3.1. Relations among the pressure gradient and the skin-friction coefficients

By recalling that the velocity statistics depend on $r$ only and by integrating the averaged radial momentum equation, the averaged streamwise pressure gradient $\varPi _x\lt 0$ is shown to be a constant. The averaged $x$ -momentum equation becomes

(3.1) \begin{align} -\varPi _x = \frac {1}{r}\frac {\mathrm{d}}{\mathrm{d} r}\left (r \overline {u'v'} - \frac {r}{Re_b} \frac {\mathrm{d} \overline {u}}{\mathrm{d} r} \right ). \end{align}

The relation among the skin-friction coefficients $C_{\kern-1pt f,i}, C_{\kern-1pt f,o}$ and the pressure gradient $\varPi _x$ is obtained by multiplying (3.1) by $r$ and by integrating along $r$ between $R_i$ and $R_o$ . As the Reynolds shear stresses $\overline {u'v'}$ vanish on the cylinders because of the no-slip condition, we obtain (Walker et al. Reference Walker, Whan and Rothfus1957)

(3.2) \begin{align} \varPi _x = \dfrac {R_o C_{\kern-1pt f,o}-R_i C_{\kern-1pt f,i}}{R_o^2 - R_i^2}. \end{align}

Henceforth, the terminology Reynolds stresses is adopted to denote $\overline {u'v'}$ . Substituting $\mathcal{D}^*=2 \pi L^* \mu ^* (R_i^* \mathrm{d} \overline {u}^*/\mathrm{d} r^*|_{r^*=R_i^*} - R_o^* \mathrm{d} \overline {u}^*/\mathrm{d} r^*|_{r^*=R_o^*} )$ (where $\mu ^*$ is the dynamic viscosity of the fluid and $L^*$ is the length of the annulus) and $\mathcal{A}^*=2 \pi L^* (R_i^*+R_o^* )$ into the definition of the average $C_{\kern-1pt f}$ leads to

(3.3) \begin{align} C_{\kern-1pt f} = \frac {R_i C_{\kern-1pt f,i} - R_o C_{\kern-1pt f,o}}{R_i+R_o}. \end{align}

By comparing (3.2) and (3.3) and using $R_o-R_i=1$ , it follows that $C_{\kern-1pt f}=-\varPi _x$ .

3.2. Laminar-flow quantities

The laminar solution is obtained by integrating the $x$ -momentum equation (3.1) along $r$ with $\overline {u'v'}=0$ and by using the no-slip conditions, $\overline {u} = 0$ at $r=R_i, R_o$ . The solution is

(3.4) \begin{align} \overline {u}_{lam}(r) = \dfrac {\varPi _{x,lam} Re_b}{4} \left [r^2 - R_i^2 + \frac {R_o^2 - R_i^2}{\ln \left (R_i/R_o\right )} \ln \left (\frac {r}{R_i}\right )\right ]\!, \end{align}

first derived by Lamb (Reference Lamb1932, p. 586). By substituting (3.4) into (2.2) and (2.3), the laminar skin-friction coefficients are found:

(3.5) \begin{align} C_{f,i,lam} = \varPi _{x,lam} \! \left [ \!R_i + \frac {R_o^2 - R_i^2}{2R_i\ln \left (R_i/R_o\right )} \!\right ], \quad C_{f,o,lam} = \varPi _{x,lam} \! \left [ \!R_o + \frac {R_o^2 - R_i^2}{2R_o\ln \left (R_i/R_o\right )}\!\right ]\!. \end{align}

Substituting the laminar solution (3.4) into the bulk-velocity equation (2.4), we obtain

(3.6) \begin{align} \varPi _{x,lam} = - C_{f,lam} = \frac {8\ln \left (R_o/R_i\right )}{Re_b\left [R_o^2 - R_i^2 - (R_o^2 + R_i^2)\ln \left (R_o/R_i\right )\right ]}. \end{align}

By combining (3.5) and (3.6), we find

(3.7) \begin{align} C_{f,i,lam} = \frac {4\left [R_i^2 - R_o^2 + 2 R_i^2 \ln (R_o/R_i)\right ]}{Re_b R_i \left [R_o^2 - R_i^2 - (R_o^2+R_i^2) \ln (R_o/R_i)\right ]}, \\[-28pt] \nonumber \end{align}

(3.8) \begin{align} C_{f,o,lam} = \frac {4\left [R_i^2 - R_o^2 + 2 R_o^2 \ln (R_o/R_i)\right ]}{Re_b R_o \left [R_o^2 - R_i^2 - (R_o^2+R_i^2) \ln (R_o/R_i)\right ]}. \\[0pt] \nonumber \end{align}

In terms of the radius ratio $\xi =R_o/R_i$ (thus using $R_i=1/(\xi -1)$ and $R_o=\xi /(\xi -1)$ ), the laminar pressure gradient and skin-friction coefficients are

(3.9) \begin{align} \varPi _{x,lam} = - C_{f,lam} = \frac {8 (\xi -1)^2 \ln \xi }{Re_b\left [\xi ^2 - 1 - (\xi ^2 + 1)\ln \xi \right ]}, \\[-12pt] \nonumber \end{align}

(3.10) \begin{align} C_{f,i,lam} = \frac {4(\xi -1)\left (1 - \xi ^2 + 2 \ln \xi \right )}{Re_b \left [\xi ^2 - 1 - \left (\xi ^2+1\right ) \ln \xi \right ]}, \\[-12pt] \nonumber \end{align}

(3.11) \begin{align} C_{f,o,lam} = \frac {4(\xi -1)\left (1 - \xi ^2 + 2 \xi ^2 \ln \xi \right )}{Re_b \xi \left [\xi ^2 - 1 - \left (\xi ^2+1\right ) \ln \xi \right ]}. \\[-6pt] \nonumber \end{align}

3.3. Identities for the pressure gradient and the skin-friction coefficients

The $x$ -momentum equation (3.1) is multiplied by $r$ , integrated from $R_i$ to $r$ and each term is divided by $r$ to single out $\mathrm{d} \overline {u}/\mathrm{d} r$ :

(3.12) \begin{align} - \frac {\varPi _x}{2}\left ( r - \frac {R_i^2}{r} \right ) = \overline {u'v'} - \frac {1}{Re_b} \frac {\mathrm{d} \overline {u}}{\mathrm{d} r} + \frac {R_i C_{\kern-1pt f,i}}{2 r}. \end{align}

Equation (3.12) is integrated from $R_i$ to $r$ to obtain $\overline {u}$ :

(3.13) \begin{align} - \frac {\varPi _x}{4} \left (r^2 - R_i^2\right ) + \frac {\varPi _x R_i^2}{2} \ln \left ( \frac {r}{R_i} \right ) = \underbrace {\int _{R_i}^r \overline {u'v'} \mathrm{d}\tilde {r}}_{\rm A} \underbrace {- \frac {\overline {u}}{Re_b}}_{\rm B} + \frac {R_i C_{\kern-1pt f,i}}{2} \ln \left ( \frac {r}{R_i} \right )\!. \end{align}

Equation (3.13) is multiplied by $r$ and integrated between $R_i$ and $R_o$ . The integrated term A in (3.13) is simplified by using integration by parts. The bulk-velocity equation (2.4) is used in the integration of term B in (3.13). Equation (3.13) becomes

(3.14) \begin{align} \begin{split} & - \varPi _x \left ( R_o^2 - R_i^2 \right )^2 + 2 R_i \left (\varPi _x R_i - C_{\kern-1pt f,i}\right ) \left [ R_i^2 - R_o^2 + 2 R_o^2\ln \left ( \frac {R_o}{R_i} \right ) \right ] \\ & = 8 \int _{R_i}^{R_o} \overline {u'v'} \left ( R_o^2 - r^2 \right )\, \mathrm{d} r + 8 \frac {R_i^2 - R_o^2}{Re_b}. \end{split} \end{align}

By integrating (3.12) between $R_i$ and $R_o$ and by using the no-slip conditions on the cylinders, one finds

(3.15) \begin{align} -\dfrac {\varPi _x}{4} \left [R_o^2 - R_i^2 - 2 R_i^2 \ln \left (\dfrac {R_o}{R_i}\right ) \right ] = \int _{R_i}^{R_o} \overline {u'v'} \,\mathrm{d}r+\dfrac {R_i C_{\kern-1pt f,i}}{2} \ln \left (\dfrac {R_o}{R_i}\right ). \end{align}

Combining (3.3), (3.14) and (3.15) yields

(3.16) \begin{align} C_{\kern-1pt f} = -\varPi _x = - \varPi _{x,lam} + \frac { 4\left ( R_o^2 - R_i^2 \right ) \int _{R_i}^{R_o} \overline {u'v'} \mathrm{d} r - 8 \ln \left (R_o/R_i\right ) \int _{R_i}^{R_o} \overline {u'v'} r^2 \mathrm{d} r }{\left [R_o^2 - R_i^2 - (R_o^2+R_i^2) \ln \left (R_o/R_i\right )\right ]\left (R_o^2 - R_i^2\right )}, \\[-28pt] \nonumber \end{align}

(3.17) \begin{align} \begin{split} C_{\kern-1pt f,i} & = C_{f,i,lam} \nonumber \\ & \quad + \frac { 4 \left [ R_i^2 - R_o^2 + 2 R_i^2 \ln \left (R_o/R_i\right ) \right ] \int _{R_i}^{R_o} \overline {u'v'}r^2 \mathrm{d} r + 2 (R_o^2-R_i^2)^2 \int _{R_i}^{R_o} \overline {u'v'} \mathrm{d} r }{ \left [R_o^2 - R_i^2 - (R_o^2+R_i^2) \ln \left (R_o/R_i\right ) \right ]\left (R_o^2 - R_i^2\right )R_i },\qquad\quad \end{split} \\[-28pt] \end{align}

(3.18) \begin{align} \begin{split} C_{\kern-1pt f,o} & = C_{f,o,lam} \nonumber \\ & \quad + \frac { 4 \left [ R_i^2 - R_o^2 + 2 R_o^2 \ln \left (R_o/R_i\right ) \right ] \int _{R_i}^{R_o} \overline {u'v'} r^2 \mathrm{d} r -2(R_o^2-R_i^2)^2\int _{R_i}^{R_o} \overline {u'v'} \mathrm{d} r } { \left [R_o^2 - R_i^2 - (R_o^2+R_i^2) \ln \left (R_o/R_i\right ) \right ] (R_o^2 - R_i^2) R_o }.\qquad\quad \end{split} \\[0pt] \end{align}

In terms of the radius ratio $\xi$ , identities (3.16)–(3.18) become

(3.19) \begin{align} C_{\kern-1pt f} = -\varPi _x = - \varPi _{x,lam} + \frac { 4\left ( \xi ^2 - 1 \right )\int _{1}^{\xi } \overline {u'v'} \mathrm{d}\tilde r - 8 \ln \xi \int _{1}^{\xi } \overline {u'v'} {\tilde r}^2 \mathrm{d}\tilde r }{(\xi + 1)\left [\xi ^2 - 1 - (\xi ^2+1) \ln \xi \right ]}, \\[-28pt] \nonumber \end{align}

(3.20) \begin{align} \begin{split} C_{\kern-1pt f,i} = C_{f,i,lam} + \frac { 4 \left ( 1 - \xi ^2 + 2 \ln \xi \right ) \int _{1}^{\xi } \overline {u'v'} \tilde r^2 \mathrm{d}\tilde r + 2 (\xi ^2-1)^2 \int _{1}^{\xi } \overline {u'v'} \mathrm{d}\tilde r }{ \left (\xi ^2 - 1\right )\left [\xi ^2 - 1 - (\xi ^2+1) \ln \xi \right ] }, \end{split} \\[-28pt] \nonumber \end{align}

(3.21) \begin{align} \begin{split} C_{\kern-1pt f,o} = C_{f,o,lam} + \frac { 4 \left ( 1 - \xi ^2 + 2 \xi ^2 \ln \xi \right ) \int _{1}^{\xi } \overline {u'v'} \tilde r ^2 \mathrm{d}\tilde r -2(\xi ^2-1)^2 \int _{1}^{\xi } \overline {u'v'} \mathrm{d}\tilde r } { \xi (\xi ^2 - 1) \left [\xi ^2 - 1 - (\xi ^2+1) \ln \xi \right ] }, \end{split} \\[0pt] \nonumber \end{align}

Figure 2.

Absolute values of the skin-friction coefficients obtained via the theoretical formulas, $|C_{f*,uv}|$ (the subscript asterisk indicates any of (3.16), (3.17) or (3.18)), versus experimental (open symbols) and numerical (filled symbols) skin-friction coefficients reported in the literature, $C_{f*}$ . The coefficients $C_{f*}$ are shown without an absolute value because all cited studies report them using an unsigned (positive) convention. The colour-coded per cent values quantify the differences between $|C_{f*,uv}|$ and $C_{f*}$ when the difference is larger than 1 %. The data of Boersma & Breugem (Reference Boersma and Breugem2011) are divided by three to avoid an excessively large range of values and preserve visual clarity. Parameter $\xi =R_o/R_i$ is the radius ratio. All data are for stationary cylinders, except those of Mamori et al. (Reference Mamori, Higashimoto, Morita, Wang, Miyazaki, Fukudome and Nabae2026), which are for oscillating cylinders. Inset: Reynolds stresses $\overline {u'v'}$ computed by Boersma & Breugem (Reference Boersma and Breugem2011) for $ \textit{Re}_b=9270$ ( $\xi =$ 10) and $ \textit{Re}_b=8900$ ( $\xi =$ 50).

where $\tilde r = r/R_i$ . The integral term $\int _{R_i}^{R_o} \overline {u'v'}\, \mathrm{d} r$ in (3.16)–(3.18) is not zero because $\overline {u'v'}$ is not symmetric with respect to the midplane between the two cylinders, as shown by the numerical data of Boersma & Breugem (Reference Boersma and Breugem2011) in the inset of figure 2 (refer also to figure 5 of Quadrio & Luchini (Reference Quadrio and Luchini2002) and figure 20 of Rodriguez-Corredor et al. (Reference Rodriguez-Corredor, Bizhani, Ashrafuzzaman and Kuru2014)). The profiles in figure 2 further reveal that, as the radius ratio $\xi$ increases and the annular flow approaches a pipe flow, the negative peak becomes more pronounced and the point where $\overline {u'v'}=0$ moves closer to the inner cylinder, thereby making the positive area under the curve larger.

Equations (3.16)–(3.18) are also valid when either or both cylinders are in motion along the azimuthal direction because the azimuthal velocity component is absent from the averaged $x\hbox{-}$ momentum equation (3.1). The Reynolds-number dependence of the skin-friction coefficients expressed in identities (3.16)–(3.18) is found explicitly in the laminar parts $C_{f,lam}$ , $C_{f,i,lam}$ , $C_{f,o,lam}$ , given in (3.6)–(3.8), and implicitly in the Reynolds-stress integrals in (3.16)–(3.18). Although the Reynolds stresses $\overline {u'v'}$ tend to agree more closely with the total stress in the core of the annular gap as the Reynolds number increases, at any Reynolds number the near-wall viscous effects exert an influence on $\overline {u'v'}$ , and therefore on $C_{f}$ , $C_{\kern-1pt f,i}$ , $C_{\kern-1pt f,o}$ through the integrals (3.16)–(3.18). The accurate quantification of the near-wall behaviour of $\overline {u'v'}$ thus establishes the Reynolds-number dependence of $C_{f}$ , $C_{\kern-1pt f,i}$ , $C_{\kern-1pt f,o}$ . The neglect of these viscous effects leads to constant $C_{f}$ , $C_{\kern-1pt f,i}$ , $C_{\kern-1pt f,o}$ at a large Reynolds number, which contradicts experimental and numerical evidence.

Figure 2 reports the absolute values of the skin-friction coefficients computed using the theoretical formulas (3.16)–(3.18) versus the corresponding coefficients reported in the literature. We selected datasets that include both $\overline {u'v'}$ and $C_{\kern-1pt f}$ . The Reynolds-stress data were digitised from the figures reported in the studies cited in figure 2. The trapezoidal rule was used for the computation of the integrals in identities (3.16)–(3.18) and in the following (3.26), (4.4) and (4.5). The agreement is excellent for most of the experimental and numerical datasets. Quadrio & Luchini (Reference Quadrio and Luchini2002), Chung et al. (Reference Chung, Rhee and Sung2002) and Boersma & Breugem (Reference Boersma and Breugem2011) (for $\xi =50$ ) (stationary cylinders) show differences below 1 %. The data of Mamori et al. (Reference Mamori, Higashimoto, Morita, Wang, Miyazaki, Fukudome and Nabae2026) present a maximum difference of 1.6 % (oscillating cylinders). The data of Boersma & Breugem (Reference Boersma and Breugem2011) for $\xi =10$ and $\xi =25$ show 9.7 % and 2.2 % differences, respectively. The experimental data of Rodriguez-Corredor et al. (Reference Rodriguez-Corredor, Bizhani, Ashrafuzzaman and Kuru2014) present a maximum difference of 3.9 % and those by Nouri et al. (Reference Nouri, Umur and Whitelaw1993) show a disparity of 15 %, arguably due to the uncertainty of the measured Reynolds stresses in their figure 2(c).

The limit of vanishingly thin inner cylinder, i.e. $R_i^*/h^*\rightarrow 0$ ( $\xi \rightarrow \infty$ , or $R_o\rightarrow 1$ and $R_i \rightarrow 0$ ), leads to

(3.22) \begin{align} C_{\kern-1pt f} & \sim -\frac {8\ln \left (R_o/R_i\right )}{Re_b\left [1 - \ln \left (R_o/R_i\right )\right ]} + \frac { 4 \int _{R_i}^{R_o} \overline {u'v'} \mathrm{d} r - 8 \ln \left (R_o/R_i\right ) \int _{R_i}^{R_o} \overline {u'v'} r^2 \mathrm{d} r }{1 - \ln \left (R_o/R_i\right )} \\[-12pt] \nonumber \end{align}

(3.23) \begin{align} & \sim \frac {8}{Re_b} + 8 \int _0^1 \overline {u'v'} r^2 \,\mathrm{d} r \sim -C_{\kern-1pt f,o}. \\[-6pt] \nonumber \end{align}

After rescaling, (3.23) coincides with equation (19) in FIK for circular pipe flow despite the streamwise velocity $\overline {u}$ satisfying different conditions at the centreline in the pipe-flow case and in the annular-flow case as $R_i \rightarrow 0$ . In this limit, although the wall-shear stress at the centreline grows unbounded, the surface area vanishes. It follows that the friction drag on the inner cylinder vanishes $\propto (\ln \xi )^{-1}$ as $\xi \rightarrow \infty$ . The total drag is thus given by the friction on the outer cylinder only.

The limit of vanishingly thin annular gap, i.e. $h^*/R_o^* \rightarrow 0$ ( $\xi \rightarrow 1$ , or $R_i, R_o \rightarrow \infty$ with $R_o-R_i=1$ ), is studied to verify that (3.16)–(3.18) simplify to the identity for fully developed planar channel flow discovered by FIK. This limit is obtained using the Taylor series $\ln (\epsilon +1) = \epsilon - \epsilon ^2/2 + \epsilon ^3/3\ldots ,$ for $|\epsilon |\ll 1,$ where $R_o/R_i=\epsilon +1$ . Substituting the resulting expansion $\ln (R_o/R_i ) \sim (6 R_i^2 - 3 R_i + 2)/(6 R_i^3)$ and the change of variable $y=r-R_i=\mathcal{O}(1)$ into (3.6)–(3.8) and (3.16)–(3.18), and using $\int _0^1 \overline {u'v'}\, \mathrm{d}y=0$ arising from the symmetry of the channel flow with respect to the channel midplane, it follows that $C_{f_i}\sim -C_{f_o}\sim C_{\kern-1pt f}$ , where

(3.24) \begin{align} C_{\kern-1pt f} \sim \frac {12}{Re_b} - 12\int _0^1 \left ( \dfrac {1}{2} - y \right ) \overline {u'v'}\, \mathrm{d}y. \end{align}

After rescaling, (3.24) coincides with equation (18) in FIK for planar channel flow.

An identity for the averaged skin-friction coefficient for annular flows has recently been derived by Mamori et al. (Reference Mamori, Higashimoto, Morita, Wang, Miyazaki, Fukudome and Nabae2026). The main differences with respect to our (3.16) are that, in the identity of Mamori et al. (Reference Mamori, Higashimoto, Morita, Wang, Miyazaki, Fukudome and Nabae2026): (i) the laminar skin-friction coefficient is not isolated; (ii) the integral terms involve the radial location $r_m$ of the maximum streamwise velocity, which appears in the integrands, in the integration limits and in other multiplicative factors; (iii) the wall-normal integration is conducted separately on different integrands from the inner cylinder to $r_m$ and from $r_m$ to the outer cylinder; (iv) the identity is implicit with respect to the skin-friction coefficients because $r_m$ is expressed as a function of $C_{\kern-1pt f,i}$ and $C_{\kern-1pt f,o}$ ; and (v) a term $C_{f,bulk}$ appears, which involves integrals of the streamwise mean velocity and finds no equivalent in either (3.16) or the classical FIK identities.

3.4. Identities for the radial location of the maximum mean velocity

It is also instructive to study how the radial location $r_m$ of the maximum mean streamwise velocity varies for different flow conditions. Figure 3 presents numerical and experimental data for the ratio $r_m/r_{m,\textit{lam}}$ , where $r_{m,\textit{lam}}$ is the radial location of the laminar streamwise velocity, i.e. $r_{m,\textit{lam}}= [ (R_o^2 - R_i^2 )/[2 \ln (R_o/R_i )] ]^{1/2}$ . In the transitional regime, the results by Rothfus and co-workers (Walker et al. Reference Walker, Whan and Rothfus1957; Croop & Rothfus Reference Croop and Rothfus1962) demonstrate that the ratio can be smaller or larger than unity. In the turbulent regime, except for the data of Rothfus and co-workers (e.g. Rothfus et al. Reference Rothfus, Sartory and Kermode1966), consensus exists on $r_m/r_{m,\textit{lam}}$ being smaller than unity and decreasing with $R_i/R_o$ , i.e. $r_m$ becomes smaller than $r_{m,\textit{lam}}$ as both quantities approach the centreline of the annular system and the ratio $R_i/R_o$ decreases. Figure 3 further indicates that the ratio becomes independent of the Reynolds number as the Reynolds number increases, a result that is particularly evident from the experimental data of Brighton & Jones (Reference Brighton and Jones1964), conducted at the highest Reynolds numbers to date.

As a first step to find a relation between the radial location $r_m$ of the maximum mean streamwise velocity and the integrated Reynolds stresses, it is useful to discuss first whether the Reynolds stresses vanish at $r_m$ , a controversial point in the study of annular flows. The experimental results of Lawn & Elliott (Reference Lawn and Elliott1972), Rehme (Reference Rehme1974), Rodriguez-Corredor et al. (Reference Rodriguez-Corredor, Bizhani, Ashrafuzzaman and Kuru2014) and Ghaemi et al. (Reference Ghaemi, Rafati, Bizhani and Kuru2015) showed that the radial locations where $\overline {u'v'}=0$ and $\mathrm{d}\overline {u}/\mathrm{d}r=0$ did not coincide, implying that, in a small range of radial locations, negative production of turbulent kinetic energy occurred. Ghaemi et al. (Reference Ghaemi, Rafati, Bizhani and Kuru2015) discussed a few physical interpretations for this occurrence, but also hypothesised that the flow might not have reached fully developed conditions. The direct numerical simulations of Chung et al. (Reference Chung, Rhee and Sung2002) reported a mismatch between the two locations, whereas, in the numerical studies of Quadrio & Luchini (Reference Quadrio and Luchini2002), Boersma & Breugem (Reference Boersma and Breugem2011) and Bagheri & Wang (Reference Bagheri and Wang2020), the two values coincided. Boersma & Breugem (Reference Boersma and Breugem2011) claimed that experimental errors might have led to the conclusions of the experimental studies and attributed the differences between the two locations found by Chung et al. (Reference Chung, Rhee and Sung2002) to insufficient spatial resolution. Highly accurate direct numerical simulations were therefore conducted by Orlandi et al. (Reference Orlandi, Soldati and Pirozzoli2025) for a wide range of radius ratios to resolve this issue. In all those cases, the locations where $\overline {u'v'}=0$ and $\mathrm{d}\overline {u}/\mathrm{d}r=0$ coincided exactly.

Figure 3.

Ratio between the radial locations $r_m$ of the maximum $\overline {u}$ and the corresponding location $r_{m,\textit{lam}}$ in the laminar case: (a) $ \textit{Re}_b\lt 12\,000$ ; (b) $ \textit{Re}_b\gt 45\,000$ . Filled circles: numerical data; open circles: experimental data. The symbol size is inversely proportional to the radius ratio $\xi =R_o/R_i$ (colour-coded, representative $\xi$ values are given). In the legend of ( $a$ ), Rothfus et al. (1962, Reference Rothfus, Sartory and Kermode1966) indicates Croop & Rothfus (Reference Croop and Rothfus1962) and Rothfus et al. (Reference Rothfus, Sartory and Kermode1966).

The location $r_m$ is thus found by setting $\overline {u'v'}=0$ and $\mathrm{d}\overline {u}/\mathrm{d}r=0$ in (3.12) and by using (3.2) (Bagheri & Wang Reference Bagheri and Wang2020):

(3.25) \begin{align} r_m = \left [R_i^2 - \dfrac {R_iC_{\kern-1pt f,i}}{\varPi _x}\right ]^{1/2} = \left [R_i R_o \dfrac {R_i C_{\kern-1pt f,o}-R_o C_{\kern-1pt f,i}}{R_o C_{\kern-1pt f,o}-R_i C_{\kern-1pt f,i}}\right ]^{1/2}. \end{align}

Substituting (3.16) and (3.17) into (3.25) gives $r_m$ as a function of the integrated Reynolds stresses:

(3.26) \begin{align} r_m= \left [ \! \frac { (R_o^2-R_i^2) \left [ 2\left (R_o^2-R_i^2\right ) +2Re_b \int _{R_i}^{R_o}\overline {u'v'} r^2 \mathrm{d}r -Re_b (R_i^2 + R_o^2) \int _{R_i}^{R_o}\overline {u'v'} \mathrm{d}r \right ] }{ 4\ln \left (R_o/R_i\right ) \left ( R_o^2-R_i^2 +Re_b \int _{R_i}^{R_o}\overline {u'v'}r^2 \mathrm{d}r \right )- 2Re_b(R_o^2-R_i^2) \int _{R_i}^{R_o}\overline {u'v'} \mathrm{d}r } \!\right ]^{1/2} \!, \end{align}

Figure 4.

Radial locations $r_{m,uv}$ of the maximum $\overline {u}$ computed by the theoretical identity (3.26) versus experimental (open symbols) and numerical (filled symbols) radial locations $r_m$ .

which reduces to the laminar $r_{m,\textit{lam}}$ when $\overline {u'v'} = 0$ . In terms of the radius ratio $\xi$ , identity (3.26) becomes

(3.27) \begin{align} r_m^2= \mathcal{D}_m \left [ 2(\xi ^2-1)^2+2Re_b (\xi +1) \int _{1}^{\xi }\overline {u'v'} \tilde r^2 \,\mathrm{d}\tilde r -Re_b(\xi ^2+1)(\xi +1) \int _{1}^{\xi }\overline {u'v'}\,{\rm d} \tilde r \right ]\!, \end{align}

where

(3.28) \begin{align} \begin{aligned} \mathcal{D}_m(\xi )= & \left [4\ln \xi \left [(\xi +1)(\xi -1)^3+Re_b(\xi -1) \int _{1}^{\xi }\overline {u'v'}\tilde r^2 \,\mathrm{d}\tilde r\right ] \right . \\ & \left . -2Re_b (\xi +1) (\xi -1)^2\int _{1}^{\xi }\overline {u'v'}\,\mathrm{d}\tilde r \right ]^{-1}. \end{aligned} \end{align}

Figure 4 compares the $r_m$ data computed via (3.26) with data in the literature. The percentage deviations from our computations of $r_m$ are smaller than or comparable with those reported in figure 2 for the skin-friction coefficients. By substituting $r_m$ into (3.13), the maximum mean velocity $u_m$ is determined as a function of $R_i, R_o, Re_b$ and $\overline {u'v'}$ .

The result that $r_m/r_{m,\textit{lam}}$ is smaller than unity in the turbulent regime, shown in figure 3, is confirmed by the asymptotic form of (3.26) in the limit $R_i\rightarrow 0$ with $R_o=O(1)$ :

(3.29) \begin{align} \begin{aligned} r_m^2 \sim & \underbrace {\dfrac {R_o^2 - R_i^2}{2\ln \!\left (R_o/R_i\right )}}_{r_{m,\textit{lam}}^2} -\underbrace {\dfrac {Re_b (R_o^4 - R_i^4) \int _{R_i}^{R_o} \overline {u'v'} \,\mathrm{d}r}{4 \left (R_o^2 - R_i^2 + Re_b \int _{R_i}^{R_o} \overline {u'v'}\, r^2 \,\mathrm{d}r\right )\ln \left (R_o/R_i\right )}}_{\mathcal{R}_m}. \end{aligned} \end{align}

The quantity $\mathcal{R}_m$ in (3.29) is positive because the integrals involving the Reynolds stresses are both positive when the flow is turbulent. This result is evident from available $\overline {u'v'}$ profiles at large radius ratio $\xi$ , such as those shown in the inset of figure 2. The Reynolds stresses $\overline {u'v'}$ are positive for a wide range of radial positions in the outer portion of the annular gap and the positive $\overline {u'v'}$ peak is more intense than the negative peak (Chung et al. Reference Chung, Rhee and Sung2002; Boersma & Breugem Reference Boersma and Breugem2011). Therefore, the second term in (3.29) renders $r_m/r_{m,\textit{lam}}\lt 1$ because of its negative sign.

4.1. Relations between the skin-friction coefficients

The averaged $x$ -momentum equation for the cylinder-driven annular flow is

(4.1) \begin{align} \frac {\mathrm{d}}{\mathrm{d} r}\left (r \overline {u'v'} - \frac {r}{Re_c} \frac {\mathrm{d} \overline {u}}{\mathrm{d} r} \right )=0. \end{align}

The relation among the skin-friction coefficients $C_{f,i,c}, C_{f,o,c}$ is obtained by integrating (4.1) along $r$ . As the Reynolds stresses $\overline {u'v'}$ vanish on the cylinders because of the no-slip condition, one finds $R_o C_{f,o,c}=R_i C_{f,i,c}$ .

4.2. Laminar-flow quantities

The laminar solution for the cylinder-driven flow is obtained by integrating (4.1) along $r$ with $\overline {u'v'}=0$ and by using the boundary conditions, $\overline {u}(R_i)=U_c$ and $\overline {u}(R_o)=0$ (where $U_c=1$ , though left unassigned for clarity). The solution is $\overline {u}_{lam}(r)=U_c \ln (r/R_o )/\ln (R_i/R_o)$ . The laminar skin-friction coefficients are

(4.2) \begin{align} C_{f,i,c,lam}=\dfrac {-2 U_c}{R_i Re_c \ln (R_o/R_i)}, \quad C_{f,o,c,lam}=\dfrac {-2 U_c}{R_o Re_c \ln (R_o/R_i)}. \\[-12pt]\nonumber \end{align}

4.3. Identities for the skin-friction coefficients

The $x$ -momentum equation (4.1) is integrated from $R_i$ to $r$ to single out $\mathrm{d} \overline {u}/\mathrm{d} r$ :

(4.3) \begin{align} \overline {u'v'} - \frac {1}{Re_c} \frac {\mathrm{d} \overline {u}}{\mathrm{d} r} + \dfrac {R_i C_{f,i,c}}{2 r}=0. \end{align}

By integrating (4.3) between $R_i$ and $R_o$ and by using (4.2), we obtain

(4.4) \begin{align} C_{f,i,c}=C_{f,i,c,lam}-\dfrac {2 \int _{R_i}^{R_o} \overline {u'v'} \mathrm{d}r}{R_i \ln \left (R_o/R_i\right )}, \quad C_{f,o,c}=C_{f,o,c,lam}-\dfrac {2 \int _{R_i}^{R_o} \overline {u'v'} \mathrm{d}r}{R_o \ln \left (R_o/R_i\right )}. \end{align}

The average skin-friction coefficient $C_{f,c}$ is found by substituting identities (4.4) into (3.3), valid for the cylinder-driven case as well.

It was numerically verified by Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019) that $\overline {u'v'}\gt 0$ $\forall r$ (and, analogously, in plane Couette flows (Kawata & Alfredsson Reference Kawata and Alfredsson2019)), as shown in the inset of figure 5. It follows that the Reynolds-stress integral terms in (4.4) quantify the increase of the skin-friction coefficients in absolute value with respect to the laminar coefficients, as demonstrated by figure 12(a) of Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019) (where their Reynolds number is $ \textit{Re}=Re_c/4$ ). Figure 5 also compares the average skin-friction coefficients computed using the theoretical formulas (4.4) and (3.3) with the average coefficients computed by Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019). The agreement is excellent for all cases, except for the coefficient for $ \textit{Re}_c=3000$ and $\xi =10$ (black circle), showing a discrepancy of about 5 %.

Figure 5.

Skin-friction coefficients computed via the theoretical formulas (4.4) and (3.3), $-C_{f,c,uv}$ , versus the numerical skin-friction coefficients computed by Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019), $-C_{f,c}$ . Inset: Reynolds stresses $\overline {u'v'}$ computed by Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019) for $ \textit{Re}_c=3000$ for $\xi =1.25$ (blue), $\xi =2$ (red) and $\xi =10$ (black).

4.4. Identities for the bulk velocity

By using (2.5) and (4.2)–(4.4), the bulk velocity is expressed as a function of the integrated Reynolds stresses:

(4.5) \begin{align} U_{b,c} = \underbrace { \dfrac {U_c \left [R_o^2 - R_i^2 - 2R_i^{2}\ln (R_o/R_i)\right ]}{2\left (R_o^{2}-R_i^{2}\right )\ln (R_o/R_i)} }_{U_{b,c,lam}} + \int _{R_i}^{R_o} \underbrace { \dfrac {Re_c \left [R_o^2-R_i^2 - 2 r^2 \ln (R_o/R_i) \right ]}{2 \left (R_o^2-R_i^2\right ) \ln (R_o/R_i)} \overline {u'v'} }_{U_i} \,\mathrm{d}r. \end{align}

In terms of the radius ratio $\xi$ , identity (4.5) becomes

(4.6) \begin{align} U_{b,c} = \dfrac {U_c \left (\xi ^2 - 1 - 2\ln \xi \right )}{2\left (\xi ^2-1\right )\ln \xi } + \int _{1}^{\xi } \dfrac {Re_c \left (\xi ^2 - 1 - 2 \tilde r ^2 \ln \xi \right ) }{2 \left (\xi -1\right ) \left (\xi ^2-1\right ) \ln \xi } \overline {u'v'} \,\mathrm{d} \tilde r. \end{align}

The integral term in (4.5) quantifies the decrease of the bulk velocity with respect to the laminar $U_{b,c,lam}$ , i.e. the Reynolds stresses have the effect of reducing the averaged streamwise flow rate produced by the motion of the inner cylinder with respect to the laminar value. This result is analogous to that observed in channel flow at constant pressure gradient when the flow transitions from the laminar to the turbulent regime (Marusic et al. Reference Marusic, Joseph and Mahesh2007). Figure 6(a), however, illustrates that the $U_{b,c,lam}$ values agree very well with the $U_{b,c}$ data in the range 1300 $\leqslant Re_c\leqslant$ 3000, computed by integrating the streamwise velocity profiles of figure 12(a) in Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019) (yellow squares). The largest discrepancy between the bulk-velocity values occurs for $R_i/R_o=0.1$ : $U_{b,c}$ is 1.7 % smaller than $U_{b,c,lam}$ . This finding reveals that the decrease of mass flow rate due to the Reynolds stresses is negligible at these Reynolds numbers. The $U_{b,c}$ data computed by integrating $\overline {u}$ show excellent agreement with the $U_{b,c}$ data computed by the theoretical formula (4.5) ( $ \textit{Re}_c=$ 3000; open coloured circles in figure 6 a). The differences are always smaller than 0.2 %, providing further evidence of the high-quality data computed by Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019). The reason for the negligible differences between $U_{b,c,lam}$ and $U_{b,c}$ at these Reynolds numbers lies in the profiles of the integrand $U_i$ in (4.5), shown in figure 6(b). These profiles are either nearly antisymmetric about the midpoint of the annular gap (for $\xi =1.25$ and $2$ ) or, if not, nevertheless exhibit nearly equal and opposite areas bounded by the curve at either side of the radial location where $U_i=0$ (for $\xi =10$ ). Since $U_i$ is proportional to $ \textit{Re}_c$ , an interesting line of future research is to investigate whether the integration of $U_i$ at larger $ \textit{Re}_c$ leads to a quantifiable difference between $U_{b,c}$ and $U_{b,c,lam}$ .

Figure 6.

(a) Bulk velocity for cylinder-driven flow versus ratio $R_i/R_o$ . The yellow squares denote data computed by integrating the $\overline {u}$ profiles by Kunii et al. (Reference Kunii, Ishida, Duguet and Tsukahara2019); the filled red circle denotes the bulk velocity for Couette flow (Kawata & Alfredsson Reference Kawata and Alfredsson2019); the open circles denote the data computed via the theoretical formula (4.5) for $ \textit{Re}_c=3000$ , coloured to match the data in (b); the solid line represents the laminar bulk velocity $U_{b,c,lam}$ , given in (4.5). (b) Integrand $U_i$ , given in (4.5), for $ \textit{Re}_c=3000$ . Parameter $\xi =R_o/R_i$ is the radius ratio.

The limits for a vanishingly thin annular gap are found following the procedure detailed in § 3.3:

(4.7) \begin{align} C_{f,i,c} \sim C_{f,o,c} \sim -\dfrac {2 U_c}{Re_c}-2 \int _0^1 \overline {u'v'}\, \mathrm{d}y, \\[-28pt] \nonumber \end{align}

(4.8) \begin{align} U_{b,c} \sim \dfrac {U_c}{2} + Re_c \int _0^1 \left ( \dfrac {1}{2} - y\right ) \overline {u'v'}\, \mathrm{d}y = \dfrac {U_c}{2}. \\[0pt] \nonumber \end{align}

After rescaling, (4.7) coincides with the identity discovered by Kawata & Alfredsson (Reference Kawata and Alfredsson2019) for planar Couette flow. The integral term in (4.8) is null because, with respect to $y=1/2$ , the integrand is odd as $\overline {u'v'}$ is even, i.e. the averaged flow rate is unaffected by the Reynolds stresses in the planar Couette flow case.