3.1. Compatibility of the scaling law in the limit a → ∞

It is readily confirmed that, for a rough flow, if we let a → ∞ with α ₀ = 0 (no intermittency correction) in (2.29), we find that it recovers Strickler's scaling law in a plane TBL flow as ${C_{f0}} = {C_f}(a \to \infty )\sim {({k_s}/\delta )^{1/3}}$ (see, for instance, Ali & Dey Reference Ali and Dey2022). Further, in a smooth flow, if we let a → ∞ with α ₀ = 0 in (2.32), it recovers Blasius’ scaling law in a plane TBL flow as ${C_{f0}}\sim Re_\delta ^{ - 1/4}$ (Ali & Dey Reference Ali and Dey2022). We recall Blasius’ empirical formula for a smooth plane TBL flow as

(3.1)\begin{equation}{C_{f0}} = 4.56 \times {10^{ - 2}}Re_\delta ^{ - 1/4}.\end{equation}

Comparing (2.32) (in the limit a → ∞ with α ₀ = 0) and (3.1), we obtain ${C_s}\sqrt {{C_0}} = 4.56 \times {10^{ - 2}}$. In the subsequent section, it is shown that (2.32) without having an intermittency correction satisfactorily complies with the experimental data and numerical simulations. Therefore, the skin-friction coefficient in a smooth ATBL flow is given by

(3.2)\begin{equation}{C_f} = 4.56 \times {10^{ - 2}}{\left[ {R{e_a}\ln \left( {1 + \frac{\delta }{a}} \right)} \right]^{ - 1/4}}.\end{equation}

The relation (3.2) unveils the functional form given in (1.1). Although (1.1) shows that C_f is a function of both Re_a and δ $/$a, these parameters can be combined into a single parameter as Re_a $\ln$(1 + δ $/$a) (see (3.2)). The above form shows that C_f follows the ‘−1$/$4’ power law scaling with Re_a $\ln$(1 + δ $/$a). In the subsequent section, the comparison of (3.2) with the experimental data and numerical simulations is discussed.

We observe that (3.2) can be derived from (3.1) by substituting δ with a $\ln$(1 + δ $/$a). Consequently, one might assume that (3.2) could have been obtained easily, without requiring an extensive mathematical analysis, by utilising the relationship Re_δ = Re_a(δ $/$a) and then replacing δ with a $\ln$(1 + δ $/$a), following the reasoning presented in (1.2) and (1.3). However, this assumption may be misleading due to the following reasons:

1. (i) First and foremost, (3.2) is derived entirely through a theoretical analysis based on the momentum balance. Blasius’ scaling law is not employed in the derivation process. However, it is revealed that the proposed scaling law derived from the theoretical analysis corroborates Blasius’ scaling law in the limit a → ∞. As a result, Blasius’ scaling law serves as a boundary condition for obtaining the proportionality constant in the limit a → ∞.

2. (ii) Additionally, this study does not consider Blasius’ scaling law as a benchmark and subsequently modifies it for an ATBL flow by merely replacing δ with a $\ln$(1 + δ $/$a). Blasius’ scaling law was not the starting point for the derivation process in any way.

3. (iii) Moreover, it may be emphasised that the proposed scaling law (3.2) of this study presents a precise functional form of (1.1). Afzal & Narasimha (Reference Afzal and Narasimha1976) did not provide an exact power law scaling behaviour similar to (3.2). Therefore, the scaling law (3.2) has not been formerly known. While it was acknowledged that the axisymmetric logarithmic law holds when y is replaced by a $\ln$(1 + y $/$a), this replacement was not specifically extended to the entire ATBL in the work of Afzal & Narasimha (Reference Afzal and Narasimha1976).

4. (iv) Furthermore, even if one were to modify Blasius’ scaling law by replacing δ with a $\ln$(1 + δ $/$a), such a modification, while producing the correct result, would lack scientific relevance. This replacement would give the impression that the ATBL thickness becomes smaller than the plane TBL thickness. However, this is inappropriate when the boundary layer thickness remains identical for both the ATBL and plane TBL flows. The notable change occurs in the contraction of the external length scale rather than a reduction in the boundary layer thickness.

5. (v) Finally, it is worth highlighting that the substitution of δ with a $\ln$(1 + δ $/$a) in Blasius’ scaling law would not offer any information regarding the intermittency correction, as described in (2.32). Therefore, the proposed replacement cannot anticipate the general scaling behaviour including the intermittency correction.

3.2. Comparison with experimental data and numerical simulations

To compare the proposed skin-friction coefficient (see (3.2)) with the experimental data and numerical simulations, we consider the available experimental and numerical database reported in the literature. The experimental data of Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976), Luxton et al. (Reference Luxton, Bull and Rajagopalan1984), Snarski & Lueptow (Reference Snarski and Lueptow1995) and Berera (Reference Berera2004), and the numerical simulations of Neves et al. (Reference Neves, Moin and Moser1994), Woods (Reference Woods2006), Tutty (Reference Tutty2008), Jordan (Reference Jordan2011, Reference Jordan2013) and Monte et al. (Reference Monte, Sagaut and Gomez2011) are used for this purpose. Table 1 shows the typical ranges of parameters Re_a, δ $/$a, Re_δ, and a ⁺ used in previous studies. The considered ranges of Re_a, δ $/$a, Re_δ and a ⁺ can capture all the axisymmetric flow regimes (regimes I, II and III). Figure 5 shows the experimental and numerical data points of the skin-friction coefficient, corresponding to each author given in table 1, on an (Re_a, δ/a) plane. The data points cover wide ranges of Re_a and δ $/$a of 140 ≤ Re_a ≤ 92 310 and 0.16 ≤ δ/a ≤ 300.

Table 1.

Typical ranges of parameters in experimental measurements and numerical simulations.

Figure 5.

Experimental and numerical database of C_f on a (Re_a, δ/a) plane.

Figure 6(a) shows the comparison of the skin-friction coefficient obtained from (3.2) with the experimental data and numerical simulations. To make the comparison, the experimental and numerical data are presented plotting C_f as a function of Re_a ln(1 + δ/a) on logarithmic scales. The ±20 % error band is also shown to understand the quantitative departure of the data from the proposed law (3.2). Overall, the theoretical prediction matches satisfactorily with the data without invoking any intermittency correction α ₀. The value of α ₀ is usually small for a large Re. This suggests that $|3{\alpha_0}/8|\ll 1/4$ (see (2.32)), and therefore, $|{\alpha_0}|\ll 2/3$, in conformity with the usual estimates of α ₀ (Frisch Reference Frisch1995). The data plots on logarithmic scales respect the ‘−1$/$4’ slope as predicted by (3.2) (figure 6a). For a given Re_a $\ln$(1 + δ/a), the relative error in C_f is calculated as $[{C_f}(\,predicted) - {C_f}(observed)]/{C_f}(observed)$. Figure 6(b) shows the relative error (in percentage) as a function of Re_a $\ln$(1 + δ $/$a) on a semi-logarithmic plot. The absolute relative errors for most of the data plots lie within 20 %. However, for a very few data, the absolute relative errors range from 25 % to 45 %.

Figure 6.

(a) Comparison of the theoretical prediction of C_f (subject to ±20 % error band) with the experimental data and numerical simulations. Woods’ (Reference Woods2006) empirical formula is also shown for the comparison. (b) Relative error in C_f versus Re_a $\ln$(1 + δ/a).

The relative error in C_f results from various uncertainties associated with the measurements. In experimental studies, the wall shear stress was usually obtained by using a Preston tube or from a Clauser plot. These techniques depend on the existence of the law of the wall in the near-wall flow region. For cylinders of larger radius, the wall shear stress was measured by using a Preston tube (see, for example, Willmarth et al. Reference Willmarth, Winkel, Sharma and Bogar1976). The calibration of the Preston tube relies on the assumption that the tube is completely submerged within the wall region of a plane TBL flow, wherein the law of the wall is preserved. To get an accurate estimate of the wall shear stress, it was necessary to ensure that the Preston tube radius was smaller than the cylinder radius. Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) found that, for a ratio of Preston tube radius to cylinder radius smaller than 0.3, the wall shear stress measured by all Preston tubes remained the same, corroborating the plane TBL calibration data of Patel (Reference Patel1965). However, they kept the ratio smaller than 0.16 to measure the wall shear stress accurately. The experimental data of Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) for cylinders of larger radius, say $R{e_a}\ln (1 + \delta /a) > {10^4}$, match well with the theoretical prediction (figure 6a), although the data slightly underestimate the theoretical prediction (maximum absolute relative error obtained from figure 6(b) is 17 %). For cylinders of smaller radius, the use of a Preston tube in measuring the wall shear stress remains a problem. This is because, for cylinders of smaller radius, the flow region consistent with the law of the wall in a plane TBL flow becomes significantly thin. To resolve this issue, Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) obtained the near-wall velocity gradient by fitting a curve to the velocity measurements recorded by a hot-wire having a shorter length compared with the cylinder radius. The method is based on the observation that, for a given a, the velocity distribution in the near-wall flow region $(y \ll a)$ follows the law of the wall in a plane TBL flow. This can be ascertained if we let $y/a \ll 1$ in (1.2) and (1.3), which subsequently produce the classical expressions $u_z^ += {y^ + }$ and $u_z^ += (1/\kappa )\ln {y^ + } + B$, respectively. As the accurate estimation of the near-wall velocity gradient is a challenging task, the experimental data of Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) in figure 6(a) for cylinders of smaller radius, say Re_a $\ln$(1 + δ $/$a) < 3000, match poorly with the theoretical prediction (maximum absolute relative error obtained from figure 6(b) is 27 %). Lueptow & Haritonidis (Reference Lueptow and Haritonidis1987) also found that C_f was overestimated by Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) for Re_a $\ln$(1 + δ $/$a) < 3000.

To estimate C_f, Luxton et al. (Reference Luxton, Bull and Rajagopalan1984) adopted a different approach. They considered the skin-friction data of Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) and found that C_f reduces with an increase in Re_θ ranging from 2.95 × 10³ to 2.23 × 10⁴, where Re_θ is the Reynolds number based on the momentum thickness. However, this variation was found to be independent of δ $/$a for δ $/$a > 9.45. These experimental results were used by Luxton et al. (Reference Luxton, Bull and Rajagopalan1984) to obtain C_f. However, Luxton et al. (Reference Luxton, Bull and Rajagopalan1984) acknowledged that such a procedure is susceptible to error. This is also reflected in figure 6(a), where the experimental data of Luxton et al. (Reference Luxton, Bull and Rajagopalan1984) largely overestimate the theoretical prediction (maximum absolute relative error obtained from figure 6(b) is almost 45 %). The skin-friction data of Snarski & Lueptow (Reference Snarski and Lueptow1995) and Berera (Reference Berera2004), obtained from wall shear stress measurements by a Preston tube, have an excellent match with the theoretical prediction (figure 6a). The maximum absolute relative errors corresponding to the data of Snarski & Lueptow (Reference Snarski and Lueptow1995) and Berera (Reference Berera2004) are only 8 % and 6 %, respectively (figure 6b).

The uncertainty in the experimentally observed skin-friction data may also arise due to the improper alignment of the cylinder with respect to the flow direction. As the boundary layer flow is extremely sensitive to the cylinder misalignment, it is important to check not only the geometric alignment of the cylinder producing an axisymmetric flow, but also the boundary layer symmetry. Even an inclination of 1° with respect to the flow direction was found to produce an asymmetric velocity distribution (see, for example, Rao & Keshavan Reference Rao and Keshavan1972). Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) observed that the sensitivity of flow past a cylinder to a small misalignment is greater for cylinders of larger radius than for those of smaller radius.

High-fidelity skin-friction data obtained from numerical simulations support the theoretical prediction reasonably well (figure 6a). Excepting a few data of Jordan (Reference Jordan2011, Reference Jordan2013) subject to a maximum absolute relative error of 34 %, all the simulation data are confined to the ±20 % error band. The maximum absolute relative errors corresponding to the data of Neves et al. (Reference Neves, Moin and Moser1994), Woods (Reference Woods2006), Tutty (Reference Tutty2008) and Monte et al. (Reference Monte, Sagaut and Gomez2011) are 12, 14, 9 and 9 %, respectively (figure 6b). For Re_a $\ln$(1 + δ $/$a) < 3000, the DNS data of Neves et al. (Reference Neves, Moin and Moser1994), Woods (Reference Woods2006) and Tutty (Reference Tutty2008), and the axisymmetric RANS data of Monte et al. (Reference Monte, Sagaut and Gomez2011) match satisfactorily with the theoretical prediction. This suggests that, for Re_a $\ln$(1 + δ $/$a) < 3000, the experimentally observed skin-friction data of Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) and Luxton et al. (Reference Luxton, Bull and Rajagopalan1984) were certainly overestimated. Woods’ (Reference Woods2006) empirical formula (1.6) is also plotted in figure 6(a). For Re_a $\ln$(1 + δ $/$a) > 3000, there is not much significant difference in C_f obtained from the theoretical prediction and Woods’ (Reference Woods2006) empirical formula. However, for Re_a $\ln$(1 + δ $/$a) < 3000, Woods’ (Reference Woods2006) empirical formula departs from the theoretical prediction. This is because of the fact that, for Re_a $\ln$(1 + δ/a) < 3000, Woods’ (Reference Woods2006) empirical formula was affected by the overestimated skin-friction data of Willmarth et al. (Reference Willmarth, Winkel, Sharma and Bogar1976) and Luxton et al. (Reference Luxton, Bull and Rajagopalan1984).

The error and correlation between the theoretically predicted and the observed (experimentally or numerically) skin-friction coefficients can be quantified by introducing the root-mean-square error $||{C_f}|{|_2}$ and the correlation coefficient r. For each dataset, table 2 summarises the $||{C_f}|{|_2}$ and r obtained from Woods’ (Reference Woods2006) empirical formula and the present study. The statistical results suggest that both Woods’ (Reference Woods2006) empirical formula and the present theoretical prediction capture the observed skin-friction coefficient satisfactorily. The comparative study shows that the computed correlation coefficients do not vary significantly. However, compared with Woods’ (Reference Woods2006) empirical formula, the present study has a better performance in capturing the experimental data of Berera (Reference Berera2004) and the numerical simulations of Tutty (Reference Tutty2008) and Monte et al. (Reference Monte, Sagaut and Gomez2011).

Table 2.

Comparison of root-mean-square error $||{C_f}|{|_2}$ and correlation coefficient r obtained from Woods’ (Reference Woods2006) empirical formula and those obtained from the present study. Here, $||{C_f}|{|_2}$ is calculated as $||{C_f}|{|_2} = \sqrt {\sum\limits_{i \,=\, 1}^N {{{(C_{f\kern-0.03em i}^{observed} - C_{f\kern-0.03em i}^{predicted})}^2}} } $, where N is the total number of data points, $C_{f\kern-0.03em i}^{observed}$ is the observed skin-friction coefficient of the ith data and $C_{f\kern-0.03em i}^{predicted}$ is the corresponding theoretical prediction. The correlation coefficient r is obtained as $r = \frac{{\sum\limits_{i \,=\, 1}^N {(C_{f\kern-0.03em i}^{observed} \,-\, \bar{C}_f^{observed})(} C_{f\kern-0.03em i}^{predicted} \,-\, \bar{C}_f^{predicted})}}{{\sqrt {\sum\limits_{i \,= \,1}^N {{{(C_{f\kern-0.03em i}^{observed} \,-\, \bar{C}_f^{observed})}^2}} \sum\limits_{i \,=\, 1}^N {{{(C_{f \kern-0.03em i}^{predicted} \,-\, \bar{C}_f^{predicted})}^2}} } }}$, where $\bar{C}_f^{observed}$ and $\bar{C}_f^{predicted}$ are the mean values of observed and predicted skin-friction coefficients, respectively.

3.3. How does a mild pressure gradient affect the proposed scaling law?

The proposed scaling law (3.2) is applicable for a zero pressure gradient ATBL flow. Indeed, the presence of a mild pressure gradient in experimental and numerical settings can introduce an error between the theoretically predicted and observed skin-friction coefficients. Establishing a zero pressure gradient in typical experimental set-ups can be challenging. Even numerical simulations enforce a mild pressure gradient to suppress the spatial growth of the ATBL (see, for instance, Neves et al. Reference Neves, Moin and Moser1994). Therefore, it is important to identify how the pressure gradient might affect (3.2) to better understand these discrepancies. Although we do not intend to modify the entire analysis by incorporating the effects of the pressure gradient, it is still valuable to present simple calculations to explore the potential amendments that could be made to the scaling law (3.2) in consideration of the pressure gradient.

As discussed in § 2.3, the total shear stress in a fully developed ATBL flow changes radially to balance the longitudinal free stream pressure gradient. Hence, the longitudinal momentum balance produces

(3.3)\begin{equation}\frac{1}{r}\frac{{\partial r\tau}}{{\partial r}} = \frac{{\textrm{d}{{\bar{p}}_\infty }}}{{\textrm{d}z}}.\end{equation}

In general, (3.3) shows that the left-hand side might be a function of r, whereas the right-hand side might be a function of z. This is possible if both sides of (3.3) are equal to a constant. Thus, for a constant free stream pressure gradient, (3.3) is integrated with limits from r = r to r = a. Subsequently, applying the boundary condition τ(r = a) = $\tau_0$ produces

(3.4)\begin{equation}{\tau _0} = \frac{r}{a}\tau - \frac{{{r^2} - {a^2}}}{{2a}}\frac{{\textrm{d}{{\bar{p}}_\infty }}}{{\textrm{d}z}}.\end{equation}

Considering the viscous shear stress to be insignificant, (3.4) is sought at the surface r = a + k_s $/$2. This gives

(3.5)\begin{equation}{\tau _0} = \left( {1 + \frac{{{k_s}}}{{2a}}} \right){\tau _t} - \frac{{{k_s}}}{2}\left( {1 + \frac{{{k_s}}}{{4a}}} \right)\frac{{\textrm{d}{{\bar{p}}_\infty }}}{{\textrm{d}z}}.\end{equation}

Introducing the pressure gradient parameter

(3.6)\begin{equation}\varTheta = \frac{\delta }{{\rho u_\ast ^2}}\frac{{\textrm{d}{{\bar{p}}_\infty }}}{{\textrm{d}z}},\end{equation}

and using ${C_f} = 2{\tau _0}/(\rho U_\infty ^2)$, (3.5) takes the following form:

(3.7) \begin{equation} {C_f} = \frac{{2\left( {1 + \dfrac{{{k_s}}}{{2a}}} \right)\dfrac{{{\tau _t}}}{{\rho U_\infty ^2}}}}{{1 + \dfrac{{{k_s}}}{{2\delta }}\left( {1 + \dfrac{{{k_s}}}{{4a}}} \right)\varTheta }}.\end{equation}

For a smooth ATBL flow with a zero pressure gradient, the numerator of (3.7) becomes identical to (3.2) as derived previously (also see (2.13)). However, for a mild pressure gradient, C_f can be derived from (3.7) by applying the limit k_s = η. Using the relationship $\eta /\delta = {c_\eta }Re_\delta ^{ - 1/4}$ (evident from (2.30)), where c_η is a constant of order one, and after some algebra, we obtain

(3.8) \begin{equation} {C_f} = \frac{{4.56 \times {{10}^{ - 2}}{{\left[ {R{e_a}\ln \left( {1 + \dfrac{\delta }{a}} \right)} \right]}^{ - 1/4}}}}{{1 + \dfrac{{{c_\eta }}}{2}Re_\delta ^{ - 1/4}\varTheta }},\end{equation}

which reveals that (3.2) needs to be corrected by a factor ${(1 + {c_\eta }Re_\delta ^{ - 1/4}\varTheta /2)^{-1}}$. In fact, (3.8) shows that C_f follows an inverse relationship with $\varTheta$. This conclusion is in qualitative agreement with the detailed analysis of Kumar & Mahesh (Reference Kumar and Mahesh2018b). Importantly, for a zero pressure gradient ($\varTheta$ = 0), (3.8) recovers the derived scaling law (3.2). It is worth noting that (3.8) remains valid as long as the pressure gradient is mild. In other words, the pressure gradient does not significantly affect the flow structure and the velocity distribution within the boundary layer, so that the axisymmetric logarithmic law is preserved. Importantly, for a favourable pressure gradient (that is, $\varTheta$ is negative), the correction factor in (3.8) is larger than unity, resulting in an increased skin-friction coefficient. This observation is in conformity with earlier studies of Patel et al. (Reference Patel, Nakayama and Damian1974) and Afzal & Narasimha (Reference Afzal and Narasimha1976), who reported that the effects of a favourable pressure gradient are qualitatively similar to those of transverse curvature. However, in the absence of a pressure gradient, (3.2) enables us to isolate the effects of transverse curvature on the skin-friction coefficient.

3.4. Sources of uncertainties

It is crucial to discuss the potential sources of uncertainties stemming from various assumptions made during the derivation. These are listed below.

1. (i) The assumption that the momentum transfer is solely governed by turbulent eddies lying in the inertial range may overlook other factors that could contribute to the overall momentum transfer process. Additionally, the determination of u_ℓ relies on the second-order transverse structure function, which obeys Kolmogorov's two-thirds law (2.21). This law is valid for an idealised scenario, particularly when the turbulence is homogeneous and isotropic. Although the literature has demonstrated the applicability of Kolmogorov's laws to the non-homogeneous and anisotropic TBL flows (Saddoughi & Veeravalli Reference Saddoughi and Veeravalli1994), their suitability for the ATBL flows needs to be scrutinised further to identify any possible errors that might arise in the derived scaling law.

2. (ii) While the axisymmetric logarithmic law (1.3) is assumed to hold in the ATBL flows, experimental evidence suggests that it may not be strictly valid in regime III, where the logarithmic layer can decay and exhibit negative curvature. The determination of Reynolds shear stress requires an estimation of the near-wall velocity gradient (see (2.15)). However, the velocity gradient (2.16) derived from the differentiation of (1.3) might not be equally valid for all the ATBL flow regimes. Using this estimation without considering potential variations in the velocity distribution among different regimes is expected to affect the final scaling law (see (2.32) or (3.2)). Indeed, it is unknown how this consideration might affect the scaling law. This is because the precise impact of the transverse curvature on the velocity distribution in the ATBL flows is not yet well understood, as there is currently no consensus on how it affects the various ATBL flow regimes.

3. (iii) The analysis successfully recovers Strickler's scaling law, which is valid for a rough plane TBL flow. In the derivation, it is assumed that ${C_u}C_\varepsilon ^{1/3}$ is independent of k_s $/$δ and remains constant in (2.26). While the former assumption is valid, it should be acknowledged that the latter assumption may not always hold true. For an ATBL flow, ${C_u}C_\varepsilon ^{1/3}$ might be a weak function of the transverse curvature, say a ⁺. In this case, as a → ∞, ${C_u}C_\varepsilon ^{1/3}$ would no longer depend on a ⁺ and becomes a constant. However, the specific form of this unknown function and its impact on the derived scaling law are not known. Consequently, this introduces some uncertainty into the derived scaling law due to the presence of this unknown function.

4. (iv) For a smooth plane TBL flow, the present formulation successfully recovers Blasius’ empirical formula. In the derivation, it is assumed that the quantity ${C_s}\sqrt {{C_0}}$ remains constant. However, for an ATBL flow, this quantity may be a weak function of a ⁺. As a → ∞, ${C_s}\sqrt {{C_0}}$ may become independent of a ⁺ attaining a constant value. This weak dependency on a ⁺ is neglected in the derivation for the sake of simplicity. The specific form relating ${C_s}\sqrt {{C_0}}$ to a ⁺ is unknown and its impact on the derived scaling law is uncertain. Therefore, the uncertainty arising from this unknown function should be taken into account when interpreting and applying the derived scaling law.

It is crucial to recognise that all the abovementioned factors, including assumptions, limitations and unknown functions, can contribute to uncertainties in the derived scaling law. Each source of error introduces its own level of uncertainty and these uncertainties can propagate to the final formula. Indeed, quantifying uncertainties in the final formula remains a challenging task. It often requires detailed sensitivity analyses and rigorous validation against experimental or numerical data to assess the magnitude and impact of each source of uncertainty.

3.5. Comparison of C_f obtained from various methods

Berera (Reference Berera2004) reported comparison of the skin-friction coefficient obtained from experimental measurements with those obtained from the integral equation and the Clauser fit. Table 3 provides the five experimental conditions of Berera (Reference Berera2004). The skin-friction coefficients, corresponding to each experimental condition, obtained from the experimental measurements of Berera (Reference Berera2004), the integral equation and the Clauser fit are tabulated. Furthermore, for the given Re_a and δ $/$a, the skin-friction coefficients obtained from Woods’ (Reference Woods2006) empirical formula (see (1.6)) and the present study (see (3.2)) are shown. The comparison shows that the skin-friction coefficients obtained from the present study are in satisfactory agreement with those obtained from the experimental measurements, the Clauser fit and Woods’ (Reference Woods2006) empirical formula. However, the skin-friction coefficients obtained from the integral equation significantly differ from those estimated using other methods.

Table 3.

Comparison of the skin-friction coefficients obtained using different methods for five experimental conditions of Berera (Reference Berera2004). The errors, expressed in percentage, are calculated as $(C_f^{observed} - C_f^{predicted})/C_f^{predicted}$.

It is interesting to identify the trend of the data plots obtained from various methods with Re_a $\ln$(1 + δ $/$a). The relationship (3.2) predicts that ${C_f}{[R{e_a}\ln (1 + \delta /a)]^{1/4}}$ remains independent of Re_a $\ln$(1 + δ $/$a) and attains a constant value (= 0.0456). Figure 7 shows the data plots of ${C_f}{[R{e_a}\ln (1 + \delta /a)]^{1/4}}$ obtained from various methods as a function of Re_a ln(1 + δ $/$a). Except for the data plots obtained from the integral equation, the other data plots respect the ‘−1/4’ power law scaling behaviour by following the horizontal line anticipated from this study. The data plots obtained from the experimental measurements of Berera (Reference Berera2004), the Clauser fit and Woods’ (Reference Woods2006) empirical formula are in satisfactory agreement with the prediction of the present study (subject to the ±5 % error band), whereas those obtained from the integral equation are prone to errors.

Figure 7.

${C_f}{[R{e_a}\ln (1 + \delta /a)]^{1/4}}$ versus Re_a $\ln$(1 + δ $/$a) obtained using different methods.

3.6. Scaling of C_f with a ⁺ and δ $/$a

The functional form (1.1) or (3.2) shows that C_f is a function of two fundamental parameters, namely Re_a and δ $/$a. Now, we recall various axisymmetric flow regimes (see § 1.2), which were classified based on a ⁺ and δ $/$a. In order to express C_f as a function of a ⁺ and δ $/$a, we use the relation $R{e_a} = {a^ + }\sqrt {2/{C_f}} $. Subsequently, (3.2) is transformed into the following expression:

(3.9)\begin{equation}{C_f} = 2.66 \times {10^{ - 2}}{\left[ {{a^ + }\ln \left( {1 + \frac{\delta }{a}} \right)} \right]^{ - 2/7}},\end{equation}

which shows that the parameters a ⁺ and δ $/$a can be combined into a single parameter as a ⁺ $\ln$(1 + δ $/$a). The above form shows that C_f follows the ‘−2$/$7’ power law scaling with a ⁺ $\ln$(1 + δ $/$a). In fact, (3.9) is not straightforward to use, because a ⁺ is in wall units. Therefore, the wall shear stress remains implicit in the parameter a ⁺ $\ln$(1 + δ $/$a). As a result, a trial-and-error method is required to obtain C_f for the given U_∞, a and δ. By contrast, (3.2) is more convenient to use, as C_f is directly related to Re_a and δ/a.

To check the performance of the predicted skin-friction coefficient (see (3.2) or (3.9)) in various axisymmetric flow regimes, the entire dataset (both experimental data and numerical simulations as considered in figure 5) is divided into three categories following the definitions of the axisymmetric flow regimes (§ 1.2). Figure 8 shows the comparison of the predicted and observed skin-friction coefficients in regimes I, II and III. The overall match between the predicted and observed skin-friction coefficients is satisfactory subject to the ±20 % error band. In this context, we remark that the proposed law of the skin-friction coefficient is capable of capturing all the axisymmetric flow regimes without making any assumption for an individual flow regime. However, it is worth noting that regime III displays the maximum error, primarily because certain data points in regime III deviate from the line of perfect agreement. This discrepancy can be attributed to the lower values of a ⁺. The ATBL laws employed in the present analysis rely on the similarity assumption and overlap argument in the inertial range, which are applicable for large Reynolds numbers. Moreover, large Reynolds numbers create a wide separation between large and small scales of motion, enabling a prolonged inertial range in the energy spectrum of turbulent eddies. Incorporating the effects of low Reynolds numbers into the proposed scaling law is beyond the scope of this work.

Figure 8.

Predicted versus observed C_f in various axisymmetric flow regimes. The line of perfect agreement and the ±20 % error band are also shown.

3.7. Behaviour of C_f for δ/a < 1

Combining (3.1) and (3.2) yields the relative skin-friction coefficient, $({C_f} - {C_{f0}})/{C_{f0}}$ as

(3.10)\begin{equation}\frac{{{C_f} - {C_{f0}}}}{{{C_{f0}}}} = {\left[ {\frac{a}{\delta }\ln \left( {1 + \frac{\delta }{a}} \right)} \right]^{ - 1/4}} - 1,\end{equation}

which represents the difference between the skin-friction coefficients in the ATBL and plane TBL flows with respect to the skin-friction coefficient in a plane TBL flow. Especially, for δ $/$a < 1 (axisymmetric flow regime I), (3.2) can be approximated using the Taylor expansion as

(3.11)\begin{equation}\left. \begin{array}{l@{}} {C_f} = 4.56 \times {10^{ - 2}}{\left[ {R{e_a}\left( {\dfrac{\delta }{a} - \dfrac{1}{2}\dfrac{{{\delta^2}}}{{{a^2}}} + \dfrac{1}{3}\dfrac{{{\delta^3}}}{{{a^3}}} - \cdots } \right)} \right]^{ - 1/4}}\\ \Rightarrow \textrm{ }{C_f} = 4.56 \times {10^{ - 2}}{\left[ {R{e_\delta }\left( {1 - \dfrac{1}{2}\dfrac{\delta }{a} + \dfrac{1}{3}\dfrac{{{\delta^2}}}{{{a^2}}} - \cdots } \right)} \right]^{ - 1/4}}\\ \Rightarrow \textrm{ }{C_f} \approx 4.56 \times {10^{ - 2}}Re_\delta^{ - 1/4}\left( {1 + \dfrac{1}{8}\dfrac{\delta }{a}} \right). \end{array} \right\}\end{equation}

Combining (3.1) and (3.11) produces the relative skin-friction coefficient for δ $/$a < 1 as follows:

(3.12)\begin{equation}\left. \begin{array}{l@{}} \dfrac{{{C_f}}}{{{C_{f0}}}} = 1 + \dfrac{1}{8}\dfrac{\delta }{a}\\ \Rightarrow \textrm{ }\dfrac{{{C_f} - {C_{f0}}}}{{{C_{f0}}}} = \dfrac{1}{8}\dfrac{\delta }{a}. \end{array} \right\}\end{equation}

The relation (3.12) shows that $({C_f} - {C_{f0}})/{C_{f0}}$ varies linearly with δ $/$a, which can be called the linear law of the relative skin-friction coefficient for δ $/$a < 1. The existence of the linear law is shown in figure 9, where $({C_f} - {C_{f0}})/{C_{f0}}$ obtained from (3.10) is plotted as a function of δ $/$a on logarithmic scales. In addition, (3.12) is plotted for δ $/$a < 1. For δ $/$a = 1, the difference between the relative skin-friction coefficients obtained from (3.10) and (3.12) is only 2.9 %. The linear law appears to strictly hold in the range δ $/$a < 0.2 (figure 9), for which the behaviour of (3.10) becomes identical to (3.12). Figure 9 shows that, for δ $/$a = 0.2, $({C_f} - {C_{f0}})/{C_{f0}}$ obtained from (3.10) and (3.12) are 2.34 % and 2.5 %, respectively. The emergence of the linear law divides the axisymmetric flow regime I into two sub-regimes. The first sub-regime corresponds to δ $/$a < 0.2, where the linear law is preserved. The second sub-regime is the transition from δ $/$a < 0.2 to δ $/$a > 1 regions. It is important to mention that Kumar & Mahesh (Reference Kumar and Mahesh2018a) obtained the relative skin-friction coefficient as

(3.13)\begin{equation}\frac{{{C_f} - {C_{f0}}}}{{{C_{f0}}}} = \frac{\theta }{a},\end{equation}

Figure 9.

Plot of $({C_f} - {C_{f0}})/{C_{f0}}$ versus δ $/$a showing the existence of the linear law $[({C_f} - {C_{f0}})/{C_{f0}}\; \sim \delta /a]$ for δ < a.

where θ is the momentum thickness. Using numerous experimental data compiled by Jordan (Reference Jordan2014a) along with the simulation database, Kumar & Mahesh (Reference Kumar and Mahesh2018a) considered δ $/$θ ≈ 7.2. Therefore, (3.13) becomes

(3.14)\begin{equation}\frac{{{C_f} - {C_{f0}}}}{{{C_{f0}}}} = \frac{1}{{7.2}}\frac{\delta }{a}.\end{equation}

The relation (3.14) also shows that $({C_f} - {C_{f0}})/{C_{f0}}$ varies linearly with δ $/$a with a data-driven proportionality constant being different from (3.12). The relationship (3.14) of Kumar & Mahesh (Reference Kumar and Mahesh2018a), also plotted in figure 9, corresponds closely with (3.12) for δ $/$a < 1. However, for δ $/$a > 1, $({C_f} - {C_{f0}})/{C_{f0}}$ obtained from (3.14) substantially departs from that obtained from (3.10). The reason is that (3.13) is legitimate for a developing ATBL flow, unlike the present study, where the boundary layer is fully developed. Moreover, while deriving (3.13), it was assumed that the boundary layer growth dδ $/$${\rm d}\kern0.7pt x$ is identical for both the plane TBL and ATBL flows. This assumption does not always hold true, as identified by Kumar & Mahesh (Reference Kumar and Mahesh2018a).

The relative skin-friction coefficients given in (3.10) and (3.12) can also be expressed in terms of the Reynolds numbers as

(3.15)\begin{equation}\frac{{{C_f} - {C_{f0}}}}{{{C_{f0}}}} = {\left[ {\frac{{R{e_a}}}{{R{e_\delta }}}\ln \left( {1 + \frac{{R{e_\delta }}}{{R{e_a}}}} \right)} \right]^{ - 1/4}} - 1,\end{equation}

and

(3.16)\begin{equation}\frac{{{C_f} - {C_{f0}}}}{{{C_{f0}}}} = \frac{1}{8}\frac{{R{e_\delta }}}{{R{e_a}}},\end{equation}

respectively. The relation (3.12) or (3.16) can also be discussed from a different perspective. Consider a thought experiment of an ATBL flow having a fixed Re_δ. If the cylinder radius a (or Re_a) keeps on increasing by maintaining Re_δ constant, the skin-friction coefficient eventually approaches the plane TBL limit as a → ∞. This poses a question: For a fixed Re_δ (< Re_a), how does the skin-friction coefficient in an ATBL flow scale with Re_a to attain the plane TBL limit? In other words, how does the relative skin-friction coefficient vary with Re_a to approach the vanishing limit? This can be answered from (3.16), which suggests that, for a given Re_δ (< Re_a), the relative skin-friction coefficient obeys an inverse scaling law with Re_a. The result of this thought experiment is shown in figure 10. The value of $({C_f} - {C_{f0}})/{C_{f0}}$ obtained from (3.15) is plotted as a function of Re_a on logarithmic scales for different values of Re_δ (5000 to 10⁵). For each value of Re_δ, the inverse scaling law is preserved in the range δ $/$a < 1. For δ $/$a = 1 (that is, Re_δ = Re_a), $({C_f} - {C_{f0}})/{C_{f0}}$ obtained from either (3.16) or (3.15) is $\ln$^−1/4 2 − 1 = 9.6 % (figure 10).

Figure 10.

Plot of $({C_f} - {C_{f0}})/{C_{f0}}$ versus Re_a for Re_δ = 5000 to 10⁵ at an interval of 5000 (bottom to top lines) showing the existence of the inverse scaling law ($({C_f} - {C_{f0}})/{C_{f0}}\sim 1/R{e_a}$ for a given Re_δ) for δ < a.