3.1. Channel flows

The skin-friction coefficient $C_{\!f}$ , in turbulent channel flows, can be defined as

(3.1) \begin{equation} C_{\!f} = \frac {2\tau _w}{\rho _bu_b^2}. \end{equation}

Here, $\tau _w= \bar {\mu }\text{d}\bar {u}/\text{d}y|_w$ is the wall shear stress, $\rho _b = 1/h\int _0^h\bar {\rho }\text{d}y$ is the bulk density, $\mu$ is the viscosity, $h$ is the half-width of the channel, and $y$ is the wall-normal coordinate. Hereafter, an overbar denotes the Reynolds average, and the subscript $w$ indicates quantities at the wall. The bulk streamwise velocity $u_b$ in channel flows is defined as

(3.2) \begin{equation} u_{b}=\frac {1}{\rho _bh}\int _0^{h}\bar {\rho } \bar {u}\,\text{d}y=\frac{u_{\tau}}{\rho_{b}h^{+}}\int _0^{h^+}\bar {\rho } \bar {u}^+\,\text{d}y^+. \end{equation}

In this paper, the superscript $+$ denotes non-dimensionalisation using $u_\tau$ , $\delta _v$ and $\mu _w$ . The viscous length scale $\delta _v$ is defined as $\delta _v=\mu _w/(u_\tau \rho _w)$ , where the friction velocity is given by $u_\tau =\sqrt {\tau _w/\rho _w}$ . According to the Prandtl relation (Zanoun et al. Reference Zanoun, Nagib and Durst2009; Schlichting & Gersten Reference Schlichting and Gersten2016), $C_{\!f}$ scales with the bulk Reynolds number, defined as

(3.3) \begin{equation} \textit{Re}_b = \frac {\rho _bu_bh}{\mu _w}. \end{equation}

It is important to note that density and viscosity remain constant in incompressible turbulent channel flows, specifically $\bar {\rho }(y)=\rho _b$ and $\bar {\mu }(y)=\mu _w$ . Thus the bulk velocity can be expressed as $u_b=1/h\int _0^h\bar {u}\text{d}y$ for incompressible turbulent channel flows.

The Prandtl relation can be derived from the logarithmic law representations of the entire mean velocity profile $\bar {u}(y)$ (Zanoun et al. Reference Zanoun, Nagib and Durst2009; Schlichting & Gersten Reference Schlichting and Gersten2016). According to the results of incompressible turbulent channel flows (Bernardini et al. Reference Bernardini, Pirozzoli and Orlandi2014; Lee & Moser Reference Lee and Moser2015; Abe & Antonia Reference Abe and Antonia2016), the total width of the viscous sublayer and buffer layer is approximately of the order of $O(10\delta _v)$ . The width of the logarithmic-law layer is typically at least of the order of $O(100\delta _v)$ , and can reach up to $O(1000\delta _v)$ at high Reynolds numbers, significantly exceeding the total width of the viscous sublayer and buffer layer. To this end, the logarithmic law of $\bar {u}$ , expressed as

(3.4) \begin{equation} y^+=\frac {\exp (\kappa \bar {u}^+)}{E}, \end{equation}

can be reasonably used to represent the entire mean velocity profile for approximating the bulk velocity (Zanoun et al. Reference Zanoun, Nagib and Durst2009). Here, $\kappa$ is the von Kármán constant, and $E$ is another constant. Indeed, $E$ is determined by $\kappa$ and the intercept $A$ of the logarithmic law for the velocity profile, specifically given by $E=\exp {(\kappa A)}$ . By substituting (3.4) into (3.2), $u_b$ of incompressible turbulent channel flows can be estimated as

(3.5) \begin{equation} u_b =\frac {u_\tau }{\textit{Eh}^+}\int _0^{u_c^+}\bar {u}^+\,{\rm d}\exp (\kappa \bar {u}^+) =\frac {u_\tau }{\textit{Eh}^+}\frac {\kappa u_c^+\exp (\kappa u_c^+)-\exp (\kappa u_c^+)+1}{\kappa }. \end{equation}

Hereafter, the subscript $c$ denotes quantities measured at the centre of the channel or pipe. Given that $\kappa u_c^+$ is typically of the order of $O(10)$ for incompressible turbulent channel flows, the ratio of the term $\exp (\kappa u_c^+)-1$ to the term $\kappa u_c^+\exp (\kappa u_c^+)$ is approximately $O(10^{-1})$ . This ratio can also be validated by the DNS data for incompressible turbulent channel flow, as presented in table 1. The ratio of the term $\exp (\kappa u_c^+)-1$ to the term $\kappa u_c^+\exp (\kappa u_c^+)$ is $0.13$ for the case with $\textit{Re}_\tau =183$ , and $0.09$ for the case with $\textit{Re}_\tau =4079$ , both of which are consistent with the order of $O(10^{-1})$ . Therefore, the term $\exp (\kappa u_c^+)-1$ in (3.5) can be reasonably neglected, allowing for a further estimation of $u_b$ as

(3.6) \begin{equation} u_b\approx \frac {u_\tau u_c^+\exp (\kappa u_c^+)}{\textit{Eh}^+}. \end{equation}

By substituting (3.6) into (3.3), $\textit{Re}_b$ for incompressible turbulent channel flows can be determined by

(3.7) \begin{equation} \textit{Re}_b=\frac {u_c^+\exp (\kappa u_c^+)}{E}. \end{equation}

Furthermore, based on (3.4), we have $h^+=\exp (\kappa u_c^+)/E$ . Thus the estimation of $u_b$ in (3.6) suggests that $u_b\approx u_c$ , indicating that the central velocity of incompressible channel flows can be approximated as the bulk velocity. It should be reiterated that $u_b\approx u_c$ is derived from the assumption of logarithmic-law representations of the entire velocity profile and the neglect of the small-value term in (3.5). Since the ratio of $\exp (\kappa u_c^+)-1$ to $\kappa u_c^+\exp (\kappa u_c^+)$ in (3.5) is of the order of $O(10^{-1})$ , the relative error $|u_c-u_b|/u_c$ is theoretically of the same order, $O(10^{-1})$ . Indeed, the order of the relative error for the approximation $u_b\approx u_c$ can be supported by the DNS data. Based on the data presented in table 1, the ratio $|u_c-u_b|/u_c$ is $0.16$ for the case with $\textit{Re}_\tau =183$ , and $0.10$ for the case with $\textit{Re}_\tau =4079$ , confirming that the relative error of the approximation $u_c\approx u_b$ is of the order of $O(10^{-1})$ . Using the definition of $C_{\!f}$ in (3.1) and the approximation $u_c\approx u_b$ , one can obtain

(3.8) \begin{equation} u_c^+\approx \sqrt {\frac {2}{C_{\!f}}} \end{equation}

for incompressible turbulent channel flows. Based on (3.7) and (3.8), $\sqrt {2/C_{\!f}}$ grows logarithmically with $\ln (\textit{Re}_b\,\sqrt {C_{\!f}/2})$ , thus the classical Prandtl relation for skin-friction law in incompressible turbulent channel flows can be derived as shown in (1.1). Moreover, given that $\textit{Re}_\tau$ is defined as $\textit{Re}_\tau =\rho _wu_\tau h/\mu _w$ , and that the mean density remains constant in incompressible turbulent channel flows, we have $\textit{Re}_\tau =\textit{Re}_b\,\sqrt {C_{\!f}/2}$ , allowing the Prandtl relation to be equivalently expressed as (1.2).

By performing a least squares fit to the data to determine the constants $\kappa _{\!f}$ and $C$ , the Prandtl relation can be quantified as

(3.9) \begin{equation} \sqrt {\frac {2}{C_{\!f}}} = \frac {1}{\kappa _{\!f}}\ln \left (\textit{Re}_b\,\sqrt {\frac {C_{\!f}}{2}}\right )+C, \end{equation}

or equivalently,

(3.10) \begin{equation} \sqrt {\frac {2}{C_{\!f}}} = \frac {1}{\kappa _{\!f}}\ln \textit{Re}_\tau +C. \end{equation}

It is important to note that the logarithmic-law coefficient of Prandtl relation $\kappa _{\!f}$ obtained by fitting $\sqrt {2/C_{\!f}}$ versus $\ln \textit{Re}_\tau$ differs slightly from the $\kappa$ appearing in the velocity logarithmic law. Indeed, this slight difference results from the approximations involved in deriving the Prandtl relation. For instance, the derivation of the Prandtl relation relies on the approximation that the entire mean velocity profile is represented by the logarithmic law. This approximation means that the Prandtl relation is not entirely equivalent to the logarithmic law of velocity, as the definition of $C_{\!f}$ in the Prandtl relation is based on the entire velocity profile, including the viscous sublayer and the buffer layer. Therefore, the value of $\kappa _{\!f}$ obtained by fitting $\sqrt {2/C_{\!f}}$ versus $\ln \textit{Re}_\tau$ differs slightly from the value of $\kappa$ obtained from the velocity profile. Through a least squares fit to the data, Abe & Antonia (Reference Abe and Antonia2016) obtained $\kappa _{\!f}=0.394$ and $C=2.41$ , resulting in a strong prediction of both experimental and DNS data for incompressible turbulent channel flows using the Prandtl relation.

Figure 1( $a$ ) shows $\sqrt {2/C_{\!f}}$ versus $\textit{Re}_\tau$ for compressible turbulent channel flows, using a logarithmic scale for $\textit{Re}_\tau$ . The incompressible data listed in table 1 are also presented in figure 1. It is evident that $\sqrt {2/C_{\!f}}$ and $\textit{Re}_\tau$ for incompressible cases agree well with the Prandtl relation. However, $\sqrt {2/C_{\!f}}$ values for compressible turbulent channel flows are lower than those predicted by the Prandtl relation, and these deviations increase with rising $\textit{Ma}_b$ at the same $\textit{Re}_\tau$ . This observation aligns with the findings of Li et al. (Reference Li, Fan, Modesti and Cheng2019) and Yao & Hussain (Reference Yao and Hussain2020). The squared Pearson correlation coefficient $R^2$ between $\sqrt {2/C_{\!f}}$ and $\ln \textit{Re}_\tau$ for compressible turbulent channel flows is $0.85$ , indicating that these two variables are not strongly linearly correlated. Here, the $R^2$ value for a pair of variables $(X,Y)$ can be calculated using the formula $R^2=\text{cov}^2(X,Y)/\sigma _X^2\sigma _Y^2$ , where $\text{cov}$ denotes the covariance, and $\sigma _X$ and $\sigma _Y$ represent the standard deviations of $X$ and $Y$ , respectively.

Figure 1. Plots of $\sqrt {2/C_{f}}$ versus ( $a$ ) $\textit{Re}_{\tau }$ and ( $b$ ) $\textit{Re}_{\tau }^*$ in compressible turbulent channel flows. Here, $\kappa _{\!f}$ and $C$ in the Prandtl relation are obtained from Abe & Antonia (Reference Abe and Antonia2016). The squared Pearson correlation coefficients $R^2$ are provided in each plot. Only compressible data are used to calculate $R^2$ .

Furthermore, Li et al. (Reference Li, Fan, Modesti and Cheng2019) and Yao & Hussain (Reference Yao and Hussain2020) empirically attempted to replace $\textit{Re}_\tau$ with $\textit{Re}_\tau ^*$ to enable the compressible $C_{\!f}$ and $\textit{Re}_\tau ^*$ to conform to the Prandtl relation. Here, the semi-local friction Reynolds number $\textit{Re}_\tau ^*$ is defined as $\textit{Re}_\tau ^*=h/\delta _v^*(h)=\textit{Re}_\tau\, \sqrt {\rho _c/\rho _w}/(\mu _c/\mu _w)$ , where $\delta _v^*(h)=\mu _c/\sqrt {\tau _w\rho _c}$ . Evidently, $\textit{Re}_\tau ^*=\textit{Re}_\tau$ for incompressible cases. As shown in figure 1( $b$ ), the values of $\sqrt {2/C_{\!f}}$ for compressible cases are higher than those predicted by the Prandtl relation at a given $\textit{Re}_\tau ^*$ , and these deviations increase with rising $\textit{Ma}_b$ . The deviations of compressible $C_{\!f}$ at a given $\textit{Re}_\tau ^*$ are also confirmed by Li et al. (Reference Li, Fan, Modesti and Cheng2019) and Yao & Hussain (Reference Yao and Hussain2020). The value of $R^2$ between $\sqrt {2/C_{\!f}}$ and $\ln \textit{Re}_\tau ^*$ is $0.89$ , thus these two variables for compressible turbulent channel are also not strongly linearly correlated. The observations above, which align with those from Li et al. (Reference Li, Fan, Modesti and Cheng2019) and Yao & Hussain (Reference Yao and Hussain2020), indicate that the effects of Mach number on the skin-friction law of compressible turbulent channel flows cannot be fully accounted for by solely using $\textit{Re}_\tau ^*$ .

The error of $C_{\!f}$ from DNS data, compared to the Prandtl relation, is provided in figure 2. Evidently, the errors increase with increasing $\textit{Ma}_b$ at the same $\textit{Re}_\tau$ or $\textit{Re}_\tau ^*$ . The maximum error is $11.12\,\%$ when using $\textit{Re}_\tau$ , and $14.40\,\%$ when using $\textit{Re}_\tau ^*$ . These results quantitatively verify that $C_{\!f}$ and $\textit{Re}_\tau$ , as well as $C_{\!f}$ and $\textit{Re}_\tau ^*$ , for compressible turbulent channel flows, do not adequately satisfy the Prandtl relation. To this end, a transformation needs to be developed to map the skin-friction law of compressible turbulent channel flows to the classical Prandtl relation.

Figure 2. The error of the skin-friction coefficient, defined as $|(\sqrt {2/C_{\!f}})_{DNS}-(\sqrt {2/C_{\!f}})_{Pr}|/(\sqrt {2/C_{\!f}})_{Pr}$ , versus ( $a$ ) $\textit{Re}_{\tau }$ and ( $b$ ) $\textit{Re}_{\tau }^*$ in compressible turbulent channel flows. Here, $(\sqrt {2/C_{\!f}})_{Pr}$ represents the value calculated using the Prandtl relation with $\kappa _{\!f}=0.394$ and $C=2.41$ .

3.2. Pipe flows

We can also define $C_{\!f}$ and $\textit{Re}_b$ in turbulent pipe flows by replacing the half-width of the channel $h$ in (3.1) and (3.3) with the radius of the pipe $R$ . For instance, the bulk Reynolds number for turbulent pipe flows can be expressed as $\textit{Re}_b=\rho _bu_bR/\mu _w$ . Due to geometric differences, the form of the bulk average in pipe flows differs from that in channel flows. For example, the bulk velocity $u_b$ in pipe flows can be expressed as

(3.11) \begin{equation} u_b=\frac {1}{\pi{\kern-1.5pt}R^2\rho _b}\iint _S\bar {\rho }\bar {u}r\,\text{d}r\,\text{d}\theta =\frac {2}{ R^2\rho _b}\int _0^R\bar {\rho }\bar {u}r\text{d}r, \end{equation}

where the cross-section of the pipe, $S$ , is described using the cylindrical coordinates $(r,\theta )$ . Using the wall-normal coordinate $y=R-r$ , $u_b$ in pipe flows can be further expressed as

(3.12) \begin{equation} u_b=\frac {2}{ R\rho _b}\int _0^R\bar {\rho }\bar {u}\left (1-\frac {y}{R}\right )\text{d}y. \end{equation}

Considering the differences in the form of the bulk average, the Prandtl relation for pipe flows is derived in detail. Similar to the derivation in channel flows, the logarithmic law of velocity (3.4) can also be used to represent the entire velocity profile for approximating $u_b$ in incompressible turbulent pipe flows (Zanoun et al. Reference Zanoun, Durst, Bayoumy and Al-Salaymeh2007; Schlichting & Gersten Reference Schlichting and Gersten2016). Thus $u_b$ in incompressible cases can be estimated as

(3.13) \begin{align} \begin{split} u_b &=\frac {2u_\tau }{ \textit{ER}^+}\int _0^{u_c^+}\bar {u}^+\left [1-\frac {\exp {(\kappa \bar {u}^+)}}{\textit{ER}^+}\right ]{\rm d}\exp (\kappa \bar {u}^+)\\[5pt] &=\frac {2u_\tau }{\textit{ER}^+}\left [\frac {\kappa u_c^+\exp (\kappa u_c^+)-\exp (\kappa u_c^+)+1}{\kappa }-\frac {2\kappa u_c^+\exp (2\kappa u_c^+)-\exp (2\kappa u_c^+)+1}{4\kappa \textit{ER}^+}\right ]\!. \end{split} \end{align}

According to the fact that $\kappa u_c^+$ is typically of the order of $O(10)$ for incompressible turbulent pipe flows (Wu & Moin Reference Wu and Moin2008; Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), the ratios of $\exp (\kappa u_c^+)-1$ to $\kappa u_c^+\exp (\kappa u_c^+)$ and $\exp (2\kappa u_c^+)-1$ to $2\kappa u_c^+\exp (2\kappa u_c^+)$ are approximately $O(10^{-1})$ . The DNS data for incompressible turbulent pipe flows, as presented in table 2, can be used to verify the estimations of these two ratios. The ratios of $\exp (\kappa u_c^+)-1$ to $\kappa u_c^+\exp (\kappa u_c^+)$ and $\exp (2\kappa u_c^+)-1$ to $2\kappa u_c^+\exp (2\kappa u_c^+)$ are $0.13$ and $0.06$ , respectively, for the case with $\textit{Re}_\tau =180$ . For the case with $\textit{Re}_\tau =6015$ , these two ratios are $0.09$ and $0.04$ . The data confirm that the estimations of these two ratios, of the order of $O(10^{-1})$ , are reasonable. Therefore, $\exp (\kappa u_c^+)-1$ and $\exp (2\kappa u_c^+)-1$ in (3.13) can be neglected. Combined with logarithmic law $R^+=\exp (\kappa u_c^+)/E$ , neglecting these enables a further estimation of $u_b$ for incompressible turbulent pipe flows as

(3.14) \begin{equation} u_b\approx \frac {u_\tau u_c^+\exp (\kappa u_c^+)}{\textit{ER}^+}. \end{equation}

By substituting (3.14) into the definition of $\textit{Re}_b$ , the value of $\textit{Re}_b$ for incompressible turbulent pipe flows can also be determined using (3.7).

Similar to channel flows, the estimation of $u_b$ in (3.14) indicates that the central velocity of incompressible pipe flows can be approximated as the bulk velocity, namely $u_c\approx u_b$ . According to the derivation from (3.13)–(3.14), the relative error of this approximation is theoretically of the order of $O(10^{-1})$ . Based on the data shown in table 2, the ratio $|u_c-u_b|/u_c$ is $0.24$ for the case with $\textit{Re}_\tau =180$ , and $0.16$ for the case with $\textit{Re}_\tau =6015$ , supporting the theoretical estimation that the relative error of the approximation $u_c\approx u_b$ is of the order of $O(10^{-1})$ . Thus the approximation in (3.8) is valid for incompressible turbulent pipe flows. Based on (3.7) and (3.8), the Prandtl relation expressed in (1.1) and (1.2) also applies to incompressible turbulent pipe flows. Moreover, as demonstrated by Abe & Antonia (Reference Abe and Antonia2016), $\kappa _{\!f}=0.407$ and $C=2.0$ show good agreement between the Prandtl relation and experimental and DNS data in incompressible pipe flows. The values of $\kappa _{\!f}$ and $C$ in pipe flows and channel flows are clearly different, attributed to geometric variations (Abe & Antonia Reference Abe and Antonia2016).

Figure 3( $a$ ) presents $\sqrt {2/C_{\!f}}$ versus $\textit{Re}_\tau$ for compressible turbulent pipe flows, utilising a logarithmic scale for $\textit{Re}_\tau$ . The incompressible data listed in table 2 are also plotted in figure 3( $a$ ) and demonstrate an elegant agreement with the Prandtl relation. Although the $R^2$ value between $\sqrt {2/C_{\!f}}$ and $\ln \textit{Re}_\tau$ is as high as $0.95$ , the $\sqrt {2/C_{\!f}}$ value for compressible data is lower than that predicted by the Prandtl relation at $\textit{Re}_\tau \gtrsim 500$ , with deviations increasing as $\textit{Ma}_b$ rises at $\textit{Re}_\tau \approx 500$ . By replacing $\textit{Re}_\tau$ with $\textit{Re}_\tau ^*$ , $\sqrt {2/C_{\!f}}$ versus $\textit{Re}_\tau ^*$ for compressible data is shown in figure 3( $b$ ). The $R^2$ value $0.87$ indicates that $\sqrt {2/C_{\!f}}$ and $\ln \textit{Re}_\tau ^*$ are not strongly linearly correlated. The value of $\sqrt {2/C_{\!f}}$ for compressible data is higher than that predicted by the Prandtl relation at low $\textit{Re}_\tau ^*\approx 150$ , and deviations increase with increasing $\textit{Ma}_b$ . The error of $C_{\!f}$ from DNS data, compared to the Prandtl relation, is provided in figure 4. The maximum error is $7.40\,\%$ when using $\textit{Re}_\tau$ , and $12.40\,\%$ when using $\textit{Re}_\tau ^*$ . Moreover, the errors increase with increasing $\textit{Ma}_b$ at the same $\textit{Re}_\tau$ or $\textit{Re}_\tau ^*$ . All the observations above quantitatively confirm that $C_{\!f}$ and $\textit{Re}_\tau$ , as well as $C_{\!f}$ and $\textit{Re}_\tau ^*$ , for compressible turbulent pipe flows, do not adequately satisfy the Prandtl relation. Therefore, similar to the compressible turbulent channel flow, the compressible turbulent pipe flow also requires a transformation that unifies its skin-friction law with the classical Prandtl relation.

Figure 3. Plots of $\sqrt {2/C_{f}}$ versus ( $a$ ) $\textit{Re}_{\tau }$ and ( $b$ ) $\textit{Re}_{\tau }^*$ in compressible turbulent pipe flows. Here, $\kappa _{\!f}$ and $C$ in the Prandtl relation are reported by Abe & Antonia (Reference Abe and Antonia2016); $R^2$ is calculated using compressible data.

Figure 4. The error of the skin-friction coefficient, defined as $|(\sqrt {2/C_{\!f}})_{DNS}-(\sqrt {2/C_{\!f}})_{Pr}|/(\sqrt {2/C_{\!f}})_{Pr}$ , versus ( $a$ ) $\textit{Re}_{\tau }$ and ( $b$ ) $\textit{Re}_{\tau }^*$ in compressible turbulent pipe flows. Here, $(\sqrt {2/C_{\!f}})_{Pr}$ represents the value calculated using the Prandtl relation with $\kappa _{\!f}=0.407$ and $C=2.0$ .

4.1. Channel flows

To account for the effect of Mach number in the velocity profile of compressible turbulent channel flows, we redefine the bulk velocity as

(4.1) \begin{equation} u_{b^*}=\frac{u_{\tau}}{\rho_{b}h^{*}}\int _0^{h^*}\bar {\rho } \bar {U}_{\!I}\,\text{d}y^*, \end{equation}

which utilises the bulk average of the transformed velocity $\bar {U}_{\!I}(y^*)$ under the semi-local wall-normal coordinate $y^*$ . Here, $y^*$ is defined as $y^*=y/\delta _v^*(y)$ , with $\delta _v^*(y)=\bar {\mu }/\sqrt {\tau _w\bar {\rho }}$ , and $h^*=h/\delta _v^*(h)$ . The transformed velocity $\bar {U}_{\!I}$ can be derived from the constant-stress-layer GFM velocity transformation (Griffin et al. Reference Griffin, Fu and Moin2021) and is expressed as

(4.2) \begin{equation} \bar {U}_{\!I}=\int _0^{y^*}\dfrac {\dfrac {1}{\bar {\mu }^+}\dfrac {\text{d}\bar {u}^+}{\text{d}y^*}}{1+\dfrac {1}{\bar {\mu }^+}\dfrac {\text{d}\bar {u}^+}{\text{d}y^*}-\bar {\mu }^+\dfrac {\text{d}\bar {u}^+}{\text{d}y^+}}\,\text{d}y^*. \end{equation}

By comparing (4.1) with (3.2), it is clear that the redefined $u_{b^*}$ mathematically represents the bulk transformed velocity under the semi-local coordinate. Moreover, according to the results of Griffin et al. (Reference Griffin, Fu and Moin2021) and Bai, Griffin & Fu (Reference Bai, Griffin and Fu2022), the profiles of transformed $\bar {U}_{\!I}(y^*)$ in the inner layer of canonical compressible wall-bounded turbulence, including fully developed channel and pipe flows, as well as the turbulent boundary layer, successfully collapse into the incompressible velocity profiles and are independent of the Mach number. To this end, the redefined $u_{b^*}$ physically represents a bulk velocity that is unaffected by the Mach number effect present in the velocity profiles of compressible turbulent channel flows.

The effects of thermodynamic property variations in compressible turbulent channel flows on the skin-friction relation can be absorbed by a transformation for the redefined skin-friction coefficient $C_{\!f^*}$ and the redefined bulk Reynolds number $\textit{Re}_{b^*}$ . Here, $C_{\!f^*}$ and $\textit{Re}_{b^*}$ are defined based on $u_{b^*}$ and are expressed as

(4.3) \begin{equation} C_{\!f^*}=\frac {2\tau _w}{\rho _bu_{b^*}^2} \end{equation}

and

(4.4) \begin{equation} \textit{Re}_{b^*} = \frac {\rho _bu_{b^*}h}{\mu _w}, \end{equation}

respectively. Moreover, the thermodynamic properties remain constant for incompressible channel flows, resulting in $y^*=y^+$ , $\bar {\rho }(y)=\rho _b$ , $\bar {U}_{\!I}=\bar {u}^+$ and $h^*=h^+$ . Hence $u_{b^*}$ in (4.1) reduces to the incompressible bulk velocity, expressed as $u_b=1/h\int _0^h\bar {u}\,\text{d}y$ , when the flow is incompressible. This ensures the reduction of the redefined $C_{\!f^*}$ and $\textit{Re}_{b^*}$ based on $u_{b^*}$ to the incompressible $C_{\!f}$ and $\textit{Re}_b$ , respectively, when the flow is incompressible and thermodynamic properties remain constant.

Subsequently, we approximate $u_{b^*}$ using asymptotic analysis to develop the skin-friction transformation and establish the skin-friction law for compressible turbulent channel flows. Given that the transformed $\bar {U}_{\!I}(y^*)$ collapses into the incompressible velocity profiles, the logarithmic law of $\bar {U}_{\!I}(y^*)$ in compressible turbulent channel flows, expressed as

(4.5) \begin{equation} y^*=\frac {\exp (\kappa \bar {U}_{\!I})}{E}, \end{equation}

can also be utilised to represent the entire mean velocity profile for approximating $u_{b^*}$ , similar to the incompressible cases. By substituting (4.5) into (4.1), $u_{b^*}$ can be determined by

(4.6) \begin{equation} u_{b^*}=\frac {u_\tau U_{\!I,c}}{\textit{Eh}^*}\int _0^1\frac {\bar {\rho }}{\rho _b} Z\,{\rm d}\exp (\kappa U_{\!I,c}Z), \end{equation}

where $Z=\bar {U}_{\!I}/U_{\!I,c}$ , and the transformed velocity at the centre is $U_{\!I,c}=\bar {U}_{\!I}(h^*)$ . The integral term in the above equation can be integrated by parts and expressed as

(4.7) \begin{align} \begin{split} \int _0^1\frac {\bar {\rho }}{\rho _b} Z\,\text{d}\exp (\kappa U_{\!I,c}Z)&=\left [\frac {\bar {\rho }}{\rho _b} Z\exp (\kappa U_{\!I,c}Z)\right ]_0^1-\int _0^1\left (\frac {Z}{\rho _b}\frac {\text{d}\bar {\rho }}{\text{d}Z}+\frac {\bar {\rho }}{\rho _b}\right )\exp (\kappa U_{\!I,c}Z)\,\text{d}Z\\[5pt] &=\frac {\rho _c}{\rho _b}\exp (\kappa U_{\!I,c})-\frac {1}{\kappa U_{\!I,c}}\int _0^1\left (\frac {Z}{\rho _b}\frac {\text{d}\bar {\rho }}{\text{d}Z}+\frac {\bar {\rho }}{\rho _b}\right )\text{d}\exp (\kappa U_{\!I,c}Z). \end{split} \end{align}

Here, the term $ [(\bar {\rho }/\rho _b)Z\exp (\kappa U_{\!I,c}Z) ]_0^1$ represents the difference in values of $ [(\bar {\rho }/\rho _b)Z\exp (\kappa U_{\!I,c}Z) ]$ at the centre ( $Z=1$ ) and the wall ( $Z=0$ ). Similarly, by repeatedly applying integration by parts to (4.7), the expression for $u_{b^*}$ in (4.6) can be further expressed as an asymptotic series of $1/(\kappa U_{\!I,c})$ as

(4.8) \begin{align} u_{b^*}=\dfrac {u_\tau U_{\!I,c}}{\textit{Eh}^*}\left \lbrace \dfrac {\rho _c}{\rho _b}\exp (\kappa U_{\!I,c})-\dfrac {\left (\dfrac {1}{\rho _b}\left .\dfrac {\text{d}\bar {\rho }}{\text{d}Z}\right |_{Z=1}+\dfrac {\rho _c}{\rho _b}\right )\exp (\kappa U_{\!I,c})-\dfrac {\rho _w}{\rho _b}}{\kappa U_{\!I,c}}+O\left [\dfrac {1}{(\kappa U_{\!I,c})^2}\right ]\right \rbrace \!. \end{align}

Since the transformed velocity profile $\bar {U}_{\!I}(y^*)$ collapses into the incompressible velocity profile $\bar {u}^+(y^+)$ , the value of $\kappa U_{\!I,c}$ is similar to that of $\kappa u_c^+$ and is typically of the order of $O(10)$ for compressible turbulent channel flows. Consequently, the ratio of the magnitude of the first-order term related to $1/(\kappa U_{\!I,c})$ to the magnitude of the zeroth-order term is of the order of $O(10^{-1})$ . This estimation can be confirmed by the data presented in tables 3 and 4. At high Mach number $\textit{Ma}_b=3.0$ , we select the case with $\textit{Re}_\tau =448$ from Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016), and the case with $\textit{Re}_\tau =1850$ from Trettel & Larsson (Reference Trettel and Larsson2016). These two cases represent the maximum and minimum Reynolds numbers at $\textit{Ma}_b=3.0$ . The ratio of the magnitude of the first-order term related to $1/(\kappa U_{\!I,c})$ in (4.8) to the magnitude of the zeroth-order term is $0.12$ for both the case with $\textit{Re}_\tau =448$ and the case with $\textit{Re}_\tau =1850$ . To this end, the higher-order terms that are smaller than the zeroth-order term in (4.8) can be reasonably neglected, allowing $u_{b^*}$ for compressible turbulent channel flows to be approximated as

(4.9) \begin{equation} u_{b^*}\approx \frac {u_\tau U_{\!I,c}\rho _c}{E\rho_{b}h^{*}}\exp (\kappa U_{\!I,c}). \end{equation}

By substituting (4.9) into (4.4), the redefined $\textit{Re}_{b^*}$ for compressible turbulent channel flows can be determined by

(4.10) \begin{equation} \textit{Re}_{b^*}=\frac {\mu _c\sqrt {\rho _c}}{\mu _w\sqrt {\rho _w}}\frac {U_{\!I,c}}{E}\exp (\kappa U_{\!I,c}). \end{equation}

Moreover, based on the logarithmic law expressed in (4.5), one can obtain $h^*=\exp (\kappa U_{\!I,c})/E$ . Hence the estimation of $u_{b^*}$ in (4.9) indicates that $U_{\!I,c}\approx u_{b^*}^+\rho _b/\rho _c$ . According to the derivation from (4.8)–(4.9), the relative error of this approximation is theoretically of the order of $O(10^{-1})$ . Indeed, at high Mach number $\textit{Ma}_b=3.0$ , the ratio $|U_{\!I,c}-u_{b^*}^+\rho _b/\rho _c|/U_{\!I,c}$ is $0.11$ for the case with a minimum $\textit{Re}_\tau =448$ , and is $0.10$ for the case with a maximum $\textit{Re}_\tau =1850$ . To this end, the data from compressible turbulent channel flows confirm that the approximation $U_{\!I,c}\approx u_{b^*}^+\rho _b/\rho _c$ is valid. By substituting this approximation into the definition of $C_{\!f^*}$ in (4.3), one can obtain

(4.11) \begin{equation} U_{\!I,c}\approx \sqrt {\dfrac {2}{C_{\!f^*}\dfrac {\rho _c\rho _c}{\rho _w\rho _b}}}. \end{equation}

Based on (4.10) and (4.11), the skin-friction law for compressible turbulent channel flows can be derived as

(4.12) \begin{equation} \sqrt {\frac {2}{C_{\!f,i}}}\propto \ln \left (\textit{Re}_{b,i}\,\sqrt {\frac {C_{\!f,i}}{2}}\right )\!. \end{equation}

In the above skin-friction law, a skin-friction transformation for $C_{\!f^*}$ and $\textit{Re}_{b^*}$ is defined as

(4.13) \begin{equation} C_{\!f,i}=F_{C^*}C_{\!f^*},\quad \textit{Re}_{b,i}=F_{b^*}\textit{Re}_{b^*}, \end{equation}

where the transformation factors are expressed as

(4.14) \begin{equation} F_{C^*}=\frac {\rho _c\rho _c}{\rho _w\rho _b},\quad F_{b^*}=\frac {\mu _w\sqrt {\rho _w}}{\mu _c\sqrt {\rho _c}}. \end{equation}

These two transformation factors capture the effects of thermodynamic property variations in compressible turbulent channel flows on the skin-friction relation. Clearly, the transformation in (4.13) accurately maps the skin-friction law for compressible turbulent channel flow in (4.12) to the classical Prandtl relation as expressed in (1.1). Furthermore, according to the definitions of $\textit{Re}_{b^*}$ , $C_{\!f^*}$ and $\textit{Re}_{\tau }^*$ , one can derive the identity expressed as

(4.15) \begin{equation} \textit{Re}_{b,i}\,\sqrt {\frac {C_{\!f,i}}{2}}=\frac {\sqrt {\tau _w\rho _c}\,h}{\mu _c}=\textit{Re}_{\tau }^*. \end{equation}

Hence the skin-friction law (4.12) for compressible turbulent channel flows can be equivalently expressed as

(4.16) \begin{equation} \sqrt {\frac {2}{C_{\!f,i}}}\propto \ln \textit{Re}_{\tau }^*, \end{equation}

which is exactly the same as the classical Prandtl relation expressed by (1.2).

For incompressible turbulent channel flows, it is evident that $F_{C^*}=1$ , $F_{b^*}=1$ , and that $C_{\!f^*}$ and $\textit{Re}_{b^*}$ reduce to the incompressible $C_{\!f}$ and $\textit{Re}_b$ , respectively. This ensures that the newly transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$ can be accurately reduced to their incompressible counterparts when the flow is incompressible. Thus the established skin-friction law can be reduced to the incompressible relation, indicating that the present novel approach theoretically generalises the classical Prandtl relation of incompressible turbulent channel flows to compressible cases. By using $\kappa _{\!f}=0.394$ and $C=2.41$ obtained from a least squares fit to the incompressible data (Abe & Antonia Reference Abe and Antonia2016), the skin-friction law for compressible turbulent channel flows can be quantified as

(4.17) \begin{equation} \sqrt {\frac {2}{C_{\!f,i}}} = \frac {1}{\kappa _{\!f}}\ln\! \left (\textit{Re}_{b,i}\,\sqrt {\frac {C_{\!f,i}}{2}}\right )+C, \end{equation}

or equivalently,

(4.18) \begin{equation} \sqrt {\frac {2}{C_{\!f,i}}} = \frac {1}{\kappa _{\!f}}\ln \textit{Re}_\tau ^*+C. \end{equation}

4.2. Pipe flows

Similar to the approach used in channel flows, the skin-friction transformation can also be developed based on $u_{b^*}$ to map the skin-friction law of compressible turbulent pipe flows to the classical Prandtl relation. Since the form of the bulk average in pipe flows differs from that in channel flows, some necessary derivations specific to compressible pipe flows will be presented. The bulk transformed velocity $u_{b^*}$ in pipe flows should be expressed as

(4.19) \begin{equation} u_{b^*}=\frac {u_\tau }{\pi R^{*2}\rho _b}\iint _{S^*}\bar {\rho }\bar {U}_{\!I}r^*\,\text{d}r^*\,\text{d}\theta =\frac {2u_\tau }{ R^{*2}\rho _b}\int _0^{R^*}\bar {\rho }\bar {U}_{\!I}r^*\,\text{d}r^*, \end{equation}

where the transformed cross-section of the pipe, $S^*$ , is described using the cylindrical transformed coordinates $(r^*,\theta )$ . Here, $r^*$ represents the semi-local radial coordinate, defined as $r^*=R^*-y^*$ , where $y^*=y/\delta _v^*(y)$ and $R^*=h/\delta _v^*(R)$ . Using the semi-local wall-normal coordinate $y^*$ , $u_{b^*}$ in pipe flows can be further expressed as

(4.20) \begin{equation} u_{b^*}=\frac {2u_\tau }{ R^*\rho _b}\int _0^{R^*}\bar {\rho }\bar {U}_{\!I}\! \left (1-\frac {y^*}{R^*}\right )\text{d}y^*. \end{equation}

Here, the transformed velocity $\bar {U}_{\!I}$ is calculated using the constant-stress-layer GFM velocity transformation, as expressed by (4.2), which has been shown to map the compressible velocity profile in pipe flows to the incompressible velocity profile (Griffin et al. Reference Griffin, Fu and Moin2021). Hence similar to channel flows, the bulk transformed velocity $u_{b^*}$ in pipe flows is also unaffected by the Mach number effect present in the velocity profiles. The definitions of $C_{\!f^*}$ and $\textit{Re}_{b^*}$ mirror those in (4.3) and (4.4), respectively, with $h$ replaced by $R$ . For instance, $\textit{Re}_{b^*}$ for pipe flows is defined as $\textit{Re}_{b^*}=\rho _bu_{b^*}R/\mu _w$ . Moreover, in incompressible cases where thermodynamic properties remain constant, $u_{b^*}$ as expressed in (4.20) can be reduced to the bulk velocity of incompressible turbulent pipe flows as shown in (3.12). This ensures that the redefined $C_{\!f^*}$ and $\textit{Re}_{b^*}$ in pipe flows can also be reduced to the incompressible $C_{\!f}$ and $\textit{Re}_b$ , respectively, when the flow is incompressible.

Similar to channel flows, $u_{b^*}$ as expressed in (4.20) can be approximated using asymptotic analysis to establish the skin-friction law for compressible turbulent pipe flows. By using the logarithmic law of $\bar {U}_{\!I}$ (4.5) to represent the entire velocity profile, $u_{b^*}$ can be determined by

(4.21) \begin{align} \begin{split} u_{b^*}&=\frac {2u_\tau U_{\!I,c}}{\textit{ER}^*}\int _0^{1}\frac {\bar {\rho }}{\rho _b}Z\left [1-\frac {\exp (\kappa U_{\!I,c}Z)}{\textit{ER}^*}\right ]{\rm d}\exp (\kappa U_{\!I,c}Z)\\[5pt] &=\frac {2u_\tau U_{\!I,c}}{\textit{ER}^*}\left [\int _0^{1}\frac {\bar {\rho }}{\rho _b}Z\,{\rm d}\exp (\kappa U_{\!I,c}Z)-\frac {1}{2\textit{ER}^*}\int _0^{1}\frac {\bar {\rho }}{\rho _b}Z\,{\rm d}\exp (2\kappa U_{\!I,c}Z)\right ]\!, \end{split} \end{align}

where the transformed velocity at the centre is given by $U_{\!I,c}=\bar {U}_{\!I}(R^*)$ . By integrating by parts, the first integral term in (4.21) can be expressed as an asymptotic series of $1/(\kappa U_{\!I,c})$ as

(4.22) \begin{align} \begin{split} &\int _0^{1}\frac {\bar {\rho }}{\rho _b}Z\,\text{d}\exp (\kappa U_{\!I,c}Z)\\ &=\dfrac {\rho _c}{\rho _b}\exp (\kappa U_{\!I,c})-\dfrac {\left (\dfrac {1}{\rho _b}\left .\dfrac {\text{d}\bar {\rho }}{\text{d}Z}\right |_{Z=1}+\dfrac {\rho _c}{\rho _b}\right )\exp (\kappa U_{\!I,c})-\dfrac {\rho _w}{\rho _b}}{\kappa U_{\!I,c}}+O\left [\dfrac {1}{(\kappa U_{\!I,c})^2}\right ]\!. \end{split} \end{align}

Similarly, the second integral term in (4.21) can be expressed as an asymptotic series of $1/(2\kappa U_{\!I,c})$ as

(4.23) \begin{align} \begin{split} \int _0^{1}&\frac {\bar {\rho }}{\rho _b}Z\,\text{d}\exp (2\kappa U_{\!I,c}Z)\\ &=\dfrac {\rho _c}{\rho _b}\exp (2\kappa U_{\!I,c})-\dfrac {\left (\dfrac {1}{\rho _b}\left .\dfrac {\text{d}\bar {\rho }}{\text{d}Z}\right |_{Z=1}+\dfrac {\rho _c}{\rho _b}\right )\exp (2\kappa U_{\!I,c})-\dfrac {\rho _w}{\rho _b}}{2\kappa U_{\!I,c}}+O\left [\dfrac {1}{(2\kappa U_{\!I,c})^2}\right ]\!. \end{split} \end{align}

Given the fact that $\bar {U}_{\!I}(y^*)$ collapses into the incompressible velocity profile $\bar {u}^+(y^+)$ , the value of $\kappa U_{\!I,c}$ is similar to that of $\kappa u_c^+$ and is typically of the order of $O(10)$ for compressible turbulent pipe flows. Consequently, the ratios of the magnitudes of the first-order term to the zeroth-order term in both (4.22) and (4.23) are of the order of $O(10^{-1})$ . Two representative cases in table 5, featuring $\textit{Ma}_b=3.0$ with a minimum $\textit{Re}_{\tau }^*=148$ , and $\textit{Ma}_b=1.5$ with a maximum $\textit{Re}_\tau ^*=669$ , are selected to verify the estimations of these two ratios. The ratios of the magnitudes of the first-order term to the zeroth-order term in both (4.22) and (4.23) are $0.12$ and $0.06$ , respectively, for the case with minimum $\textit{Re}_{\tau }^*=148$ . These two ratios are $0.05$ and $0.03$ for the case with maximum $\textit{Re}_{\tau }^*=669$ . Clearly, these two ratios are of the order of $O(10^{-1})$ , indicating that the higher-order terms in both (4.22) and (4.23) can be reasonably neglected in comparison to the zeroth-order terms. Combined with logarithmic law $R^*=\exp (\kappa U_{\!I,c})/E$ , neglecting these leads to a further estimation of $u_b^*$ for compressible turbulent pipe flows, given by

(4.24) \begin{equation} u_{b^*}\approx \frac {u_\tau U_{\!I,c}\rho _c}{E\rho _b R^*}\exp (\kappa U_{\!I,c}), \end{equation}

which is similar to the estimation of $u_{b^*}$ in compressible turbulent channel flows as expressed in (4.9).

By substituting (4.24) into the definition of $\textit{Re}_{b^*}$ , the value of $\textit{Re}_{b^*}$ for compressible turbulent pipe flows can also be determined by (4.10). Similar to channel flows, the estimation of $u_{b^*}$ in (4.24) indicates that the approximation $U_{\!I,c}\approx u_{b^*}^+\rho _b/\rho _c$ holds for compressible turbulent pipe flows, with the relative error of this approximation theoretically of the order of $O(10^{-1})$ . In fact, the ratio $|U_{\!I,c}-u_{b^*}^+\rho _b/\rho _c|/U_{\!I,c}$ is $0.18$ for the case with $\textit{Ma}_b=3.0$ and a minimum $\textit{Re}_{\tau }^*=148$ , as well as the case with $\textit{Ma}_b=1.5$ and a maximum $\textit{Re}_\tau ^*=669$ . These two cases confirm that the ratio $|U_{\!I,c}-u_{b^*}^+\rho _b/\rho _c|/U_{\!I,c}$ is of the order of $O(10^{-1})$ . Thus the approximation $U_{\!I,c}\approx u_{b^*}^+\rho _b/\rho _c$ for compressible turbulent pipe flows is valid. By substituting this approximation into the definition of $C_{\!f^*}$ , it is evident that (4.11) for channel flows is also applicable to pipe flows. Since (4.10) and (4.11) are applicable to pipe flows, the newly developed transformation for channel flows, as expressed in (4.13), can also be applied to pipe flows. This means that the transformed $C_{\!f,i}$ , $\textit{Re}_{b,i}$ and $\textit{Re}_{\tau }^*$ for compressible turbulent pipe flows adhere to the skin-friction law expressed by (4.12) or (4.16). To this end, the proposed transformation in (4.13) can theoretically map the skin-friction law of compressible turbulent pipe flows to the classical Prandtl relation. By utilising $\kappa _{\!f}=0.407$ and $C=2.0$ , applicable to incompressible pipe flows as reported by Abe & Antonia (Reference Abe and Antonia2016), the quantified skin-friction law expressed by (4.17) or (4.18) can be applied to compressible turbulent pipe flows.