2.1. The original model

We first revisit the original spectrum-based attached eddy model of Perry et al. (Reference Perry, Henbest and Chong1986), focusing on the streamwise velocity in a turbulent boundary layer, the thickness of which is given by $\delta$. The modelling efforts start with the streamwise turbulence intensity given in terms of the power-spectral density:

(2.1)\begin{equation} \overline{u'u'}(z)=\int_0^\infty \varPhi_{uu}(k_x,z)\,\mathrm{d}k_x, \end{equation}

where $\varPhi _{uu}(k_x,z)$ is the power-spectral density of streamwise velocity at each wall-normal location $z$, and $k_x$ is the streamwise wavenumber. As in Townsend (Reference Townsend1976), only the flow above the thin viscous sublayer was considered, assuming that the flow is inviscid with finite slip velocity at the boundary. Under this assumption, the power-spectral density $\varPhi _{uu}$ is expected to be a function of $u_\tau$, $k_x$, $z$ and $\delta$. Further to this, $z\ll \delta$ was assumed, given the wall-normal location of interest (i.e. the logarithmic layer).

Under the two assumptions (i.e. $Re_\tau \rightarrow \infty$ and $z\ll \delta$), the model yields a power-spectral density of streamwise velocity schematically depicted in figure 1. At a given wall-normal location $z\ll \delta$, energy-containing eddies of outer scale would contribute to the wavenumbers of $k_x \sim O(1/\delta )$ through their inactive component (Perry & Abel Reference Perry and Abel1977; Perry et al. Reference Perry, Henbest and Chong1986). In this range of wavenumbers, the assumption of $z\ll \delta$ implies that the power-spectral density can further be assumed to be independent of $z/\delta$. Using (2.1), the power-spectral density for $k_x \sim O(1/\delta )$ is then given by

(2.2a)\begin{equation} \frac{\varPhi_{uu}(k_x,z)}{ u_\tau^2}=\frac{\delta\,\varPhi_{uu}(k_x \delta)}{u_\tau^2}= \delta\, g_1(k_x \delta). \end{equation}

The premultiplied power-spectral density is subsequently written as

(2.2b)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2}=k_x \delta\,g_1(k_x \delta)=h_1(k_x \delta), \end{equation}

as sketched in the $\delta$-scaling region of figure 1. For the wavenumbers of $k_x \sim O(1/z)$, the power-spectral density would not be a function of $\delta$, as only eddies scaling in $z$ would contribute to these wavenumbers. Therefore, at $z\ll \delta$, the power-spectral density for $k_x \sim O(1/z)$ becomes

(2.3a)\begin{equation} \frac{\varPhi_{uu}(k_x,z)}{u_\tau^2}=\frac{z\,\varPhi_{uu}(k_x z)}{u_\tau^2}=z\,g_2(k_x z), \end{equation}

and the corresponding premultiplied power-spectral density is

(2.3b)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2}=k_x z\,g_2(k_x z)=h_2(k_x z). \end{equation}

This is sketched in the $z$-scaling region in figure 1. The power spectrum for very high $k_x$ should obviously be related to the energy cascade (the Kolmogrov scaling region in figure 1). Therefore it follows the scaling of inertial subrange ($k_x^{-5/3}$ spectrum) and the dissipation at the Kolmogorov length scale $\eta$ (Perry et al. Reference Perry, Henbest and Chong1986). The details of this part of the spectrum will not, however, be pursued here because its contribution to turbulence intensity would be small.

Figure 1. A schematic diagram of the streamwise velocity spectrum for the attached eddy model of Perry et al. (Reference Perry, Henbest and Chong1986). Here, the overlap region (i.e. inertial subrange) between $z$-scaling and dissipation (Kolmogorov) scaling and the dissipation scaling region are merged into a single region for simplicity (see text). Note that this is only a conceptual sketch introduced to explain the model of Perry et al. (Reference Perry, Henbest and Chong1986). Therefore it does not necessarily reflect the experimentally measured spectrum of streamwise velocity, especially when the Reynolds number is not sufficiently high.

We now consider the wavenumber region of $1/\delta \ll k_x \ll 1/z$ (the overlap region in figure 1). In this region, it was argued that both of the scalings in (2.2b) and (2.3b) would simultaneously be valid. Here, note that $h_1$ in (2.2b) is a function of only $k_x \delta$, while $h_2$ in (2.3b) is a function of $k_x z$. Therefore matching between (2.2b) and (2.3b) leads to

(2.4)\begin{equation} h_1(k_x z)=h_2(k_x \delta)=A, \end{equation}

where $A$ is a constant independent of both $k_x z$ and $k_x \delta$, resulting in the following power spectrum:

(2.5)\begin{equation} \frac{\varPhi_{uu}(k_x,z)}{u_\tau^2}=\frac{A}{k_x}. \end{equation}

Using (2.1), (2.2b), (2.3b) and (2.5), the streamwise turbulence intensity, which is the area below the curve for the power-spectral density in figure 1, is obtained as

(2.6a)$$\begin{align} \frac{\overline{u^\prime u^\prime}}{u_\tau^2} &= \int_{-\infty}^\infty \frac{k_x\, \varPhi_{uu}(k_x;z)}{u_\tau^2}\,\mathrm{d}(\ln k_x) = \int_{\ln (a/\delta)}^{\ln (b/z)} \frac{k_x\, \varPhi_{uu}(k_x)}{u_\tau^2}\,\mathrm{d}(\ln k_x)+C(z;Re_\tau)\nonumber\\ &={-}A\ln\left(\frac{z}{\delta}\right)+B(z;Re_\tau), \end{align}$$

where $a$ and $b$ are the constants defining the wavenumber boundaries of the $k_x^{-1}$ region (blue-shaded region in figure 1) and

(2.6b)\begin{equation} B(z;Re_\tau)=A\ln \left(\frac{a}{b}\right)+C(z;Re_\tau), \end{equation}

(2.6c)\begin{equation} C\left(z;Re_\tau\right)=\frac{\overline{u^\prime u^\prime}}{u_\tau^2}-\int_{\ln (a/\delta)}^{\ln(b/z)} \frac{k_x\, \varPhi_{uu}(k_x)}{u_\tau^2}\,\mathrm{d}(\ln k_x). \end{equation}

Here, $C(z;Re_\tau )$ depicts the contribution from the remainder of the latter integral in (2.6a), and it should be a weak function of $z$ and $Re_\tau$ due to the small contribution made from the Kolmogorov-scaling part of the spectrum. (Note that under the original assumptions of Perry et al. (Reference Perry, Henbest and Chong1986), the contribution from the outer-scaling part for $k_x \leq a/\delta$ to $C(z;Re_\tau )$ does not depend on $z$ and $Re_\tau$; see (2.2a).) If this contribution is ignored, then $B$ and $C$ become constants, leading (2.6a) to be identical to (1.1) from Townsend (Reference Townsend1976). We note that (1.1) in Townsend (Reference Townsend1976) was obtained by ignoring the contribution from small-scale eddies for energy cascade and dissipation. Therefore the two models by Townsend (Reference Townsend1976) and by Perry et al. (Reference Perry, Henbest and Chong1986), the former of which was built in physical space and the latter in spectral space, become consistent.

2.2. Scaling of streamwise velocity spectra

Now we examine the spectrum-based attached eddy model of Perry et al. (Reference Perry, Henbest and Chong1986) using the experimental data from Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). These data were taken from the high Reynolds number boundary layer wind tunnel located at the University of Melbourne. The boundary-layer thickness $\delta$ and friction velocity $u_\tau$ were estimated by fitting the measured experimental data to a composite law of the wall/wake curve in Chauhan, Monkewitz & Nagib (Reference Chauhan, Monkewitz and Nagib2010). Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) compared their estimates for $u_\tau$ from the composite fit to those measured directly by Baars et al. (Reference Baars, Squire, Talluru, Abbassi, Hutchins and Marusic2016) with a floating element drag balance, in the same facility, and matched $x$ and $U_\infty$ (free-stream velocity) to the two highest Reynolds numbers of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018), finding agreement to within $\pm 1$% The near-wall region is fully resolved using the nanoscale thermal anemometry probes (NSTAPs) (Vallikivi & Smits Reference Vallikivi and Smits2014). The Reynolds numbers considered are $Re_\tau =6123,10\,100,14\,680,19\,680$. For further details on the experiment, the reader may refer to Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018).

Figure 2 shows the contours of the premultiplied streamwise power-spectral density of streamwise velocity at $Re_\tau =19\,680$, in which the two red straight lines indicate $\lambda _x=z$ and $\lambda _x=10\delta$. We first define the conventional logarithmic layer with a relatively conservative limit: $z \in [z_{i},z_o]$ where $z_i^+=200$ and $z_o=0.1\delta$ (the superscript $^+$ denotes the normalisation with $\delta _\nu (=\nu /u_\tau )$ and $u_\tau$). As was proposed previously (Afzal Reference Afzal1982; Sreenivasan & Sahay Reference Sreenivasan and Sahay1997; Wei et al. Reference Wei, Fife, Klewicki and Mcmurtry2005; Klewicki Reference Klewicki2013), the logarithmic layer may be partitioned further into two sublayers with the boundary at $z_m^+=3.6 Re_\tau ^{1/2}$: i.e. layer I for $z \in [z_{m},z_o]$, and layer  II for $z \in [z_{i},z_m]$. Here, we note that the location of $z_m$ is a little below the empirical wall-normal location of the outer peak in the spectra ($z_m^+=3.9 Re_\tau ^{1/2}$) proposed by Mathis, Hutchins & Marusic (Reference Mathis, Hutchins and Marusic2009).

1. (i) Layer I (inertial sublayer): in this layer, the inertial effect would dominate the viscous wall effect. Given the inviscid-flow assumption of Perry et al. (Reference Perry, Henbest and Chong1986), this is the location where their model is supposed to be directly applicable. Indeed, the contour lines for $\lambda _x \gtrsim 10\delta$ in figure 2 change little along the $z$-direction and remain mostly parallel to $\lambda _x=10 \delta$, indicating that (2.2b) would be a good approximation for $\lambda _x \gtrsim 10\delta$. Similarly, the contour lines for $10^{-2} \delta \leq \lambda _x \leq \delta$ are approximately parallel to $\lambda _x=z$, consistent with (2.3b). These scaling behaviours are more precisely confirmed in figure 3 – the spectra in layer I follow the scaling of (2.2b) for $\lambda _x \gtrsim 10 \delta$ (figure 3$a$) and they do so with (2.3b) for $z \lesssim \lambda _x \lesssim 10 z$ (figure 3$b$). Despite the scaling behaviours being fully consistent with (2.2b) and (2.3b), the spectra in between ($10 z\leq \lambda _x \leq 10 \delta$) do not seem to exhibit a well-developed $k_x^{-1}$ behaviour. Furthermore, the values of the spectra vary non-negligibly with $z$, indicating that there is an issue in comparing (2.5) with the experimental data (e.g. Morrison et al. Reference Morrison, Jiang, McKeon and Smits2001, Reference Morrison, McKeon, Jiang and Smits2004; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). These issues will be discussed in depth in § 3.

2. (ii) Layer II (mesolayer): given the nature of the mean momentum balance in this layer (Afzal Reference Afzal1982; Sreenivasan & Sahay Reference Sreenivasan and Sahay1997; Wei et al. Reference Wei, Fife, Klewicki and Mcmurtry2005; Klewicki Reference Klewicki2013), the viscous wall effect would be more important than the inertial effect. In particular, the premultiplied power-spectral density for $\lambda _x \gtrsim 10 \delta$ appears to become weaker as $z\rightarrow 0$, and the related contour lines in figure 2 are not parallel to $\lambda _x=10 \delta$. On the contrary, the power-spectral density for $10^{-2} \delta \leq \lambda _x \leq 10^{-1} \delta$ still appears to follow (2.3b), as the contour lines in figure 2 are approximately parallel to $\lambda _x=z$ (see also figure 7). This is also confirmed in figure 3 – the spectra in layer II do not precisely follow the scaling of (2.2b) for $\lambda _x \gtrsim 10 \delta$ (figure 3$c$), while they show a behaviour consistent with (2.3b) for $z \lesssim \lambda _x \lesssim 10 z$ (figure 3$d$). These observations will be the basis of the model for layer II in § 4, obtained by extending the one in § 3.

Figure 2. Premultiplied streamwise power-spectral density of streamwise velocity ($k_x^+\, \varPhi _{uu}^+(z;k_x)$) at $Re_\tau \simeq 19\,680$ (data from Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). The contour lines are $k_x^+\,\varPhi _{uu}^+(z;k_x)=0.25, 0.5, 0.75,$ $1, 1.25, 1.5, 1.75, 2.0$. Here, the logarithmic layer from $z^+= 200$ to $z/\delta =0.1$ is divided into two sublayers: layer I (inertial sublayer) and layer  II (mesolayer). The two red solid lines indicate $\lambda _x=z$ and $\lambda _z=10\delta$, as labelled.

Figure 3. Premultiplied streamwise power-spectral density of streamwise velocity in $(a{,}b)$ layer  I and $(c{,}d)$ layer  II at $Re_\tau =19\,680$: $(a{,}c)$ $\delta$-scaling, and $(b{,}d)$ $z$-scaling. The arrows indicate the directions of increasing $z$.

We note that the classification of the logarithmic layer into layers I and  II in the present study is based on the streamwise velocity spectra, not on the mean velocity. Given the streamwise mean momentum equation for a turbulent boundary layer, the streamwise mean velocity is mostly related to the Reynolds shear stress like channel and pipe flows. However, the fluctuations of the streamwise velocity carry a substantial amount of motions that contain little Reynolds shear stress. Such motions have been referred to as ‘inactive’ motions (Townsend Reference Townsend1976), which appear to be particularly important in the region close to the wall (Hwang Reference Hwang2015, Reference Hwang2016; Cho, Hwang & Choi Reference Cho, Hwang and Choi2018; Baars & Marusic Reference Baars and Marusic2020a; Deshpande, Monty & Marusic Reference Deshpande, Monty and Marusic2020). These motions do not necessarily have strong effect on the mean velocity. Therefore the empirical scaling of $z_m^+(=3.6 Re_\tau ^{1/2})$ used here is not necessarily the same as the wall-normal location (say $z_R$) that has been used to partition the inertial sublayer and the mesolayer based on the Reynolds shear stress (i.e. $z_R$ is the location where the Reynolds shear stress is maximum). Indeed, an empirical value of $z_R^+$ is given by $z_R^+=1.9 Re_\tau ^{1/2}$ (Afzal & Yajnik Reference Afzal and Yajnik1973; Afzal Reference Afzal1976, Reference Afzal1982; Sreenivasan & Sahay Reference Sreenivasan and Sahay1997; Wei et al. Reference Wei, Fife, Klewicki and Mcmurtry2005; Jiménez & Moser Reference Jiménez and Moser2007; Klewicki Reference Klewicki2013), indicating that $z_m>z_R$. This implies that the Reynolds shear stress would slightly decrease with $z$ in layer I and that the peak Reynolds shear stress is located in layer II.

3.1. Spectrum at $1/\delta \ll k_{x} \ll 1/z$ for small and finite $z/\delta$

If $z/\delta$ is finite in the region of interest, then the power-spectral density for $k_x \sim O(1/\delta )$ and $k_x \sim O(1/z)$ would not strictly be independent of $z/\delta$. Such a behaviour is particularly pronounced in the region above layer I where $z/\delta$ is not small: indeed, the contour lines of the premultiplied spectra in figure 2 are not parallel to $\lambda _x=10\delta$ and $\lambda _x=z$. Therefore, without loss of generality, the premultiplied spectra in (2.2b) and (2.3b) should be replaced with

(3.1a)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2}=\frac{k_x \delta\,\varPhi_{uu}(k_x \delta ,z/\delta)}{u_\tau^2}=\tilde{h}_1\left(k_x \delta,\frac{z}{\delta}\right) \end{equation}

and

(3.1b)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2}=\frac{k_x z\,\varPhi_{uu}(k_x z,z/\delta)}{u_\tau^2}=\tilde{h}_2\left(k_x z,\frac{z}{\delta}\right), \end{equation}

respectively. Considering a $z/\delta$-dependence of the spectrum for $k_x \sim O(1/\delta )$ and $k_x \sim O(1/z)$ is physically more sound and offers more modelling flexibility. Indeed, (3.1a) would allow one to account for any wall-normal variation of outer-scaling structures such as superstructures/very-large-scale motions, while (3.1b) allows one to consider directly the small wall-normal variation in the $z$-scaling spectrum caused by the energy cascade.

Now (3.1a) and (3.1b) are considered for $1/\delta \ll k_{x} \ll 1/z$. For $1/\delta \ll k_{x} \ll 1/z$, the spectrum should satisfy $\tilde {h}_1(k_x \delta,{z/\delta })=\tilde {h}_2(k_x z,z/\delta )$. Here, $k_x z$ and $k_x \delta$ must be treated as two independent variables, because the two scaling laws in (3.1a) and (3.1b) from the Buckingham ${\rm \pi}$ theorem are introduced to cover different values of $k_x$ at all admissible small $z/\delta$ (Hinch Reference Hinch1991). While this may be a useful feature to proceed further to identify the form of spectrum for $1/\delta \ll k_{x} \ll 1/z$, here we shall write the following form of the premultiplied spectrum without loss of generality:

(3.2)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2}=h\left(k_x l_{I},\frac{z}{\delta}\right), \end{equation}

leading to

(3.3)\begin{equation} \frac{\varPhi_{uu}(k_x,z)}{u_\tau^2}=\frac{h(k_x l_{I},z/\delta)}{k_x}, \end{equation}

where $l_{I}$ is an appropriate length scale that can be chosen for $z\ll l_{I} \ll \delta$ in the range of $1/\delta \ll k_{x} \ll 1/z$. Here, we note that only the condition of $z/\delta \rightarrow 0$ is relaxed compared to Perry et al. (Reference Perry, Henbest and Chong1986), as we are concerned with finite Reynolds number and turbulence intensity for layer I. While we assume that there is no explicit form of the spectrum available for $1/\delta \ll k_{x} \ll 1/z$, the form of (3.3) raises several non-trivial issues to be discussed as follows.

1. (i) The spectrum for $O(1/\delta ) < k_x < O(1/z)$: while (3.3) provides a general form of the spectrum for $1/\delta \ll k_x \ll 1/z$, this part of the spectrum should not be interpreted as an outcome of simultaneous ‘physical’ contribution of $\delta$- and $z$-scaling motions, as was argued in Perry et al. (Reference Perry, Henbest and Chong1986). We note that the logarithmic wall-normal dependence of the $u$ variance in the original theory of Townsend (Reference Townsend1976) is due to the wall-reaching inactive motions of each attached eddy, the size of which varies from $O(z)$ to $O(\delta )$. In other words, the non-zero spectrum for $O(1/\delta ) < k_x < O(1/z)$ should be primarily from the contribution of the inactive motions of those attached eddies, as was also directly confirmed by the recent work of Deshpande et al. (Reference Deshpande, Monty and Marusic2020).

2. (ii) Relation to the original model: now let us assume $Re_\tau \rightarrow \infty$, so that the lower boundary of layer I reaches the wall (i.e. $z_m \rightarrow 0$). In this case, using the Taylor series expansion about $z=0$, the premultiplied power-spectral intensity in (3.1a), (3.1b) and (3.3) can further be approximated with

   (3.4a)\begin{equation} \tilde{h}_1(k_x \delta, z/\delta)=\tilde{h}_1(k_x \delta,0)+\frac{\partial \tilde{h}_1(k_x \delta, 0)}{\partial (z/\delta)}\left(\frac{z}{\delta}\right)+O\left(\frac{z^2}{\delta^2}\right) \end{equation}
   for $k_x \sim O(1/\delta )$,
   (3.4b)\begin{equation} \tilde{h}_2(k_x z, z/\delta)=\tilde{h}_2(k_x z,0)+\frac{\partial \tilde{h}_2(k_x z,0)}{\partial (z/\delta)}\left(\frac{z}{\delta}\right)+O\left(\frac{z^2}{\delta^2}\right) \end{equation}
   for $k_x \sim O(1/z)$, and
   (3.4c)\begin{equation} {h(k_x l_{I}, z/\delta)=h(k_x l_{I}, 0)+\frac{\partial h(k_x l_{I},0)}{\partial (z/\delta)}\left(\frac{z}{\delta}\right)+O\left(\frac{z^2}{\delta^2}\right)} \end{equation}
   for $1/\delta \ll k_x \ll 1/z$, respectively. If the prediction (2.5) given by Perry et al. (Reference Perry, Henbest and Chong1986) is correct, it is now expected that $h(k_x l_{I}, 0) \rightarrow A$ in the limit as $z/\delta \rightarrow 0$. Although the detailed convergence to this possible limiting behaviour would be answered only by additional measurements at higher Reynolds numbers, it is evident that in this limit, (3.1a), (3.1b) and (3.3) become identical to (2.2b), (2.3b) and (2.5) by setting $\tilde {h}_1(k_x \delta, 0)=h_1(k_x \delta )$, $\tilde {h}_2(k_x z, 0)=h_2(k_x z)$ and $h(k_x l_{I}, 0)=A$. This implies that the original model of Perry et al. (Reference Perry, Henbest and Chong1986) would be strictly valid in the limit as $z/\delta \rightarrow 0$. As such, the emergence of a well-developed $k_x^{-1}$ spectrum in the sense of Perry et al. (Reference Perry, Henbest and Chong1986) might be observed only for very small $z/\delta$ at extremely high Reynolds numbers, as was recently proposed by Baars & Marusic (Reference Baars and Marusic2020a). Care therefore needs to be taken in interpreting (2.5) obtained from Perry et al. (Reference Perry, Henbest and Chong1986) for the experimental data where $z/\delta$ is not infinitesimal but small. In fact, (3.4c) indicates that the wall-normal variation of the spectral intensity for $1/\delta \ll k_x \ll 1/z$ would be of $O(z/\delta )$. Since $z/\delta \sim O(10^{-1})$ in layer I, the premultiplied spectrum for $O(1/\delta ) < k_x < O(1/z)$ is expected to change with $z/\delta$ at least by $O(10^{-1})$. This is exactly seen in the experimental data in figure 3 as well as in other previous papers (Morrison et al. Reference Morrison, Jiang, McKeon and Smits2001, Reference Morrison, McKeon, Jiang and Smits2004; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013), and it is due to the ‘finite $z/\delta$’ of the measurement locations at finite Reynolds numbers.

3. (iii) Townsend–Perry constant: The general form of spectrum given by (3.3) suggests that in practice (at finite Reynolds number and finite $z/\delta$), the premultiplied spectrum in the absence of a well-developed $k_x^{-1}$ spectrum in layer I would offer little insight into the Townsend–Perry constant $A$ (see (2.5)). This also implies that the log-linear behaviour of the streamwise turbulence intensity, previously reported to emerge in the form of (1.1) (e.g. Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013), is not the one expected directly from the model of Perry et al. (Reference Perry, Henbest and Chong1986), because the spectrum intensity varies with $z/\delta$ without necessarily exhibiting a well-developed $k_x^{-1}$ spectrum. In this respect, it is now important to understand how the logarithmic wall-normal dependence of streamwise turbulence intensity would emerge without having a $k_x^{-1}$ spectrum.

3.2. Turbulence intensity

From the discussion given above, a schematic diagram of the premultiplied power-spectral density for the model in the present study is sketched in figure 4. This schematic diagram is also consistent with the spectra observed in figure 3 (compare figure 3 with figure 4). The primary differences from the schematic diagram in figure 1 are: (1) a non-trivial form of spectrum for $1/\delta \ll k \ll 1/z$; (2) the intensity dependent on $z/\delta$ for all $k_x$; (3) the spectrum with order unity intensity for $k_x \in [a_{I}/\delta, b_{I}/\delta ]$ from (3.1a) and (3.1b) (red-shaded region in figure 4), where $a_{I}$ and $b_{I}$ are $O(1)$ constants to be defined below. In this model, the spectrum for $a_{I}/\delta \leq k_x \leq b_{I}/z$ is set to take the following form without loss of generality:

(3.5a)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2} = h\left(k_x l_{I},\frac{z}{\delta}\right)\quad\mathrm{for}\ a_{I}/\delta \ll k_x \ll b_{I}/z, \end{equation}

and $a_{I}$ and $b_{I}$ are given such that

(3.5b)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x,z)}{u_\tau^2} \sim O(A_{{I},0})\quad\mathrm{for}\ a_{I}/\delta \leq k_x \leq b_{I}/z, \end{equation}

where $A_{{I},0}$ is a constant of $O(1)$, and such that $\Delta A_{{I},0}(k_x l_{I},z/\delta )(\equiv k_x\,\varPhi (k_x,z)/u_\tau ^2- A_{{I},0})$ satisfies

(3.5c)\begin{equation} \underbrace{\int_{\ln (a_{I}/\delta)}^{\ln (b_{I}/z)} \Delta A_{{I},0}\left(k_x l_{I},\frac{z}{\delta}\right)\mathrm{d}(\ln k_x)}_{{}\equiv A_{{I},1}(z/\delta)} \sim O\left(\frac{z}{\delta}\right). \end{equation}

In other words, with the spectrum, $A_{{I},0}$, $a_{I}$ and $b_{I}$ can be chosen such that the area below the premultiplied spectrum for $a_{I}/\delta \leq k_x \leq b_{I}/z$ is approximated by the red-shaded region in figure 4 with an error of $O(z/\delta )$. We note that such a choice of $A_{{I},0}$, $a_{I}$ and $b_{I}$ must always be possible, because the mean-value theorem for integrals ensures the existence of $A_{{I},M}(z/\delta )$ satisfying

(3.6a)\begin{equation} {A}_{{I},M}(z/\delta)=\left[\ln\left(\frac{b_{{I}}}{z}\right)-\ln\left(\frac{a_{{I}}}{\delta}\right)\right]^{{-}1} \int_{\ln (a_{I}/\delta)}^{\ln (b_{I}/z)} \frac{k_x\,\varPhi_{uu}(k_x,z/\delta)}{u_\tau^2}\,\mathrm{d}(\ln k_x). \end{equation}

Here, $A_{{I},M}(z/\delta )$ can further be written such that

(3.6b)\begin{equation} A_{{I},M}(z/\delta)=A_{{I},0}+A_{{I},1}(z/\delta), \end{equation}

where $A_{{I},1}(z/\delta )$ remains at $O(z/\delta )$ by applying the Taylor series expansion to $A_{{I},M}(z/\delta )$ at any wall-normal location in layer I.

Figure 4. A schematic diagram of the proposed model in the present study for the spectra in layer I.

From (2.1) and (3.6), the streamwise turbulence intensity is now written as

(3.7a)\begin{align} \frac{\overline{u^\prime u^\prime}}{u_\tau^2} &= \int_{-\infty}^\infty \frac{k_x\, \varPhi_{uu}(k_x;z)}{u_\tau^2}\,\mathrm{d}(\ln k_x) = \int_{\ln (a_{I}/\delta)}^{\ln (b_{I}/z)} \frac{k_x\,\varPhi_{uu}(k_x)}{u_\tau^2}\,\mathrm{d}(\ln k_x)+C_{{I}}(z;Re_\tau) \nonumber\\ & = \left[A_{{I},0}+A_{{I},1}\left(\frac{z}{\delta}\right)\right]\left[\ln\left(\frac{b_{{I}}}{z}\right)-\ln\left(\frac{a_{{I}}}{\delta}\right)\right]+C_{{I}}(z;Re_\tau) \nonumber\\ &={-}\left[A_{{I},0}+A_{{I},1}\left(\frac{z}{\delta}\right)\right]\ln\left(\frac{z}{\delta}\right)+B_{{I}}(z;Re_\tau), \end{align}

where

(3.7b)\begin{gather} C_{{I}}\left(z;Re_\tau\right)=\frac{\overline{u^\prime u^\prime}}{u_\tau^2}-\int_{\ln (b_{I}/\delta)}^{\ln(a_{I}/z)} \frac{k_x\,\varPhi_{uu}(k_x)}{u_\tau^2}\,\mathrm{d}(\ln k_x), \end{gather}

(3.7c)\begin{gather} B_{{I}}\left(z;Re_\tau\right)=C_{{I}}\left(z;Re_\tau\right)-\left[A_{{I},0}+A_{{I},1}\left(\frac{z}{\delta}\right)\right]\ln \left(\frac{b_{I}}{a_{I}}\right). \end{gather}

Here, $C_{{I}}(z;Re_\tau )$ represents the contribution of the spectrum for $k_x< a_{I}/\delta$ and $k_x>b_{I}/z$ to the turbulence intensity, and it should depend weakly on $z$ and $Re_\tau$ due to the Kolmogorov-scaling part in figure 4 and the weak $(z/\delta )$-dependence of (3.4a) and (3.4b) within layer I. In (3.7a) and (3.7c), the Taylor series expansion allows $A_{{I},1}(z/\delta )$ to be approximated further as

(3.8)\begin{equation} A_{{I},1}\left(\frac{z}{\delta}\right)=A_{{I},1}\left(\frac{z_{I,m}}{\delta}\right)+\frac{\mathrm{d} A_{{I},1}}{\mathrm{d} (z/\delta)}\bigg|_{z=z_{{I},m}} \frac{(z-z_{{I},m})}{\delta} +O\left(\frac{|z-z_{{I},m}|^2}{\delta^2}\right), \end{equation}

where $z_{{I},m}=\sqrt {z_m z_o}$ is chosen to be the geometric mean of $z_m$ and $z_o$, so that the right-hand side of (3.8) becomes a good approximation to $A_{{I},1}$ over the entire layer I in the ‘logarithmic’ wall-normal coordinate. The same approximation can be applied to $C_{{I}}(z;Re_\tau )$ in (3.7b). Since $z_{I,m}/\delta$ is a function of $Re_\tau$ from $z_m/\delta = 3.6 Re_\tau ^{-0.5}$, the streamwise turbulence intensity is finally given by

(3.9)\begin{equation} \frac{\overline{u^\prime u^\prime}}{u_\tau^2} ={-}A_{{I}}(Re_\tau)\ln\left(\frac{z}{\delta}\right)+B_{{I}}(z_{{I},m};Re_\tau)+O\left(\frac{z}{\delta}\right), \end{equation}

where $A_{{I}}(Re_\tau )=A_{{I},0}+A_{{I},1}(z_{{I},m}/\delta )$ and the error at $O(z/\delta )$ stems from approximation of $C_{{I}}(z;Re_\tau )$ in (3.7c) for layer I. We also note that in layer I, the Reynolds shear stress would evidently be $-\overline {u'w'}/{u_\tau ^2}= \mathrm {const} + O(z/\delta )$, where $w'$ is the wall-normal velocity fluctuation. Therefore (3.9) is consistent with the original attached eddy model of Townsend (Reference Townsend1976) with an error of $O(z/\delta )$.

Now, the logarithmic wall-normal dependence of the streamwise turbulence intensity is retrieved in (3.9) as in the classical theories (Townsend Reference Townsend1976; Perry & Chong Reference Perry and Chong1982; Perry et al. Reference Perry, Henbest and Chong1986) with a small error at $O(z/\delta ) \sim 10^{-1}$, indicating that the classical result is indeed a reliable first approximation to the streamwise turbulence intensity in layer I. Importantly, the Townsend–Perry constant $A_{{I}}$ in this case emerges as a function of $Re_\tau$, and, to our knowledge, the present study provides the first rigorous derivation for this feature – it would depend weakly on $Re_\tau$, given $A_{{I},1} \sim O(z/\delta )$ from (3.6b). We note that this is a combined consequence of (3.6b) and the nature of $z_{m}$ (or $z_{{I},m}$) depending on $Re_\tau$. Since $z_{m}$ is a function of $Re_\tau$ due to the viscous wall effect, the Reynolds-number-dependent nature of $A_{{I}}(Re_\tau )$ in (3.9) essentially originates from the role of viscosity ignored in the original theories (e.g. Townsend Reference Townsend1976; Perry & Chong Reference Perry and Chong1982; Perry et al. Reference Perry, Henbest and Chong1986). This is evident from $z_{m}/\delta \rightarrow 0$ in the limit as $Re_\tau \rightarrow \infty$: that is, the velocity slip condition (i.e. non-zero spectrum at $z=z_m$) reaches all the way down to the wall in such a limit. In this case, as $z/\delta \rightarrow 0$, $A_{{I}}$ becomes constant from (3.6b), and the error of $O(z/\delta )$ in (3.9) vanishes. Therefore (3.9) consequently becomes identical to (2.6) from the original model of Perry et al. (Reference Perry, Henbest and Chong1986) in the limit as $Re_\tau \rightarrow \infty$ and $z/\delta \rightarrow 0$.

It should, however, be mentioned that the use of the mean-value theorem in (3.6) for the derivation of (3.9) is the key difference from that of (2.6) in Perry et al. (Reference Perry, Henbest and Chong1986). The derivation here is more general and inclusive than that in Perry et al. (Reference Perry, Henbest and Chong1986), because it does not rely on the existence of a $k_x^{-1}$ spectrum. Instead, it suggests that a more general condition for the existence of an approximate logarithmic wall-normal dependence of the streamwise turbulence intensity is the existence of the premultiplied power-spectral density of $O(1)$ for $O(1/\delta ) \leq k_x \leq O(1/z)$ (red-shaded region in figure 4), which is entirely consistent with the experimental data in figure 3. Here, it is important to note that this behaviour essentially originates from the spectrum scalings in (3.1a) and (3.1b), the general versions of (2.2) and (2.3) for finite $z/\delta$, because (3.9) is simply a consequence of applying the mean-value theorem in (3.6) with them. Since (2.2) and (2.3) depict the $\delta$-scaling inactive motions of large eddies and the $z$-scaling self-similar eddies, the theoretical development here is also well within the framework of the attached eddy hypothesis. Lastly, it is worth mentioning that a more accurate description for $\Delta A_{{I},0}(z_{{I},m}/\delta )$ may well be possible from a viewpoint of scaling with $Re_\tau$. The recent effort made by Vassilicos et al. (Reference Vassilicos, Laval, Foucaut and Stanislas2015) can be interpreted in such a context, as it is based on a prescribed semi-empirical description for the spectrum lying between the wavenumber ranges for (2.2b) and (2.5).

3.3. Determination of $A_{I}$ and $B_{I}$ from experimental data

While the analysis in §§ 3.1 and 3.2 provides an insight into the relationship between the power-spectral density and the streamwise turbulence intensity at finite Reynolds numbers, there is a practical issue in applying it to the data obtained from laboratory experiments and numerical simulations. This issue is related to how one would choose $A_{{I},0}$, $a_{{I}}$ and $b_{{I}}$ robustly. To this end, here we propose a simple semi-empirical modelling framework for the streamwise turbulence intensity by leveraging the mean-value theorem (3.6). We start by writing the turbulence intensity as

(3.10a)\begin{align} \frac{\overline{u^\prime u^\prime}}{u_\tau^2} & = \int_{-\infty}^\infty \frac{k_x\, \varPhi_{uu}(z;k_x;Re_\tau)}{u_\tau^2}\,\mathrm{d}(\ln k_x) \nonumber\\ &= \left[\ln\left(\frac{b_{{I},s}}{z}\right)-\ln\left(\frac{a_{{I},s}}{\delta}\right)\right] \varPi_{I}(z;Re_\tau) \nonumber\\ &= \left[\ln\left(\frac{b_{{I},s}}{a_{{I},s}}\right)-\ln\left(\frac{z}{\delta}\right)\right] \varPi_{I}(z;Re_\tau), \end{align}

where

(3.10b)\begin{equation} \varPi_{I}(z;Re_\tau)=\left[\ln\left(\frac{b_{{I},s}}{a_{{I},s}}\right)-\ln\left(\frac{z}{\delta}\right)\right]^{{-}1} \int_{0}^{\infty} \frac{k_x \varPhi_{uu}}{u_\tau^2} \,\mathrm{d}(\ln k_x). \end{equation}

Here, $a_{{I},s}$ and $b_{{I},s}$ are constants that play roles similar to those of $a_{{I}}$ and $b_{{I}}$ in figure 4, but their values are not necessarily the same. However, given the analysis in § 3.2, they need to be chosen from the wavenumbers of $\delta$- and $z$-scaling regions of the premultiplied spectra, and have to be constant at all Reynolds numbers. While this sets out a condition for $a_{{I},s}$ and $b_{{I},s}$ to be met, it is also important to note that they cannot be chosen arbitrarily. Although (3.10) is supposed to automatically yield a streamwise turbulence intensity in the form of (3.7a) for any values of $a_{{I},s}$ and $b_{{I},s}$, the last line of (3.10a) implies that it has a single degree of freedom for the determination of $A_{{I}}$ and $B_{{I}}$ in (3.7a) (i.e. the value of $b_{{I},s}/a_{{I},s}$). Indeed, if $b_{{I},s}/a_{{I},s}<1$ is chosen, then $B_{{I}}$ in (3.7a) becomes negative from (3.10a), which would obviously be non-physical. Therefore the last condition for $a_{{I},s}$ and $b_{{I},s}$ to be met is that they must be chosen such that $\ln (b_{{I},s}/a_{{I},s})\,\varPi _{I}(z;Re_\tau )$ would represent $B_{{I}}$ in (3.7a) well, while ensuring $b_{{I},s}/a_{{I},s}>1$.

In the present study, $a_{{I},s}={\rm \pi} /5$ and $b_{{I},s}=2{\rm \pi}$ are chosen so that $\lambda _{x,{I}\,a}(\equiv 2{\rm \pi} \delta /a_{{I},s})=10\delta$ and $\lambda _{x,{I}\,b}(\equiv 2{\rm \pi} z/b_{{I},s})=z$. These values are obtained by trial and error. To do so, a range of candidate values of $a_{{I},s}$ and $b_{{I},s}$ are first selected from the spectra in figure 3, so that $\lambda _{x,{I}\,a}$ and $\lambda _{x,{I}\,b}$ lie in the $\delta$- and $z$-scaling regions, respectively. Indeed, figure 3 shows that the premultiplied spectra in layer I scale well with $\delta$ for $\lambda _x\simeq \lambda _{x,{I}\,a}$, and with $z$ for $\lambda _x\simeq \lambda _{x,{I}\,b}$, and that $b_{{I},s}/a_{{I},s}>1$, consistent with the purpose of $a_{{I},s}$ and $b_{{I},s}$ introduced here. Once this is ensured, the final values of $a_{{I},s}$ and $b_{{I},s}$ are determined by monitoring the form of $\varPi _{I}(z)$ within the layer. If a set of sensible values is chosen, then $\varPi _{I}(z)$ must exhibit a plateau around $z=z_{I,m}$ in layer I. Indeed, figure 5 demonstrates that, for the given $a_{{I},s}$ and $b_{{I},s}$, $\varPi _{I}(z)$ remains approximately constant in layer I. Once $a_{{I},s}$ and $b_{{I},s}$ are determined, the resulting streamwise turbulence intensity is approximated around $z=z_{{I},m}$ (see also (3.9)):

(3.11a)\begin{equation} \frac{\overline{u^\prime u^\prime}}{u_\tau^2} \simeq{-}A_{{I}}(Re_\tau)\ln\left(\frac{z}{\delta}\right)+B_{{I}}(Re_\tau), \end{equation}

where

(3.11b)\begin{gather} A_{{I}}(Re_\tau)=\varPi_{{I}}(z_{{I},m};Re_\tau), \end{gather}

(3.11c)\begin{gather} B_{{I}}(Re_\tau)=\varPi_{{I}}(z_{{I},m};Re_\tau)\ln\left(\frac{b_{{I},s}}{a_{{I},s}}\right). \end{gather}

The values of $A_{{I}}(Re_\tau )$ and $B_{{I}}(Re_\tau )$ are then obtained from (3.11b) and (3.11c), respectively. In the present study, the values of $a_{{I},s}={\rm \pi} /5$ and $b_{{I},s}=2{\rm \pi}$ are first determined at $Re_\tau =19\,680$ and are subsequently used for the other Reynolds numbers. Figure 6 shows the streamwise turbulence intensity and its approximation from (3.11). With the given $a_{{I},s}$ and $b_{{I},s}$, the proposed framework in (3.10) determines both $A_{{I}}$ and $B_{{I}}$ well, so that (3.11a) becomes a good fit to the turbulence intensity in layer I at all the Reynolds numbers. The computed values of $A_{{I}}(Re_\tau )$ and $B_{{I}}(Re_\tau )$ are also reported in table 1 with the approximation error of the model (3.11a) in layer I. The maximum error is found to be less than about 4 % at all the Reynolds numbers considered.

Figure 5. $\varPi _{{I}}(z)$ (solid line) from (3.10b), and $A_{{I}}$ (dashed line) from (3.11b): ($a$) $Re_{\tau }=6123$; ($b$) $Re_{\tau }= 10\,100$; ($c$) $Re_{\tau }=14\,680$; ($d$) $Re_{\tau }=19\,680$. Here, $a_{{I},s}=2{\rm \pi}$ ($\lambda _{x,{I}\,a}=10$) and $b_{{I},s}={\rm \pi} /5$ ($\lambda _{x,{I}\,b}=1$).

Figure 6. Streamwise turbulence intensity (solid line) and its approximation given by (3.11) (dashed line) in layer I: ($a$) $Re_{\tau }=6123$; ($b$) $Re_{\tau }=10\,100$; ($c$) $Re_{\tau }=14\,680$; ($d$) $Re_{\tau }=19\,680$. Here, $a_{{I},s}={\rm \pi} /5$ ($\lambda _{x,{I}\,a}/\delta =10$) and $b_{{I},s}=2{\rm \pi}$ ($\lambda _{x,{I}\,b}/z=1$).

Table 1. The Reynolds-number dependence of $A_{I}$ and $B_{I}$ from (3.11) with $a_{{I},s}={\rm \pi} /5$ ($\lambda _{x,{I}\,a}/\delta =10$) and $b_{{I},s}=2{\rm \pi}$ ($\lambda _{x,{I}\,b}/z=1$). Here, $\text {Error}\equiv \max _z |\overline {u^\prime u^\prime }_{{model}}-\overline {u^\prime u^\prime }|/\overline {u^\prime u^\prime }$ for $z \in [z_m,z_o]$, where $\overline {u^\prime u^\prime }_{{model}}$ is from (3.11), and it indicates the maximum error of the proposed model in layer I.

4.1. Scaling of spectra

The model developed in § 3 is further extended to layer II where $z_i \leq z \leq z_m$ with $z_i^+=200$ and $z_m^+=3.6 Re_\tau ^{1/2}$ ($z_m^+=505$ at $Re_\tau =19\,680$). In particular, here we shall take an empirical approach for the information that cannot be retrieved solely with scaling arguments. Given the importance of viscous forces in this layer, the power-spectral density should be a function of $u_\tau$, $k_x$, $z$, $\delta$ and $\delta _\nu (\equiv \nu /u_\tau )$. Without loss of generality, the outer-scaling part of the spectrum for $k_x \sim O(1/\delta )$ would therefore be written as a function of $z^+$ and $Re_\tau (=\delta /\delta _\nu )$ as

(4.1a)\begin{equation} \frac{\varPhi_{uu}(k_x \delta,z^+,Re_\tau)}{u_\tau^2}=\frac{\varPhi_{uu}(k_x,z^+,Re_\tau)}{\delta u_\tau^2}=g_{1,{II}}(k_x \delta,z^+,Re_\tau), \end{equation}

with the corresponding premultiplied spectrum

(4.1b)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x)}{u_\tau^2}=k_x \delta\,g_{1,{II}}(k_x \delta,z^+,Re_\tau)=h_{1,{II}}(k_x \delta,z^+,Re_\tau). \end{equation}

Similarly, the $z$-scaling part of the spectrum is written as

(4.2a)\begin{equation} \frac{\varPhi_{uu}(k_x z,z^+,Re_\tau)}{u_\tau^2}=\frac{\varPhi_{uu}(k_x,z^+,Re_\tau)}{z u_\tau^2}=g_{2,{II}}(k_x z,z^+,Re_\tau), \end{equation}

and the premultiplied spectrum is

(4.2b)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x)}{u_\tau^2}=k_x z\,g_{2,{II}}(k_x z,z^+,Re_\tau)=h_{2,{II}}(k_x z,z^+,Re_\tau). \end{equation}

Here, we note that the introduction of a possible dependence on $z^+$ and $Re_\tau$ in (4.1) and (4.2) is to describe the viscous wall effect on $z$- and $\delta$-scaling motions, the scaling and the corresponding energy balance of which were discussed in detail in Hwang (Reference Hwang2016) and Cho et al. (Reference Cho, Hwang and Choi2018) using a linear theory and numerical simulation data. While the contribution of $\delta$-scaling motions is sometimes accounted for separately (Baars & Marusic Reference Baars and Marusic2020a,Reference Baars and Marusicb; Deshpande et al. Reference Deshpande, Monty and Marusic2020), here we shall consider the contributions of all the energy-containing motions simultaneously.

The spectra given in (4.1) and (4.2) suggest that the dependence of $h_{1,{II}}$ and $h_{2,{II}}$ on $z^+$ and $Re_\tau$ would be the major complication in extending the analysis in § 3 to layer II. Therefore, from here, we shall further proceed by empirically modelling the experimental data of Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) in layer II. As discussed in § 2, the contour lines for $\lambda _x\gtrsim 10 \delta$ in figure 2 are not parallel to $\lambda _x=10 \delta$. They rather appear to fall off approximately linearly towards the wall in the logarithmic coordinates, indicating that the outer-scaling part of the spectra would empirically follow a mixed scale of $\lambda _x/\delta \sim (z^+)^p$, where $p$ is a positive number to be determined. Based on this observation, (4.1) may be written as

(4.3a)\begin{equation} h_{1,{II}}(k_x \delta,z^+,Re_\tau)\simeq \tilde{h}_{1,{II}}\left(k_x \delta_{{II}},\frac{z}{\delta_{{II}}}\right), \end{equation}

where $\delta _{{II}}=\delta (z^+)^p$ is the empirical similarity length scale formed by $\delta$ and $z^+$, and the $z/\delta _{{II}}$-dependence is introduced to admit the small deviation of the spectra in layer II similarly to the case in layer I (§ 3.1). As for the inner-scaling part of the spectra, the contour lines for $10^{-2} \delta \leq \lambda _x \leq 10^{-1} \delta$ in layer II (figure 2) are still approximately parallel to $\lambda _x=z$. Therefore $h_{2,{II}}(k_x z,z^+)$ in (4.2) is also written as

(4.3b)\begin{equation} h_{2,{II}}(k_x z,z^+,Re_\tau)\simeq \tilde{h}_{2,{II}}\left(k_x z,\frac{z}{\delta_{{II}}}\right), \end{equation}

where $z/\delta _{{II}}$ is introduced to allow for a small variation in the wall-normal location. Figure 7 confirms that the empirical scalings in (4.3a) and (4.3b) are reasonably good: the outer-scale part of the premultiplied spectra scales well with $\delta _{{II}}$ for $p=0.41$ obtained empirically (figure 7$a$), while the inner-scaling part shows good scaling with $z$ (figure 7$b$).

Figure 7. Premultiplied streamwise power-spectral density of streamwise velocity in layer II ($Re_\tau = 19\,680$): ($a$)  $\delta _{{II}}$-scaling and ($b$)  $z$-scaling. Here, $\delta _{{II}}=\delta (z^+)^p$ with $p=0.41$.

The form of (4.3a) and (4.3b) is now identical to that of (3.1a) and (3.1b), if $\delta$ is replaced by $\delta _{II}$. The procedure in § 3.1 can therefore be applied exactly in the same manner. Without loss of generality, this yields

(4.4)\begin{equation} \frac{k_x\,\varPhi_{uu}(k_x)}{u_\tau^2}=h_{{II}}\left(k_x l_{{II}}, \frac{z}{\delta_{{II}}}\right) \end{equation}

for $1/\delta _{{II}} \ll k_x \ll 1/z$ with $z \ll l_{II} \ll \delta _{II}$, and the resulting premultiplied power-spectral density for $1/\delta _{{II}} \ll k_x \ll 1/z$ is given by

(4.5)\begin{equation} \frac{\varPhi_{uu}(k_x)}{u_\tau^2}=\frac{h_{{II}}(k_x l_{{II}},z/\delta_{{II}})}{k_x}. \end{equation}

Equation (4.5) suggests that a $k_x^{-1}$ spectrum would also be observed at each wall-normal location in layer II. As shown in figure 7, such a spectrum does seem to appear for $50 z \lesssim \lambda _x \lesssim 0.5 \delta _{{II}}$ in the form of a peak or a narrow plateau. It also indicates that the intensity of the $k_x^{-1}$ spectrum would vary in the wall-normal direction by $O(z/\delta _{II})$ in layer II. Given the definition of layer II, $z/\delta _{II} \sim O(10^{-2})$. Figure 7 shows that the variation of the peak intensity of each premultiplied spectrum is much smaller than that in layer I (compare with figure 3), supporting the relevance of (4.5).

4.2. Turbulence intensity

Given the form of spectra shown in (4.3a), (4.3b) and (4.5), the turbulence intensity in layer II can also be obtained by applying the same procedure in § 3.2. In Appendix A, it is shown that the streamwise turbulence intensity in layer II takes the following form:

(4.6)\begin{equation} \frac{\overline{u^\prime u^\prime}}{u_\tau^2} \simeq{-}A_{{II}}(z_{{II},m};Re_\tau)\left[\ln\left(\frac{z}{\delta}\right)-p\ln(z^+)\right]+B_{{II}}(z_{{II},m};Re_\tau), \end{equation}

where $z_{{II},m}=\sqrt {z_i z_m}$, $A_{{II}}(z_{{II},m};Re_\tau )$ and $B_{{II}}(z_{{II},m};Re_\tau )$ are defined in Appendix A. Here, we note that the term with $z^+$ now emerges, and it incorporates the effect of the viscosity into the original attached eddy model. Also, similarly to layer I, $A_{{II}}$ and $B_{{II}}$ in (4.6) are expected to depend weakly on $Re_\tau$ (see Appendix A). Lastly, following the modelling procedure for layer I in § 3.3, $A_{{II}}(z_{{II},m};Re_\tau )$ and $B_{{II}}(z_{{II},m};Re_\tau )$ can be semi-empirically determined, such that

(4.7a)\begin{gather} A_{{II}}(Re_\tau)=\varPi_{II}(z_{{II},m};Re_\tau), \end{gather}

(4.7b)\begin{gather} B_{{II}}(Re_\tau)=\varPi_{II}(z_{{II},m};Re_\tau)\ln\left(\frac{b_{{II},s}}{a_{{II},s}}\right), \end{gather}

where

(4.7c)\begin{equation} \varPi_{II}(z;Re_\tau)=\left[\ln\left(\frac{b_{{II},s}}{z}\right)-\ln\left(\frac{a_{{II},s}}{\delta_{{II}}}\right)\right]^{{-}1} \int_{0}^{\infty} \frac{k_x \varPhi_{uu}}{u_\tau^2} \,\mathrm{d}(\ln k_x), \end{equation}

and $a_{{II},s}$ and $b_{{II},s}$ are the fitting constants for the model in (4.6), similarly to $a_{{I},s}$ and $b_{{I},s}$ in § 3.3. In the present study, $a_{{II},s}=2{\rm \pi} /3$ and $b_{{II},s}=2{\rm \pi}$ are obtained by trial and error with the experimental data for all the Reynolds numbers. The determination procedure of $a_{{II},s}=2{\rm \pi} /3$ and $b_{{II},s}=2{\rm \pi}$ follows that in § 3.3 exactly. This leads to $\lambda _{x,II\,a}(\equiv 2{\rm \pi} \delta _{II}/a_{{II},s})=3\delta _{II}$ and $\lambda _{x,{I}\,b}(\equiv 2{\rm \pi} z/b_{{II},s})=z$. We note that the premultiplied spectra in layer II scale well with $\delta _{II}$ for $\lambda _x\simeq \lambda _{x,II\,a}$ (figure 7$a$) and with $z$ for $\lambda _x\simeq \lambda _{x,II\,b}$ (figure 7$b$), serving the introduced purpose of $a_{{II},s}$ and $b_{{II},s}$. Figure 8 shows $\varPi _{II}(z)$ and the corresponding $A_{{II}}(Re_\tau )$ determined at all the Reynolds numbers. Similar to the case of layer I, $\varPi _{II}(z)$ remains approximately constant in layer II with the given $a_{{II},s}$ and $b_{{II},s}$, and the resulting $A_{{II}}(Re_\tau )$ is also found to depend weakly on the Reynolds number. The streamwise turbulence intensity and its approximation from (3.11) are shown in figure 9. For the given $a_{{II},s}$ and $b_{{II},s}$, the model in (4.6) and (4.7) well approximates the streamwise turbulence intensity in layer  II at all the Reynolds numbers. Finally, the values of $A_{{II}}(Re_\tau )$ and $B_{{II}}(Re_\tau )$ are listed in table 2. The maximum error of the model of (4.6) remains at less than  3 %.

Figure 8. $\varPi _{{II}}(z)$ (solid line) from (4.7c) and $A_{{II}}$ (dashed line) from (4.7a): ($a$) $Re_{\tau }=6123$; ($b$) $Re_{\tau }=10\,100$; ($c$) $Re_{\tau }=14\,680$; ($d$) $Re_{\tau }=19\,680$. Here, $a_{{II},s}=2{\rm \pi} /3$ ($\lambda _{x,II\,a}/\delta _{{II}}=3$) and $b_{{II},s}=2{\rm \pi}$ ($\lambda _{x,II\,b}/z=1$).

Figure 9. Streamwise turbulence intensity (solid line) and its approximation given by (4.6) (dashed line) in layer II: ($a$) $Re_{\tau }=6123$; ($b$) $Re_{\tau }=10\,100$; ($c$) $Re_{\tau }=14\,680$; ($d$) $Re_{\tau }=19\,680$. Here, $a_{{II},s}=2{\rm \pi} /3$ ($\lambda _{x,II\,a}/\delta _{{II}}=3$) and $b_{{II},s}=2{\rm \pi}$ ($\lambda _{x,II\,b}/z=1$). The streamwise turbulence intensity from model (3.11a) for layer I (dashed-dotted line) is also plotted for comparison.

Table 2. The Reynolds-number dependence of $A_{II}$ and $B_{II}$ from (4.7) with $a_{{II},s}=2{\rm \pi} /3$ and $b_{{II},s}=2{\rm \pi}$. Here, $\mathrm {Error}=\max _z |\overline {u^\prime u^\prime }_{{model}}-\overline {u^\prime u^\prime }|/\overline {u^\prime u^\prime }$ for $z \in [z_i,z_m]$, where $\overline {u^\prime u^\prime }_{{model}}$ is from (4.6), and it indicates the maximum error of the proposed model in layer II.