2. Intermediate scaling for mean velocity and variance

The present analysis is based on the nanoscale thermal anemometry probe (NSTAP) data measured in the Princeton Superpipe (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012). Figure 1 shows scaling of the streamwise variance with length scale, $y_m^+ = 3.5 \sqrt {Re_{\tau }}$, and velocity scale, $u_m = \sqrt {\overline {u^2}} (y=y_m)$. As can be seen, there is an excellent collapse of the profiles in the region around $y / y_m = 1$ for two decades in $Re_\tau$, $1985 \le Re_{\tau } \le 98\,190$: this is the motivation for using $u_m$ as the intermediate velocity scale. At the highest Reynolds numbers, the scaled region extends from $0.15 \leq y/y_m \leq 15$ and, as expected, the profiles peel off from this trend on both sides of $y/y_m = 1$ as $Re_{\tau }$ decreases. The choice of constant used in the definition of $y_m^+$ is guided by the coefficients for $\sqrt {Re_{\tau }}$ used in previous definitions of the mesolayer location, e.g. $2\sqrt {Re_{\tau }}$ (Sreenivasan & Sahay Reference Sreenivasan and Sahay1997), or in determining the lower bound of the inertial sublayer, $3\sqrt {Re_{\tau }}$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). Here, a slightly higher value, 3.5, is chosen to provide a better $Re_{\tau }$ scaling of the variance profiles for the pipe as well as boundary layer data (not shown); see Diwan & Morrison (Reference Diwan and Morrison2019). Note that the qualitative (and, to certain extent, quantitative) nature of the results is unaffected by the precise choice of this constant. Taking $U_m$ as the mean velocity at $y = y_m$, figure 2 shows, in ‘defect’ form, the corresponding mean velocity profiles scaled on $u_m$. Excellent scaling is apparent around $y/y_m = 1$ for mean velocity also. This scaling for the mean velocity and variance suggests the existence of a distinct, asymptotic intermediate layer lying between the classical inner and outer layers. This implies that there exists two overlap regions: one between the inner and intermediate layers (‘Overlap Layer I’) and the other between the intermediate and outer layers (‘Overlap Layer II’); see also Afzal (Reference Afzal1982). We choose the velocity scales in the inner and outer layers as $u_i$ and $u_o$, respectively, which in general need not be the same as $u_m$. This is in contrast to the earlier formulations (Afzal Reference Afzal1982; Sreenivasan & Sahay Reference Sreenivasan and Sahay1997; Klewicki Reference Klewicki2013), which used the same velocity scale, $u_{\tau }$, for all the layers considered. The use of $u_m$ instead of $u_{\tau }$ in our formulation enables the demonstration of Reynolds-number similarity in the intermediate layer of the pipe (figures 1, 2). Note that the five-layer description based on spectral characteristics of the streamwise velocity proposed by Vallikivi, Ganapathisubramani & Smits (Reference Vallikivi, Ganapathisubramani and Smits2015) is different in spirit to the present formulation which, including the two overlap layers, also proposes a total of five layers. Next, we carry out a matching analysis using the method used by Zagarola & Smits (Reference Zagarola and Smits1998). For Overlap Layer I, the inner and intermediate scaling laws are written as

(2.1a,b)\begin{equation} U^+= f(y^+), \quad \frac {U-U_m} {u_m} = g(\zeta), \end{equation}

where $\zeta = y/y_m$. Asymptotic matching of the velocity gradient for the inner and intermediate layers gives

(2.2)\begin{equation} y^+ f^{\prime} (y^+) = \varLambda_{I} \zeta g^{\prime}(\zeta), \end{equation}

where $\varLambda _{I} = u_m / u_{\tau }$ and $(^\prime )$ indicates the derivative with respect to the corresponding independent variable. When $\varLambda _I$ is Reynolds-number dependent, (2.2) does not imply log-law scaling. For this case, Reynolds-number similarity can be achieved by simultaneously matching both velocity and velocity gradient in the overlap region. This results in a power law for the mean velocity:

(2.3)\begin{equation} \left.\begin{gathered} f(y^+) = U^+= C (y^+) ^ \gamma, \\ \frac{U_m}{u_m} + g(\zeta) = \frac{U}{u_m} = C_m \left(\frac{y}{y_m} \right) ^ \gamma, \end{gathered}\right\} \end{equation}

where $\gamma$, $C$ and $C_m$ are constants. Alternatively, when $\varLambda _I$ = constant, a log law is obtained in the overlap region.

Figure 1.

Streamwise variance profiles in a smooth pipe scaled on the intermediate variables $y_m$ and $u_m$; data from Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012). The solid line is the log-law fit.

Figure 2.

Streamwise mean velocity deficit scaled on the intermediate variables. The magenta line indicates a power-law fit and the green line indicates the log-law fit. The inset shows the variation without the power and log law lines.

Figure 3(a) shows the variation of $\varLambda _I$ with Reynolds number, where it continues to increase even at the highest $Re_{\tau }$. This implies that Overlap Layer I is governed by a power law up to $Re_\tau \approx 10^5$. The ratio $U_m/u_m$ shows a much weaker variation with $Re_{\tau }$ than $\varLambda _I$, without a monotonic trend (figure 3b): there is only a $6\,\%$ variation in $U_m/u_m$ for two decades of change in $Re_{\tau }$; this justifies the power law in (2.3). Fitting a power-law curve to $U/u_m$ for $0.06 \leq y/y_m \leq 0.8$, $Re_{\tau } = 98\,190$ gives the power-law constants as $\gamma = 0.14$ and $C_m = 8.51$. Note that $\gamma$ is independent of $Re_{\tau }$, whereas $C_m$ shows a weak $Re_{\tau }$-dependence reflecting the weak trend of $U_m/u_m$ with $Re_{\tau }$ (figure 3b). Using these parameters, the variation of $(U - U_m) / u_m$ is plotted in figure 2 as a magenta line which fits the data quite well in Overlap Layer I. To determine $C$, we separately fit a power law to the inner-scaled data (not shown here) for $Re_{\tau } = 98\,190$ in the corresponding range, $65 \leq y^+ \leq 880$ (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012). This yields the same value of $\gamma =0.14$, with $C = 8.47$. These are close to the values $\gamma = 0.142$ and $C = 8.48$ reported by McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2004). Overlap Layer II is bounded by the intermediate layer and the outer layer. The scaling law for the mean velocity deficit in the outer layer can be written as

(2.4)\begin{equation} \frac {U_{CL}-U} {u_o} = h\left(\frac{y}{R}\right), \end{equation}

where $U_{CL}$ is the pipe centre-line velocity. It can be seen that the character of the mean velocity variation in Overlap Layer II is determined by the velocity-scale ratio, $\varLambda _{II} = u_o / u_m$ (corresponding to $\varLambda _{I}$ in 2.2). Note that although (2.4) is written in a Reynolds-number-invariant form, the appropriate velocity scale, $u_o$, that would result in Reynolds-number similarity in the outer region is still unknown (Morrison et al. Reference Morrison, McKeon, Jiang and Smits2004). The two alternatives for $u_o$ that have been used so far, for the pipe, are $u_{\tau }$ and $U_{CL}-U_b$ (Zagarola & Smits Reference Zagarola and Smits1998), where $U_b$ is the bulk velocity. Choosing $u_o = u_{\tau }$, $\varLambda _{II} = 1 / \varLambda _{I}$, implying (figure 3a) that Overlap Layer II is also governed by a power law for the entire $Re_{\tau }$ range. This would be a surprising result as there has been overwhelming support in favour of the log law. If, on the other hand, we choose $u_o = (U_{CL} - U_b)$, we get $\varLambda _{II} = (U_{CL} - U_b) / u_m$, see figure 3(c), which shows that $\varLambda _{II}$ is a strong function of $Re_{\tau }$, but where, for $Re_{\tau } \gtrsim 10^4$, it shows a much weaker dependence on $Re_{\tau }$. If we assume $\varLambda _{II}$ to be approximately constant for $Re_{\tau } \gtrsim 10^4$, we recover the log law for the mean velocity in Overlap Layer II, which, in terms of $u_m$ and $y_m$, can be written as

(2.5)\begin{equation} \frac{U - U_m}{u_m} = \frac{1}{\kappa_m} \ln \left(\frac{y}{y_m}\right) + A_m. \end{equation}

To obtain $\kappa _m$ and $A_m$, we fit a least-square straight line through the mean velocity data (green solid line in figure 2) for the two highest Reynolds numbers, $Re_{\tau } = 68\,370$ and $Re_{\tau } = 98\,190$, and for $1.2 \leq (y/y_m) \leq 13$, equivalent to $4.2\sqrt {Re_{\tau }} \leq y^+ \leq 0.145 Re_{\tau }$ for $Re_{\tau } = 98\,190$, which is broadly consistent with the range used in Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013). The fit gives the following values for the constants:

(2.6a,b)\begin{equation} \kappa_m = 1.034, \quad A_m = 0.0084. \end{equation}

These represent the Reynolds-number-invariant log-law constants in terms of the intermediate variables for $Re_{\tau } \gtrsim 10^4$. Note that, provided the Reynolds-number similarity in the intermediate layer is ensured, the value of $\kappa _m$ is independent of small changes in the numerical value of the coefficient in the definition of $y_m$. The value of $A_m$, however, is directly dependent on this choice (2.5). Moreover, the range $1.2 \leq (y/y_m) \leq 13$ is valid only for the two highest $Re_{\tau }$. As $Re_{\tau }$ decreases, the range of $y/y_m$ over which a log variation is observed continues to decrease, although $\kappa_m$ and $A_m$ remain the same, as clearly seen in figure 2. To further support this observation, we plot the variation of an ‘indicator function’ (McKeon et al. Reference McKeon, Li, Jiang, Morrison and Smits2004) with $Re_{\tau }$, which is presented as supplementary material to this paper available at https://doi.org/10.1017/jfm.2021.71.

Figure 3.

Variation with $Re_{\tau }$ of (a) $\varLambda _I = u_m / u_{\tau }$, (b) $U_m/u_m$, (c) $\varLambda _{II} = (U_{CL} - U_b) / u_m$, (d) $\kappa$ obtained from (3.1a,b); the values of $\kappa$ for $Re_{\tau } \lesssim 10^4$ do not correspond to the log law and are shown by crosses.

In figure 1, the Reynolds-number similarity of the streamwise variance for $y \approx y_m$ implies that, for $y > y_m$, there should exist a Reynolds-number-invariant log law for the variance as well (scaled on the intermediate variables), which is

(2.7)\begin{equation} \frac {\overline{u^2}} {u_{m}^2} = B^m_1 - A^m_1 \ln \left( \frac{y}{y_m} \right). \end{equation}

To determine $A^m_1$ and $B^m_1$, we fit a least-square straight line through the points in figure 1 (shown as a solid line) in the region $1.2 \leq y/y_m \leq 13$ for the two highest Reynolds numbers; this range is the same as that chosen for fitting the log law for the mean velocity in Overlap Layer II (figure 2). Again the extent of the log region for the variance decreases with decrease in $Re_{\tau }$; see the supplementary material for the indicator-function plots. (The behaviour of $\overline {u^2}$ in Overlap Layer I is beyond the scope of the present work.) This line fit gives the $Re$-invariant log-law constants for the streamwise variance as

(2.8a,b)\begin{equation} A^m_1 = 0.178, \quad B^m_1 = 1.005. \end{equation}

We do not attempt to estimate the uncertainty bounds for $A^m_1$ and $B^m_1$, nor for $\kappa _m$ and $A_m$, (2.6a,b) here, as their precise numerical values are not relevant for our key conclusions.

3. Consequences for the von Kármán and Townsend–Perry ‘constants’

The classical log-law constants for the mean velocity and variance can be readily expressed in terms of the constants obtained from the intermediate-scaled log laws. For the mean velocity, this relation can be written as (1.1), (2.5)

(3.1a,b)\begin{equation} \kappa = \frac{\kappa_m} {(u_m / u_{\tau})},\quad A = \frac{u_m}{u_{\tau}} \left\{\left[\frac{U_m}{u_m} + A_m\right] - \frac{1}{\kappa_m} \ln (y_m^+) \right\}. \end{equation}

The value of $\kappa$ obtained from (3.1a,b) is plotted in figure 3(d); see also table 1. For $Re_{\tau } > 2 \times 10^4$, $\kappa$ falls within the range $0.4 \pm 0.02$ (Bailey et al. Reference Bailey, Vallikivi, Hultmark and Smits2014), shown as dashed lines in the figure; the trend exhibited by $\kappa$ within the band reflects the variation of $u_m/u_{\tau }$ with $Re_{\tau }$ (figure 3a; 3.1a,b). For $Re_{\tau } < 10^4$, the values of $\kappa$ are found to be much higher than those which could be reasonably associated with a log law. These are denoted as crosses in figure 3(d), including $\kappa =0.43$ at $Re_{\tau } = 10\,480$ which falls beyond the band of $0.4 \pm 0.02$. This suggests that the mean velocity profile in Overlap Layer II is better described by a power law at lower $Re_{\tau }$, as implied by the strong Reynolds-number dependence of $\varLambda _{II}$ for $Re_{\tau } \lesssim 10^4$ (figure 3c). Note that Overlap Layer I is governed by power law for all $Re_{\tau }$ considered here. For $Re_{\tau } < 10^4$, the two overlap layers may not be entirely distinct and therefore the two power-law profiles may appear indistinguishable (figure 2). These results are consistent with those in Zagarola & Smits (Reference Zagarola and Smits1998) and McKeon et al. (Reference McKeon, Li, Jiang, Morrison and Smits2004).

Table 1.

The log-law constants for the mean velocity ($\kappa$ and $A$) and variance ($A_1$ and $B_1$) obtained from (3.1a,b) and (3.2a,b) using the log fits in (2.5) and (2.7). For $Re_{\tau } \leq 10,480$, $\kappa$ and $A$ are shown as ‘$\times$’ as log law is not expected in Overlap Layer II (figure 3d).

The presence of a power law in Overlap Layer I and of the log law in Overlap Layer II, for $Re_{\tau } > 2 \times 10^4$, supports the observation by Zagarola & Smits (Reference Zagarola and Smits1998) (see also Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012) that, at high $Re_{\tau }$, the mean velocity initially follows a power law, with a log law at larger $y$. Note that in their analysis, the power and log laws share the same overlap region, whereas in the present three-layer formulation they occupy two different overlap regions. This provides an explanation for the coexistence of the power and log-law profiles in the pipe flow at a given (and sufficiently large) $Re_{\tau }$. Furthermore, since the length scale for the intermediate layer is $\propto \sqrt {Re_{\tau }}$, the lower limit for the log law for the mean velocity should be Reynolds-number dependent, rather than constant in wall variables: see the discussion in Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) and Bailey et al. (Reference Bailey, Vallikivi, Hultmark and Smits2014).

For the variance, the classical constants, $A_1$ and $B_1$ (1.2), can be expressed in terms of $A^m_1$ and $B^m_1$ (2.7) as

(3.2a,b)\begin{equation} A_1 = A_1^m \left( \frac {u_m^2} {u_{\tau}^2} \right),\quad B_1 = \left[A_1^m \ln \left(\frac {y_m} {R} \right) + B_1^m \right] \left( \frac {u_m^2} {u_{\tau}^2} \right). \end{equation}

Values of $A_1$ and $B_1$ calculated from (3.2a,b) (with $A^m_1 = 0.178$ and $B^m_1 = 1.005$) are included in table 1; a clear trend in $A_1$ and $B_1$ with respect to $Re_{\tau }$ is evident. Figure 4 shows the log-law fits to the variance, in wall variables, obtained by using $A_1$ and $B_1$ from table 1. As can be seen, the log fits inferred from (3.2a,b) show a good match with the measured profiles in the intermediate region, over the entire $Re_{\tau }$ range. This leads us to conclude that the Townsend–Perry ‘constant’, $A_1$, actually shows a systematic dependence on $Re_{\tau }$ even for $Re_{\tau } > 10^4$. This is due to the fact that $A^m_1$ and $B^m_1$ are $Re_{\tau }$-invariant and that $u_m/u_{\tau }$ (figure 3a) and $y_m/R\ (=3.5/\sqrt {Re_{\tau }})$ show a continuous dependence on $Re_{\tau }$. Furthermore, the values of $A_1$ in table 1 are entirely consistent, at corresponding Reynolds numbers, with those in Perry et al. (Reference Perry, Henbest and Chong1986) ($A_1 = 0.9$ for $Re_{\tau } \leq 3900$) and Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012) ($A_1 = 1.25$ for $Re_{\tau } = 98\,190$; Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) reported $A_1 = 1.23\pm 0.05$ for the same $Re_{\tau }$). Hence, the use of the three-layer formulation and intermediate scaling enable us to explain the $Re_{\tau }$-dependence of $A_1$, providing both an explanation for the low values of $A_1$ obtained by Perry et al. (Reference Perry, Henbest and Chong1986), as well as reproducing $A_1 \approx 1.25$ for the high-$Re$ data. Another advantage of this framework is that the extent of the log region for different $Re_{\tau }$ in the classical scaling (figure 4) is automatically determined once the extent of the intermediate-scaled log law (2.7), (3.2a,b) is known. This removes the subjectivity of fitting a log law to the variance data, which has a direct bearing on the determination of $A_1$. The same considerations also apply to the determination of $\kappa$ for the mean velocity (table 1).

Figure 4.

Streamwise variance profiles for the pipe; solid lines are the classical log-law fits using $A_1$ and $B_1$ from table 1.

Note that the values of $A_1$ and $B_1$ are listed for all $Re_{\tau }$ in table 1, whereas those for $\kappa$ and $A$ are listed only for $Re_{\tau } > 2 \times 10^4$, below which we expect a power law. However, there is no general principle to rule out the presence of log law in $\overline {u^2}$ at $Re_{\tau } \lesssim 10^4$ apart from low-Reynolds-number effects. In addition, indicator functions for $\overline {u^2}$ suggest the presence of a log law for the entire range of $Re_{\tau }$. Of course, the numerical values of $\kappa$ and $A_1$ (and also $A$ and $B_1$) in table 1 and their precise variation with $Re_{\tau }$ are contingent on the accuracy of the data used for fitting the intermediate-scaled log laws. Vallikivi (Reference Vallikivi2014) estimates uncertainties in the NSTAP measurement, $U$, to be ${\pm }2.2\,\%$, the corresponding uncertainty in $\overline {u^2}$ being ${\pm }3.0\,\%$. Using the mean momentum equation, the associated error in $\overline {uv}$ (where $v$ is the radial velocity component) is expected to be approximately ${\pm }5\,\%$. These estimates can be usefully compared to the Pitot probe data from McKeon (Reference McKeon2004) which estimates the corresponding uncertainties in $U$ to be ${\pm }0.3\,\%$. However, here obviously one does not have the higher-order statistics provided by the NSTAP probe. Nonetheless, as table 1 shows, the present analysis using the NSTAP data provides estimates of $\kappa$ and $A$ that are consistent with the earlier Pitot probe data from the Superpipe. We therefore expect the three-layer structure to be apparent for both data sets.