2 Zero-advection condition and log scaling in TBL flows

In a seminal work, Coles (Reference Coles1955) proposed the so-called streamline hypothesis for TBLs according to which a necessary and sufficient condition for a universal law of the wall for the mean velocity in the inner region is that the ratio $\langle u/u_{\unicode[STIX]{x1D70F}}\rangle$ be constant on streamlines of the mean flow (i.e. streamwise self-similarity). This implies that the iso- $y_{+}$ lines would be identical to the mean streamlines in the inner region. Coles averred that ‘this result must surely be considered in any search for a fundamental order and unity in the description of turbulent shear flows’ (Coles Reference Coles1956). In the present work, we build on this description to obtain a generalized streamline hypothesis (streamwise self-similarity or zero advection) for higher-order moments of fluctuating velocity. In particular, we shall consider advection of a scaled mean quantity $Q_{+}$ in a steady two-dimensional TBL flow,

(2.1) $$\begin{eqnarray}\text{ADV}(Q_{+})\triangleq \langle \boldsymbol{u}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}Q_{+}=\langle u\rangle \frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}x}+\langle v\rangle \frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}y}.\end{eqnarray}$$

Here, $\langle u\rangle$ and $\langle v\rangle$ are the components of the mean velocity vector $\langle \boldsymbol{u}\rangle$ in the $x$ and $y$ directions respectively. It should be noted that $\langle \boldsymbol{u}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}Q_{+}$ – dimensional advection of non-dimensional $Q_{+}$ – represents the spatial rate of change of $Q_{+}$ along a mean-flow streamline; $Q_{+}$ could be the normalized mean velocity $\langle u_{+}\rangle$ or generalized mixed moments $\langle u_{+}^{\prime m}v_{+}^{\prime n}\rangle ^{2/(m+n)}$ of velocity fluctuations in the $x$ and $y$ directions; mixed moments may be considered as generalizations of the even moments of $u_{+}^{\prime }$ studied in the literature (Meneveau & Marusic Reference Meneveau and Marusic2013).

In the inner region (including the ISL), the law-of-the-wall scaling for the mean velocity in the $x$ direction,

(2.2) $$\begin{eqnarray}\langle u\rangle =u_{\unicode[STIX]{x1D70F}}f(y_{+}),\end{eqnarray}$$

along with the continuity equation, $\unicode[STIX]{x2202}\langle u\rangle /\unicode[STIX]{x2202}x+\unicode[STIX]{x2202}\langle v\rangle /\unicode[STIX]{x2202}y=0$ , leads to

(2.3) $$\begin{eqnarray}\langle v\rangle =-yf\frac{\text{d}u_{\unicode[STIX]{x1D70F}}}{\text{d}x}.\end{eqnarray}$$

Assuming that the law of the wall for the mean velocity holds, i.e. (2.2) and (2.3) are valid, we proceed in the following sections to show that

1. (i) the zero-advection requirement of $Q_{+}$ follows from its log variation with $y$ (or $y_{+}$ ) and conversely

2. (ii) the log variation of $Q_{+}$ with $y$ (or $y_{+}$ ) follows from its zero-advection requirement.

Table 1. Various parameters for the Princeton (Vallikivi et al. Reference Vallikivi, Hultmark and Smits2015, flow code VHS), Melbourne (Harun et al. Reference Harun, Monty, Mathis and Marusic2013; Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014; Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015, flow codes HMMM, TBHM and MCKH respectively) and Stanford (DeGraaff & Eaton Reference DeGraaff and Eaton2000, flow code DE) ZPG TBL data sets. It should be noted that the Melbourne data have $Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FF}_{c}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}=3020,4945,8280,10\,489,13\,386$ and $15\,000$ respectively, where $\unicode[STIX]{x1D6FF}_{c}$ is a measure of the boundary layer thickness different from $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{99}$ ; $\unicode[STIX]{x1D6FF}_{c}>\unicode[STIX]{x1D6FF}$ (Jones, Marusic & Perry Reference Jones, Marusic and Perry2001; Monkewitz, Chauhan & Nagib Reference Monkewitz, Chauhan and Nagib2008). We have used $\unicode[STIX]{x1D6FF}_{+}=\unicode[STIX]{x1D6FF}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}=\unicode[STIX]{x1D6FF}_{99}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D6FF}$ has been obtained from the mean velocity profile. Original data sets have been used for the analysis of the Melbourne experiments, whereas mean velocity profiles have been digitized from Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015) and DeGraaff & Eaton (Reference DeGraaff and Eaton2000) for the analysis of the Princeton and Stanford experiments respectively. Here, $x_{s}=-u_{\unicode[STIX]{x1D70F}}/(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)$ is the distance of the virtual wall source upstream of the measurement location according to the kinematic description of Coles (Reference Coles1955); see figure 2 of § 2.2.

2.1 Zero advection of $Q_{+}$ follows from its log variation with $y$

For $Q_{+}=\langle u_{+}\rangle$ , (2.2) gives $\unicode[STIX]{x2202}\langle u_{+}\rangle /\unicode[STIX]{x2202}x=[y(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)/\unicode[STIX]{x1D708}]\text{d}f/\text{d}y_{+}$ and $\unicode[STIX]{x2202}\langle u_{+}\rangle /\unicode[STIX]{x2202}y=(u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708})\text{d}f/\text{d}y_{+}$ . Substitution of these expressions along with (2.2) and (2.3) in (2.1) yields

(2.4) $$\begin{eqnarray}\text{ADV}(\langle u_{+}\rangle )=\langle u\rangle \frac{\unicode[STIX]{x2202}\langle u_{+}\rangle }{\unicode[STIX]{x2202}x}+\langle v\rangle \frac{\unicode[STIX]{x2202}\langle u_{+}\rangle }{\unicode[STIX]{x2202}y}=0\end{eqnarray}$$

in the entire inner region (including the ISL, i.e. the log scaling region) for arbitrary values of $\unicode[STIX]{x1D6FF}_{+}$ . This is consistent with the experimental observation of log variation for mean velocity occurring even at moderate Reynolds numbers ( $\unicode[STIX]{x1D6FF}_{+}$ of order $10^{3}$ ; see, for example, Fernholz & Finley Reference Fernholz and Finley1996; DeGraaff & Eaton Reference DeGraaff and Eaton2000). The slope of a mean streamline in the inner region is $\text{d}y/\text{d}x=\langle v\rangle /\langle u\rangle =-y(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)/u_{\unicode[STIX]{x1D70F}}$ and is equal to the slope of an iso- $y_{+}$ line there; the latter may be obtained from the condition $\text{d}(yu_{\unicode[STIX]{x1D70F}})=0$ . Thus, mean streamlines and iso- $y_{+}$ lines coincide in the inner region (including the ISL), and there is no mean exchange (‘entrainment’) of fluid between the outer wake region and the inner region. This, in essence, is Coles’ streamline hypothesis for mean velocity (Coles Reference Coles1955).

Now, we generalize this to higher-order moments of velocity fluctuations, such as variance. Velocity fluctuations in the near-wall region are known to be influenced by outer-scaled motions through the so-called amplitude modulation effects. Hence, in this region, velocity fluctuation in the $x$ direction can be expressed (Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010a ) as

(2.5) $$\begin{eqnarray}u_{+}^{\prime }(y_{+},\unicode[STIX]{x1D702},t)=g_{1,u}(y_{+},t)g_{2,u}(\unicode[STIX]{x1D702},t)+g_{3,u}(\unicode[STIX]{x1D702},t),\end{eqnarray}$$

where $t$ is the time coordinate, $y_{+}=yu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$  ( $\unicode[STIX]{x1D702}=y/\unicode[STIX]{x1D6FF}$ ) is the wall-normal distance in inner (outer) coordinates, $g_{1,u}$ is the statistically universal near-wall component and $g_{2,u}$ and $g_{3,u}$ are respectively the amplitude modulation and linear superposition effects of outer-scaled motions (Marusic et al. Reference Marusic, Mathis and Hutchins2010a ). Since the wall-normal velocity fluctuation $v^{\prime }$ is also modulated in a similar fashion to $u^{\prime }$ (Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014), an expression for $v_{+}^{\prime }$ may be written similarly to that for $u_{+}^{\prime }$ given above. For a generalized mixed moment $Q_{+}=\langle u_{+}^{\prime m}v_{+}^{\prime n}\rangle ^{2/(m+n)}$ in the inner region, these expressions lead to the functional form $Q_{+}=Q_{+}(y_{+},\unicode[STIX]{x1D702})$ or $Q_{+}(y_{+},\unicode[STIX]{x1D6FF}_{+})$ , where, in general, the variables are not separable. Indeed, experiments show that the log variations of variance of $u_{+}^{\prime }$ at different Reynolds numbers in a TBL do not collapse in inner as well as outer scaling but exhibit a systematic shift with $\unicode[STIX]{x1D6FF}_{+}$ (see figures 4a, 6a and 6c in Vallikivi et al. Reference Vallikivi, Hultmark and Smits2015). Substitution of $Q_{+}=Q_{+}(y_{+},\unicode[STIX]{x1D6FF}_{+})$ in the definition of $\text{ADV}(Q_{+})$ yields

(2.6) $$\begin{eqnarray}\text{ADV}(Q_{+})=\text{ADV}(\langle u_{+}^{\prime m}v_{+}^{\prime n}\rangle ^{2/(m+n)})=u_{\unicode[STIX]{x1D70F}}f\frac{\text{d}\unicode[STIX]{x1D6FF}_{+}}{\text{d}x}\frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}_{+}},\end{eqnarray}$$

which could become zero when

(2.7) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6FF}_{+}}{\text{d}x}\rightarrow 0.\end{eqnarray}$$

This shows that even a complicated functional form such as $Q_{+}(y_{+},\unicode[STIX]{x1D6FF}_{+})$ can result in $\text{ADV}(Q_{+})=0$ provided that the condition $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ is satisfied; the role of $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}_{+}$ towards explaining experimental observations of even and odd moments will be discussed in § 4. Physically, the condition $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ implies that the mean streamlines of the flow in the inner region (including the ISL) of a TBL flow tend to be straight lines ( $\text{d}y/\text{d}x=\langle v\rangle /\langle u\rangle =\text{const}.$ along a mean streamline) and concomitantly become orientated parallel to the wall (the slope $\text{d}y/\text{d}x\rightarrow 0$ , see figure 4 b). Thus, the condition $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ effectively reduces to $[\langle v_{A}\rangle /\langle u_{A}\rangle =\langle v_{B}\rangle /\langle u_{B}\rangle ]_{y_{+}}$ , where $\langle u\rangle$ and $\langle v\rangle$ are measured at the same value of $y_{+}$ in the inner layer (i.e. on the same mean streamline) but at different measurement stations $A$ and $B$ along the $x$ direction.

In order to see that the condition $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ is closely approximated in canonical ZPG TBLs only at high Reynolds numbers, we consider the so-called skin-friction law

(2.8) $$\begin{eqnarray}S=\unicode[STIX]{x1D705}^{-1}\ln (\unicode[STIX]{x1D6FF}_{+})+D,\end{eqnarray}$$

where $S=u_{\infty }/u_{\unicode[STIX]{x1D70F}}$ and $D$ is a constant. Differentiation of both sides with respect to $x$ and rearrangement yield

(2.9) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6FF}_{+}}{\text{d}x}=\unicode[STIX]{x1D705}\frac{u_{\infty }}{\unicode[STIX]{x1D708}}\left(\frac{\unicode[STIX]{x1D6FF}}{S}\frac{\text{d}S}{\text{d}x}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ , $u_{\infty }$ and $\unicode[STIX]{x1D708}$ are constants in a given ZPG TBL flow developing in the $x$ direction. Jones, Nickels & Marusic (Reference Jones, Nickels and Marusic2008) have shown that $\unicode[STIX]{x1D6FF}/S(\text{d}S/\text{d}x)\rightarrow 0$ as $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ in ZPG TBL flows, implying that $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ in the limit of infinite Reynolds number. Section 3 presents evaluation of different ZPG TBL experiments in the literature with respect to the zero-advection condition.

As an interim summary, so far we have shown that in the ISL of a TBL flow, (i) the functional form of the dimensionless mean velocity satisfies the corresponding zero-advection condition (Coles Reference Coles1955) for even moderate values of the Reynolds number $\unicode[STIX]{x1D6FF}_{+}$ , (ii) the functional form of the dimensionless generalized mixed moments satisfies the corresponding zero-advection condition when $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ , and this is satisfied only at high Reynolds numbers in canonical ZPG TBLs.

2.2 Log variation of $Q_{+}$ with $y$ follows from its zero advection

Starting from the premise $\text{ADV}(Q_{+})=0$ , one obtains

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}y}=-\frac{\langle u\rangle }{\langle v\rangle }\frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}x},\end{eqnarray}$$

where the ratio $\langle u\rangle /\langle v\rangle$ , from (2.2) and (2.3), at any $y$ in the inner region is

(2.11) $$\begin{eqnarray}\frac{\langle u\rangle }{\langle v\rangle }=-\frac{u_{\unicode[STIX]{x1D70F}}/(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)}{y}=\frac{x_{s}}{y_{s}}.\end{eqnarray}$$

Here, $x_{s}=-u_{\unicode[STIX]{x1D70F}}/(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)$ and $y_{s}=y$ are length scales in the $x$ and $y$ directions respectively; $y_{s}=y$ is predicated on the existence of an ISL (Tennekes & Lumley Reference Tennekes and Lumley1972). If $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}x$ is independent of $y$ , as we shall confirm a posteriori, then (2.10) and (2.11) immediately lead to log scaling of $Q_{+}$ .

Before proceeding further, it is instructive to consider a kinematic interpretation of the streamwise length scale $x_{s}$ . According to Coles (Reference Coles1955), $x_{s}$ geometrically represents the distance, upstream of a streamwise measurement station, of the common point of intersection (located on the solid wall) of all mean velocity vectors in the inner region at that station (see figure 2). These vectors can be imagined to be emanating from a virtual source point on the wall located a distance $x_{s}$ upstream. Coles (Reference Coles1955) showed that the kinematic picture depicted in figure 2 immediately follows from the law of the wall for the mean velocity. In a ZPG TBL, $x_{s}$ increases with the Reynolds number $\unicode[STIX]{x1D6FF}_{+}$ (see table 1), i.e. the virtual source point moves further upstream of the measurement location. Therefore, mean streamlines in the inner region (including the ISL) would tend to straight lines emanating from the virtual source (dashed lines in figure 2) and would be nearly parallel to the wall (see figure 4 b) at high Reynolds numbers as $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ .

Figure 2. Geometrical interpretation of the streamwise length scale $x_{s}$ in the inner layer of a ZPG TBL following Coles (Reference Coles1955). The dashed lines represent the directions of the mean velocity vectors $\langle \boldsymbol{u}_{1}\rangle$ and $\langle \boldsymbol{u}_{2}\rangle$ at two representative heights $y_{1}$ and $y_{2}$ in the inner layer. The dashed–dotted line indicates the mean velocity profile.

Next, one may write the asymptotic expansion (Van Dyke Reference Van Dyke1975) for $Q_{+}$ in the ISL as

(2.12) $$\begin{eqnarray}Q_{+}(y_{+};\unicode[STIX]{x1D6FF}_{+})=\unicode[STIX]{x1D716}_{0}(\unicode[STIX]{x1D6FF}_{+})h_{0}(y_{+})+\unicode[STIX]{x1D716}_{1}(\unicode[STIX]{x1D6FF}_{+})h_{1}(y_{+})+\cdots \,,\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{0},\unicode[STIX]{x1D716}_{1},\ldots$ are gauge functions (capturing $\unicode[STIX]{x1D6FF}_{+}$ dependence of $Q_{+}$ , if any) in decreasing order such that $Q_{+}/\unicode[STIX]{x1D716}_{0}h_{0}\rightarrow 1$ and $\unicode[STIX]{x1D716}_{1}/\unicode[STIX]{x1D716}_{0}\rightarrow 0$ as $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ , and $h_{0},h_{1},\ldots$ are coefficients (capturing $y_{+}$ dependence of $Q_{+}$ ), where $h_{0}$ is a constant of $O(1)$ . It should be noted that for mean velocity $Q_{+}=\langle u_{+}\rangle$ , $\unicode[STIX]{x1D716}_{0},\unicode[STIX]{x1D716}_{1},\ldots$ are constants in view of the law of the wall (2.2). For ZPG TBLs, the lower end of the log region occurs at $y_{+,1}=y_{1}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}\approx 3\sqrt{\unicode[STIX]{x1D6FF}_{+}}$ and the upper end is located at $\unicode[STIX]{x1D702}_{2}=y_{2}/\unicode[STIX]{x1D6FF}=0.15$ , i.e. $y_{+,2}=0.15\unicode[STIX]{x1D6FF}_{+}$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). The mean velocity profile in the log region is given by $\langle u_{+}\rangle =\unicode[STIX]{x1D705}^{-1}\ln (y_{+})+B$ , where $\unicode[STIX]{x1D705}^{-1}$ is the slope in the semi-logarithmic inner co-ordinates and $B$ is the intercept. The maximum slope of the mean velocity profile in the log region occurs at $y_{+,1}$ , and in the limit of $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ it is given by

(2.13) $$\begin{eqnarray}\left.\lim _{\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty }\frac{\text{d}\langle u_{+}\rangle }{\text{d}y_{+}}\right|_{y_{+,1}}=\frac{1}{\unicode[STIX]{x1D705}y_{+,1}}=\lim _{\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty }\frac{1}{3\unicode[STIX]{x1D705}\sqrt{\unicode[STIX]{x1D6FF}_{+}}}\rightarrow 0.\end{eqnarray}$$

Therefore, $\langle u_{+}\rangle =\unicode[STIX]{x1D716}_{0}h_{0}$ is the lowest-order approximation for $\langle u_{+}\rangle$ in the ISL and is independent of $y_{+}$ as well as $\unicode[STIX]{x1D6FF}_{+}$ . For variance of $u_{+}^{\prime }$ , i.e. $Q_{+}=\langle u_{+}^{\prime 2}\rangle$ , a recent work (Monkewitz & Nagib Reference Monkewitz and Nagib2015) suggests that $\langle u_{+}^{\prime 2}\rangle$ in the inner region is likely to be bounded and tends to a constant value (independent of $y_{+}$ ) as $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ . Therefore, to the lowest order, $\langle u_{+}^{\prime 2}\rangle =\unicode[STIX]{x1D716}_{0}h_{0}$ in the ISL. Extending this to the generalized moments $\langle u_{+}^{\prime m}v_{+}^{\prime n}\rangle ^{2/(m+n)}$ , one may expect them to remain bounded and tend to their respective constant values as $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ ; it should be noted that $h_{0}$ and $\unicode[STIX]{x1D716}_{0}$ will, in general, be different for different moments. With this, $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}x\sim \unicode[STIX]{x1D716}_{0}h_{0}/x_{s}$ to the lowest order in the ISL and (2.10) and (2.11) yield

(2.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}y}\sim \frac{x_{s}}{y_{s}}\frac{\unicode[STIX]{x1D716}_{0}h_{0}}{x_{s}}\sim \frac{\unicode[STIX]{x1D716}_{0}h_{0}}{y}.\end{eqnarray}$$

Integration of (2.14) yields $Q_{+}\sim \ln (y)$ , which may be written in inner and outer coordinates respectively as

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{+}=A_{m,n}\ln (y_{+})+B_{m,n}, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{+}=A_{m,n}\ln (\unicode[STIX]{x1D702})+C_{m,n}. & \displaystyle\end{eqnarray}$$

Here, $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ are coefficients that, in general, depend on the Reynolds number $\unicode[STIX]{x1D6FF}_{+}$ . For the mean velocity, $A_{m,n}$ is $\unicode[STIX]{x1D705}^{-1}$ and is a constant for TBLs (Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010b ). Lastly, we may now check a posteriori whether $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}x$ is indeed independent of $y$ . Towards this, we write $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}x=\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}y_{+}[y(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)/\unicode[STIX]{x1D708}]+\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}_{+}[\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x]$ . For the mean velocity, the second term is identically zero since $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}_{+}=0$ due to (2.2); for generalized mixed moments, this term tends to zero at high Reynolds numbers due to $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x\rightarrow 0$ . Use of (2.15) in the first term leads to $\unicode[STIX]{x2202}Q_{+}/\unicode[STIX]{x2202}x=-A_{m,n}/x_{s}$ , which is independent of $y$ .

To summarize, for TBL flows, if one starts with the assertion of zero advection of $Q_{+}$ in the ISL, log variation of $Q_{+}$ follows, and vice versa.

3 Appraisal of experimental data from canonical ZPG TBLs

The physical basis for the analysis presented in § 2 is invariance of a generalized scaled mean-flow quantity $Q_{+}$ along a mean streamline in a given flow. This leads to the consideration of dimensional advection $\text{ADV}(Q_{+})$ being zero. However, in order to compare the zero-advection conditions in different ZPG TBLs, a dimensionless framework is required.

We consider dimensional and dimensionless advection operators $\text{ADV}=u\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x+v\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y$ and $\text{ADV}_{+}=u_{+}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{+}+v_{+}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y_{+}$ respectively. Here, $u_{+}=u/u_{\unicode[STIX]{x1D70F}}$ , $v_{+}=v/u_{\unicode[STIX]{x1D70F}}$ , $x_{+}=xu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ and $y_{+}=yu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}$ ; $x$ is the distance from a suitable virtual origin measured along the wall in the flow direction. For $x/u_{\unicode[STIX]{x1D70F}}(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)\ll 1$ , these operators are related as

(3.1) $$\begin{eqnarray}\text{ADV}=\left(\frac{u_{\unicode[STIX]{x1D70F}}^{2}}{\unicode[STIX]{x1D708}}\right)\text{ADV}_{+},\end{eqnarray}$$

and $\text{ADV}(Q_{+})=0$ is equivalent to $\text{ADV}_{+}(Q_{+})=0$ . Using (3.1) in (2.6), one obtains

(3.2) $$\begin{eqnarray}\text{ADV}_{+}(Q_{+})=f\frac{\text{d}\unicode[STIX]{x1D6FF}_{+}}{\text{d}x_{+}}\frac{\unicode[STIX]{x2202}Q_{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}_{+}},\end{eqnarray}$$

which tends to zero if $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}\rightarrow 0$ . Using (2.9), one obtains

(3.3) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6FF}_{+}}{\text{d}x_{+}}=\unicode[STIX]{x1D705}\left(\unicode[STIX]{x1D6FF}\frac{\text{d}S}{\text{d}x}\right).\end{eqnarray}$$

Jones et al. (Reference Jones, Nickels and Marusic2008) have shown that $\unicode[STIX]{x1D6FF}(\text{d}S/\text{d}x)\rightarrow 0$ as $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ in ZPG TBLs, implying that $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}\rightarrow 0$ as well in the limit of infinite Reynolds number.

Figure 3. (a) Variation of $C_{f}$ and $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}$ with Reynolds number $R_{\unicode[STIX]{x1D6FF}}$ and (b) variation of $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}$ with Reynolds number $\unicode[STIX]{x1D6FF}_{+}$ for the experimental ZPG TBL data of table 1. The lines indicate least-squares power-law curve fits to the data. The vertical dashed–dotted lines indicate demarcations between different regimes discussed in detail in § 3.

We now consider mean velocity data from Princeton, Melbourne and Stanford experiments covering a wide range of Reynolds numbers $1000<\unicode[STIX]{x1D6FF}_{+}<73\,000$ (see table 1) and initially focus on the requirement $x/u_{\unicode[STIX]{x1D70F}}(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)\ll 1$ for (3.1) to hold. To estimate $x$ from experiments, variations of $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}$ and $C_{f}$ with $R_{\unicode[STIX]{x1D6FF}}$ are required. Here, $\unicode[STIX]{x1D703}$ is the momentum thickness, $C_{f}=2(u_{\unicode[STIX]{x1D70F}}/u_{\infty })^{2}$ is the skin-friction coefficient and $R_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x1D6FF}u_{\infty }/\unicode[STIX]{x1D708}$ is the Reynolds number based on $\unicode[STIX]{x1D6FF}$ and $u_{\infty }$ . Figure 3(a) shows that each of $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}$ and $C_{f}$ follows two distinct power-law curve fits corresponding to two different regimes that cover the entire range of Reynolds numbers; these regimes will be discussed in some detail shortly. Substitution of the equations of these curve fits (depending on the regime) into the momentum integral equation $C_{f}/2=\text{d}\unicode[STIX]{x1D703}/\text{d}x$ for ZPG TBLs (White Reference White1994) and integration with the condition $\unicode[STIX]{x1D6FF}=0$ at $x=0$ (virtual origin) yield the expressions $x/\unicode[STIX]{x1D6FF}=7.4375(S\unicode[STIX]{x1D6FF}_{+})^{0.1981}$ and $x/\unicode[STIX]{x1D6FF}=47.0556(S\unicode[STIX]{x1D6FF}_{+})^{0.0508}$ for regimes 1 and 2 respectively (figure 3); the values of $x$ in table 1 are computed using these expressions. Further, $\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x$ may be computed using $\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x=-(u_{\infty }/\unicode[STIX]{x1D6FF}S^{2})\unicode[STIX]{x1D6FF}(\text{d}S/\text{d}x)$ , wherein $\unicode[STIX]{x1D6FF}(\text{d}S/\text{d}x)$ may be estimated using (3.27) from Jones et al. (Reference Jones, Nickels and Marusic2008),

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\frac{\text{d}S}{\text{d}x}=\frac{S}{\unicode[STIX]{x1D705}C_{1}S^{2}-\unicode[STIX]{x1D705}C_{2}S+C_{2}}.\end{eqnarray}$$

It should be noted that $C_{1}$ and $C_{2}$ in (3.4) are integrals of $(u_{\infty }-\langle u\rangle )/u_{\unicode[STIX]{x1D70F}}$ and $(u_{\infty }-\langle u\rangle )^{2}/u_{\unicode[STIX]{x1D70F}}^{2}$ from $\unicode[STIX]{x1D702}=0$ to $\unicode[STIX]{x1D702}=1$ and are universal constants for defect profile similarity. Indeed, for all of the data of table 1, $C_{1}$ and $C_{2}$ are found to be fairly constant to within $\pm 7\,\%$ and $\pm 10\,\%$ of the respective average values of $4.118$ and $27.555$ . Table 1 shows that all flows meet the condition $x/u_{\unicode[STIX]{x1D70F}}(\text{d}u_{\unicode[STIX]{x1D70F}}/\text{d}x)\ll 1$ quite well, i.e. (3.1) is valid.

Next, $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}$ is computed using (3.3) and (3.4) with $\unicode[STIX]{x1D705}=0.4$ and is plotted in figure 3(b) against $\unicode[STIX]{x1D6FF}_{+}$ . Again, two distinct power-law curve fits characterize the regimes,

(3.5) $$\begin{eqnarray}\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FF}_{+}}{\text{d}x_{+}} & = & \displaystyle 0.0345\unicode[STIX]{x1D6FF}_{+}^{-0.1396},\quad 1000\leqslant \unicode[STIX]{x1D6FF}_{+}<7000,\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & = & \displaystyle 0.0224\unicode[STIX]{x1D6FF}_{+}^{-0.0902},\quad 7000\leqslant \unicode[STIX]{x1D6FF}_{+}<70\,000,\end{eqnarray}$$

with an unmistakable switchover around $\unicode[STIX]{x1D6FF}_{+}=7000$ , which corresponds very well with the onset of a discernible log scaling region for even moments; see figures 4(a) and 11(a,c,e) from Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015) and figures 1 and 3 from Meneveau & Marusic (Reference Meneveau and Marusic2013). The regimes of figure 3(a,b) are identical and indicate consistent behaviour of data. The first regime, characterized by (3.5), begins around $\unicode[STIX]{x1D6FF}_{+}=1000$ , where a discernible extent of mean velocity log scaling becomes apparent and continues to grow with the Reynolds number. Variances, higher-order moments and the Reynolds shear stress do not exhibit log scaling in this regime. This regime is characterized by a set of power-law curve fits that describe variations of $C_{f}$ , $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}$ and $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}$ , with Reynolds number $\unicode[STIX]{x1D6FF}_{+}$ (figure 3). Around $\unicode[STIX]{x1D6FF}_{+}=7000$ , discernible extents of log scaling of even moments and the Reynolds shear stress become apparent, in addition to the mean velocity, marking the onset of the second regime. Consistent with this, the power-law curve fits of $C_{f}$ , $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}$ and $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}$ also show a switchover, as seen in figure 3. Finally, the third regime begins around $\unicode[STIX]{x1D6FF}_{+}=70\,000$ , beyond which odd-order moments also start to follow log scaling in addition to already established log variations of other quantities. The evidence towards the occurrence of the third regime is very limited (see the question marks in figure 3) since the present upper limit of the Reynolds number in laboratory experiments is close to $\unicode[STIX]{x1D6FF}_{+}=70\,000$ and therefore only one dataset (VHS7 from table 1 and figure 1 b) shows the emergence of the log scaling of the third-order moment. Therefore, further high- $Re$ TBL experiments are required to confirm and quantify the asymptotic log scaling of all moments in extreme-Reynolds-number situations.

In regimes 1 and 2 of figure 3(b), $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}$ decreases with $\unicode[STIX]{x1D6FF}_{+}$ , indicating that $\text{d}\unicode[STIX]{x1D6FF}_{+}/\text{d}x_{+}\rightarrow 0$ as $\unicode[STIX]{x1D6FF}_{+}\rightarrow \infty$ , which in turn leads to $\text{ADV}_{+}(Q_{+})\rightarrow 0$ .