}(\boldsymbol{x}+\boldsymbol{i}r)-u_{+}(\boldsymbol{x})]^{2}\rangle \rightarrow 2\langle u_{+}^{2}\rangle$ . For $p=2$ , the relationship is $\langle [u_{+}(\boldsymbol{x}+\boldsymbol{i}r)-u_{+}(\boldsymbol{x})]^{4}\rangle \rightarrow 2\langle u_{+}^{4}\rangle +6\langle u_{+}^{2}\rangle ^{2}$ . Using this in the context of the generalised logarithmic laws, for even moments only, we expect

(5.4) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle \rightarrow \mathop{\sum }_{k=0}^{p}\left(\begin{array}{@{}c@{}}2p\\ 2k\end{array}\right)\left(B_{p-k}-A_{p-k}\ln (z/{\it\delta})\right)^{p-k}\left(B_{k}-A_{k}\ln (z/{\it\delta})\right)^{k}.\end{eqnarray}$$

To simplify, it is useful to absorb the coefficient $B_{k}$ inside the logarithm, by setting $c_{k}=\exp (B_{k}/A_{k})$ , then we obtain

(5.5) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle \rightarrow \mathop{\sum }_{k=0}^{p}\frac{(2p)!}{(2k)!(2p-2k)!}\left[A_{p-k}\ln (c_{p-k}{\it\delta}/z)\right]^{p-k}\left[A_{k}\ln (c_{k}{\it\delta}/z)\right]^{k}.\end{eqnarray}$$

Next, we make the reasonable assumption that $B_{p}$ (as well as $c_{p}$ ) does not depend much on the moment order (e.g. that the intercepts of all moment log-laws occur roughly at the same outer length scale $c_{p}{\it\delta}\approx c{\it\delta}$ , and based on the measurements of $B_{1}\approx 1.7$ and $A_{1}\approx 1.25$ we may estimate $c\approx 4$ ); this seems to be supported by the data. We can then write

(5.6) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle \rightarrow \left[\ln (c{\it\delta}/z)\right]^{p}\mathop{\sum }_{k=0}^{p}\frac{(2p)!}{(2k)!(2p-2k)!}A_{p-k}^{p-k}A_{k}^{k},\end{eqnarray}$$

and we obtain a logarithmic dependence of the structure function asymptote at $r\gg {\it\delta}$ :

(5.7) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle ^{1/p}\rightarrow G_{p}\ln \!\left(\frac{c{\it\delta}}{z}\right),\end{eqnarray}$$

where

(5.8) $$\begin{eqnarray}G_{p}=\left(\mathop{\sum }_{k=0}^{p}\frac{(2p)!}{(2k)!(2p-2k)!}A_{p-k}^{p-k}A_{k}^{k}\right)^{1/p}.\end{eqnarray}$$

Given numeric values for $A_{p}$ for $p=1{-}5$ measured in Meneveau & Marusic (Reference Meneveau and Marusic2013), numerical values for $G_{p}$ can be obtained. For instance, for second-order structure functions, $p=1$ , we obtain

(5.9) $$\begin{eqnarray}G_{1}=2A_{1}\approx 2\times 1.25\approx 2.5\end{eqnarray}$$

and for fourth-order ones, $p=2$ , we have

(5.10) $$\begin{eqnarray}G_{2}=[2(A_{2}^{2}+3A_{1}^{2})]^{1/2}\approx [2(1.92^{2}+3\times 1.25^{2})]^{1/2}\approx 4.09.\end{eqnarray}$$

A comparison of these coefficients against experimental results will be presented in § 6.

5.2. The attached-eddy hypothesis and higher-order moments

In this section, we compare the observed statistical behaviour of the structure functions with that predicted by modelling the flow from elementary physical principles. To this end, we derive various properties of the structure functions in the logarithmic region from the attached-eddy hypothesis. This hypothesis states that the logarithmic region is dominated and characterised by the presence of a hierarchy of self-similar coherent structures, or ‘eddies’, whose sizes scale with their distances from the wall. It is in this loose sense that the eddies are said to be ‘attached’ to the wall. This will enable us to explore the validity of the current manifestation of the attached-eddy hypothesis as a potential predictive model for structure functions in the logarithmic region of turbulent wall flows.

The attached-eddy hypothesis, as it is applied here, assumes that the locations of the eddies on the wall are perfectly random and independent of each other. This has allowed the statistical properties of the flow to be derived via Campbell’s theorem. This model has been used by Woodcock & Marusic (Reference Woodcock and Marusic2015) to derive the moments of the velocity in the logarithmic region. The model treats the flow as a superposition of the velocity fields corresponding to each of a multitude of geometrically equivalent eddies. The locations of the eddies are perfectly random and are unaffected by the presence of nearby eddies. (This means that the eddies are not prevented from overlapping, and two or more eddies may occupy the same region of space.) The eddies are identical once scaled by their heights, and their shapes are unaffected by the presence of nearby eddies. The reasoning behind the attached-eddy hypothesis, as well as the derivation extended here, has been further elaborated upon by Woodcock & Marusic (Reference Woodcock and Marusic2015).

The height of any particular eddy, $h$ , is assumed to be between $h_{min}$ and $h_{max}$ . We can therefore equate $h_{max}$ with ${\it\delta}$ , the thickness of the boundary layer. In order to simplify the subsequent equations, we will introduce some new symbols. Since each eddy is characterised by its height, it is sensible to scale all lengths by $h$ . Hence, we scale the location vector, $\boldsymbol{x}$ , and the displacement via

(5.11) $$\begin{eqnarray}\boldsymbol{X}\equiv \left(X,Y,Z\right)=\frac{\boldsymbol{x}}{h},\quad R=\frac{r}{h}.\end{eqnarray}$$

When deriving the velocity moments, Campbell’s theorem reduces the derivation to an integral over the velocity field corresponding to a single eddy. Hence, if the vector $\boldsymbol{Q}(\boldsymbol{X})$ represents the velocity field corresponding to a single eddy, then we can express all moments of the velocity in terms of $\boldsymbol{Q}(\boldsymbol{X})$ .

Since each eddy is spatially bounded, the velocity field corresponding to each eddy will inevitably be negligible at a sufficient distance from the location of the eddy, and this distance is proportional to the height of the eddy. Using this, Woodcock & Marusic (Reference Woodcock and Marusic2015) showed that the log-law for the flow profile follows from Campbell’s theorem.

When applied to the derivation of the structure function, Campbell’s theorem reduces the derivation to an integral over a pair of eddies identical in magnitude but of opposite sign, a distance of $r$ apart. We therefore introduce a new function, ${\rm\Delta}I_{k}\left(\boldsymbol{X},R\right)$ , which we call the eddy difference contribution. It relates the flow field corresponding to a single eddy via

(5.12) $$\begin{eqnarray}{\rm\Delta}I_{k}(\boldsymbol{X},R)=\iint _{-\infty }^{\infty }\left[Q_{x}\left(\boldsymbol{X}+\boldsymbol{i}R\right)-Q_{x}\left(\boldsymbol{X}\right)\right]^{k}\text{d}X\text{d}Y.\end{eqnarray}$$

This we can relate to $\langle {\rm\Delta}u_{+}^{n}\rangle$ through a mathematical intermediary, which we call a cumulant. These cumulants, denoted by ${\it\Lambda}_{k}(\boldsymbol{x},r)$ , are defined by

(5.13) $$\begin{eqnarray}{\it\Lambda}_{k}(z,r)={\it\beta}\int _{h_{min}}^{h_{max}}{\rm\Delta}I_{k}\left(Z,R\right)h^{2}P(h)\text{d}h,\end{eqnarray}$$

where ${\it\beta}$ denotes the density of the eddies on the wall and $P(h)$ is the probability density function for the height of an eddy. Woodcock & Marusic (Reference Woodcock and Marusic2015) have shown that this is given by

(5.14) $$\begin{eqnarray}P(h)=2\left(h_{min}^{-2}-h_{max}^{-2}\right)^{-1}\frac{1}{h^{3}}.\end{eqnarray}$$

The structure functions can be shown to relate to the cumulants via

(5.15a ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{2}\rangle & = & \displaystyle {\it\Lambda}_{2},\end{eqnarray}$$

(5.15b ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{3}\rangle & = & \displaystyle {\it\Lambda}_{3},\end{eqnarray}$$

(5.15c ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{4}\rangle & = & \displaystyle {\it\Lambda}_{4}+3{\it\Lambda}_{2}^{2},\end{eqnarray}$$

(5.15d ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{5}\rangle & = & \displaystyle {\it\Lambda}_{5}+10{\it\Lambda}_{2}{\it\Lambda}_{3},\end{eqnarray}$$

(5.15e ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{6}\rangle & = & \displaystyle {\it\Lambda}_{6}+15{\it\Lambda}_{2}{\it\Lambda}_{4}+10{\it\Lambda}_{3}^{2}+15{\it\Lambda}_{2}^{3},\end{eqnarray}$$

(5.15f ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{7}\rangle & = & \displaystyle {\it\Lambda}_{7}+21{\it\Lambda}_{2}{\it\Lambda}_{5}+35{\it\Lambda}_{3}{\it\Lambda}_{4}+105{\it\Lambda}_{2}^{2}{\it\Lambda}_{3},\end{eqnarray}$$

(5.15g ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{8}\rangle & = & \displaystyle {\it\Lambda}_{8}+28{\it\Lambda}_{2}{\it\Lambda}_{6}+56{\it\Lambda}_{3}{\it\Lambda}_{5}+35{\it\Lambda}_{4}^{2}+210{\it\Lambda}_{2}^{2}{\it\Lambda}_{4}\nonumber\\ \displaystyle & & \displaystyle +\,280{\it\Lambda}_{2}{\it\Lambda}_{3}^{2}+105{\it\Lambda}_{2}^{4},\end{eqnarray}$$

(5.15h ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{9}\rangle & = & \displaystyle {\it\Lambda}_{9}+36{\it\Lambda}_{2}{\it\Lambda}_{7}+84{\it\Lambda}_{3}{\it\Lambda}_{6}+126{\it\Lambda}_{4}{\it\Lambda}_{5}+378{\it\Lambda}_{2}^{2}{\it\Lambda}_{5}+1260{\it\Lambda}_{2}{\it\Lambda}_{3}{\it\Lambda}_{4}\nonumber\\ \displaystyle & & \displaystyle +\,280{\it\Lambda}_{3}^{3}+1260{\it\Lambda}_{2}^{3}{\it\Lambda}_{3},\end{eqnarray}$$

(5.15i ) $$\begin{eqnarray}\displaystyle \langle {\rm\Delta}u_{+}^{10}\rangle & = & \displaystyle {\it\Lambda}_{10}+45{\it\Lambda}_{2}{\it\Lambda}_{8}+120{\it\Lambda}_{3}{\it\Lambda}_{7}+210{\it\Lambda}_{4}{\it\Lambda}_{6}+630{\it\Lambda}_{2}^{2}{\it\Lambda}_{6}+126{\it\Lambda}_{5}^{2}\nonumber\\ \displaystyle & & \displaystyle +\,2520{\it\Lambda}_{2}{\it\Lambda}_{3}{\it\Lambda}_{5}+1575{\it\Lambda}_{2}{\it\Lambda}_{4}^{2}+2100{\it\Lambda}_{3}^{2}{\it\Lambda}_{4}+3150{\it\Lambda}_{2}^{3}{\it\Lambda}_{4}\nonumber\\ \displaystyle & & \displaystyle +\,6300{\it\Lambda}_{2}^{2}{\it\Lambda}_{3}^{2}+945{\it\Lambda}_{2}^{5}.\end{eqnarray}$$

The derivation of the structure functions is analogous to that presented by Woodcock & Marusic (Reference Woodcock and Marusic2015) for the moments of the velocity. The details of this derivation can be found in appendix A. These derivations apply where

(5.16) $$\begin{eqnarray}z\ll r\ll {\it\delta}.\end{eqnarray}$$

(It should be noted that we have specifically made use of the fact that we equate $h_{max}$ with the height of the boundary layer, ${\it\delta}$ in the above equation.) In this limit, the even-ordered cumulants will obey

(5.17) $$\begin{eqnarray}{\it\Lambda}_{k}(z,r)=\mathscr{A}_{k}\log \frac{r}{z}+\mathscr{B}_{k},\end{eqnarray}$$

where $\mathscr{A}_{k}$ and $\mathscr{B}_{k}$ are constants. The constant $\mathscr{A}_{k}$ is given by

(5.18) $$\begin{eqnarray}\mathscr{A}_{k}=\frac{4{\it\beta}}{h_{min}^{-2}-h_{max}^{-2}}\iint _{-\infty }^{\infty }Q_{x}^{k}(X,Y,0)\text{d}X\text{d}Y.\end{eqnarray}$$

(We could give a similar expression for $\mathscr{B}_{k}$ . However, since the attached-eddy model contains different boundary conditions from real turbulent flows, any $\mathscr{B}_{k}$ derived in this way would not be physically significant.)

We can express the range of eddy sizes present as a Reynolds number. This Reynolds number can be conventionally defined by

(5.19) $$\begin{eqnarray}Re_{{\it\tau}}=100\frac{h_{max}}{h_{min}},\end{eqnarray}$$

which is based on the assumption that $h_{max}={\it\delta}$ and $h_{min}=100$ following Kline et al. (Reference Kline, Reynolds, Schraub and Runstadler1967). By an analogous derivation to that given by Woodcock & Marusic (Reference Woodcock and Marusic2015), it can be shown that (5.18) can be re-expressed as

(5.20) $$\begin{eqnarray}\mathscr{A}_{k}={\displaystyle \frac{16}{9}}{\displaystyle \frac{1}{k_{x}k_{y}}}{\displaystyle \frac{\left(1-10^{6}Re_{{\it\tau}}^{-3}\right)^{2}}{\left(1-10^{4}Re_{{\it\tau}}^{-2}\right)^{3}}}\iint _{-\infty }^{\infty }Q_{x}^{k}(X,Y,0)\text{d}X\text{d}Y,\end{eqnarray}$$

where $k_{x}$ is a constant defined such that the expected distance to the next closest eddy in the positive $x$ direction will always be $k_{x}h$ if the height of the next closest eddy were known to be $h$ . (More specifically, $k_{x}h$ represents the distance in a strip of height $h^{\prime }$ to the nearest eddy of height between $h$ and $h+\text{d}h$ divided by $h^{\prime }$ and $\text{d}h$ .) The constant $k_{y}$ is its spanwise equivalent.

In the large- $Re_{{\it\tau}}$ limit, this asymptotes to

(5.21) $$\begin{eqnarray}\mathscr{A}_{k}\rightarrow {\displaystyle \frac{16}{9}}{\displaystyle \frac{1}{k_{x}k_{y}}}\left(1+3\times 10^{4}Re_{{\it\tau}}^{-2}\right)\iint _{-\infty }^{\infty }Q_{x}^{k}(X,Y,0)\text{d}X\text{d}Y\quad \text{as }Re_{{\it\tau}}\rightarrow \infty .\end{eqnarray}$$

As we can easily see in (5.15), ${\it\Lambda}_{2}$ is equivalent to the second-order structure function. We can therefore return (2.9), the logarithmic law for $\langle {\rm\Delta}u_{+}^{2}\rangle$ , by stating that $\mathscr{A}_{2}\equiv D_{1}$ and $\mathscr{B}_{2}\equiv E_{1}$ .

It can also be observed that by considering (5.13) the higher even-ordered structure functions will be dominated by their final term (which is the component containing the highest power of ${\it\Lambda}_{2}$ ) in the limit as ${\it\beta}\rightarrow \infty$ . Given the relationship between ${\it\Lambda}_{2}$ and the second-order structure function, we can easily see that (5.17) returns a logarithmic law for the second-order structure function given in (2.9). Furthermore, for large ${\it\beta}$ ( ${\it\beta}\rightarrow \infty$ ), it can be seen that the even-numbered higher-ordered structure functions will obey

(5.22) $$\begin{eqnarray}\langle {\rm\Delta}u_{+}^{2p}\rangle ^{1/p}=D_{p}\log \frac{r}{z}+E_{p},\end{eqnarray}$$

where $D_{p}$ and $E_{p}$ are constants which are equivalent to $\mathscr{A}_{2p}$ and $\mathscr{B}_{2p}$ respectively. Importantly, this concurs with experimental results, as will be discussed in more detail in § 6. We also note that carrying out the above derivations for the other velocity components shows a similar logarithmic relation for the spanwise velocity, but not for the wall-normal velocity.

The odd-ordered cumulants have also been derived in appendix A. Their behaviour is given in (A 25). They are likely to be of much lower magnitude than their even-ordered counterparts wherever (5.16) applies, so their contributions to the even-ordered structure functions should be negligible. Their contributions to the odd-ordered structure functions are more significant, but since these are of less interest, we do not reproduce them here.

Therefore, we can see that the attached-eddy hypothesis produces predictions for the structure functions in the logarithmic region that are in qualitative agreement with experimental evidence. In order to extend this to a quantitative prediction for the structure functions, it would be necessary to produce a specific function for $\boldsymbol{Q}(\boldsymbol{x})$ , the velocity field corresponding to a single eddy. This is, however, beyond the scope of this present work.

Figure 8. Plots of $\langle {\rm\Delta}u_{+}^{2p}\rangle ^{1/p}(r/z)^{{\it\xi}_{2p}/p}$ versus $r/z$ on a linear–log scale at $p=1{-}5$ . The symbols represent different datasets: ●, HRNBLWT – high-mag PIV; ♢, HRNBLWT – hot-wire; ▵, SLTEST – hot-wire. Wall-normal locations: ♢,  $z^{+}\approx 800$ ; ▵,  $z^{+}\approx 1.6\times 10^{4}$ ; ●,  $z^{+}\approx 800$ . The horizontal dashed line indicates the computed multiplicative constant $M_{p}$ in the inertial subrange ${\it\eta}\ll r\ll z$ based on the SLTEST dataset (▵ symbols). The vertical dashed line corresponds to the transitional length scale $L_{s}$ .

6.1. Modelling of coefficients in the $\ln (r/z)$ range

So far we have examined the inertial subrange and $\ln (r/z)$ range of the streamwise structure function independently. However, the results shown in figures 4 and 6 suggest that the scaling fits employed in each region intersect at approximately $r/z\approx 1$ . This suggests the possibility of using this information to reduce the number of unknown constants. (Such matching approaches have been attempted previously, including in the more recent study by Davidson & Krogstad (Reference Davidson and Krogstad2009).) Therefore, using this purely empirical observation we equate the two fits and model/estimate the coefficients for the $\ln (r/z)$ range based on the scaling fits in the inertial subrange alone. To perform this, the two scaling fits given by (2.8) and (5.22) are equated to obtain

(6.3) $$\begin{eqnarray}M_{p}\mathscr{X}^{{\it\xi}_{2p}}=E_{p}^{\prime }+D_{p}^{\prime }\ln \mathscr{X},\end{eqnarray}$$

where $E_{p}^{\prime }$ and $D_{p}^{\prime }$ are modelled equivalents of $E_{p}$ and $D_{p}$ , and $\mathscr{X}$ corresponds to the magnitude of $r/z$ at the estimated point of intersection between the two scaling fits. By further matching gradients at $\mathscr{X}$ we obtain

(6.4) $$\begin{eqnarray}M_{p}{\it\xi}_{2p}\mathscr{X}^{{\it\xi}_{2p}-1}=\frac{D_{p}^{\prime }}{\mathscr{X}},\end{eqnarray}$$

which can be rearranged to give

(6.5) $$\begin{eqnarray}D_{p}^{\prime }=M_{p}{\it\xi}_{2p}\mathscr{X}^{{\it\xi}_{2p}}.\end{eqnarray}$$

Substituting (6.5) into (6.3) we can now estimate $E_{p}^{\prime }$ using

(6.6) $$\begin{eqnarray}E_{p}^{\prime }=M_{p}\ln \mathscr{X}^{{\it\xi}_{2p}}-M_{p}{\it\xi}_{2p}\mathscr{X}^{{\it\xi}_{2p}}\ln \mathscr{X}\rightarrow M_{p}\mathscr{X}^{{\it\xi}_{2p}}(1-{\it\xi}_{2p}\ln \mathscr{X}).\end{eqnarray}$$

Visual inspection of the results presented for $\langle {\rm\Delta}u_{+}^{2p}\rangle ^{1/p}$ in figures 4 and 6 shows that $\mathscr{X}\approx 1$ ; however, upon closer inspection a linear decrease in $\mathscr{X}$ is observed across all datasets, as shown in figure 11, with increasing $p$ . Therefore, following estimates of $\mathscr{X}$ from the experimental datasets, results for $E_{p}^{\prime }$ and $D_{p}^{\prime }$ are summarised in table 5 and are compared with prior experimental results in figure 12. The results show that the magnitude and trend of the modelled coefficients closely match those computed directly from the experimental datasets.

Figure 11. Variation of $\mathscr{X}$ computed from the SLTEST dataset of Kunkel & Marusic (Reference Kunkel and Marusic2006) at $Re_{{\it\tau}}=3.1\times 10^{6}$ and $z^{+}\approx 1.6\times 10^{4}$ . The solid line represents a linear fit which is used to obtain the results shown in figure 12.

Figure 12. The same as figure 10, but with the inclusion of the modelled coefficients (▿ symbols) for the SLTEST dataset of Kunkel & Marusic (Reference Kunkel and Marusic2006), as summarised in table 5.

Table 5. Comparison of the coefficients $D_{p}$ and $E_{p}$ in the region $z<r<{\it\delta}$ from different datasets. For the HRNBLWT – hot-wire datasets results are presented only at $Re_{{\it\tau}}\approx 19\,000$ .