2. Scaling law of $\boldsymbol{C}_\boldsymbol{f}$ for compressible TBLs over a flat plate

The skin-friction coefficient, defined as $C_f=2\tau _w/\rho _\infty u_\infty ^2$ with wall shear stress $\tau _w=\overline {\mu }\,\text{d}\overline {u}/\text{d}y|_w$ , free-stream density $\rho _\infty$ , free-stream velocity $u_\infty$ , viscosity $\overline {\mu }$ and wall-normal coordinate $y$ , is a crucial parameter in the design of supersonic and hypersonic aircraft. Hereafter, an overline denotes the Reynolds average, and subscripts $w$ and $\infty$ represent quantities at the wall and in the free stream, respectively. Coefficient $C_f$ is widely recognised to exhibit a functional relation with the Reynolds number $Re_\theta =\rho _\infty u_\infty \theta /\mu _\infty$ based on the momentum thickness $\theta$ . According to Spalding & Chi (Reference Spalding and Chi1964), for compressible TBLs over a flat plate, $C_f$ and $Re_\theta$ can be linearly transformed to $C_{f,i}$ and $Re_{\theta ,i}$ by multiplying by $F_C$ and $F_\theta$ , respectively, i.e.

(2.1) \begin{equation} C_{f,i}=F_C C_f,\quad Re_{\theta ,i}=F_\theta\, Re_\theta . \end{equation}

Here, $C_{f,i}$ and $Re_{\theta ,i}$ should obey the incompressible scaling for $C_f$ , i.e. (1.1). The transformation factor $F_C$ is the same in both vD-II (van Driest Reference van Driest1951) and SC (Spalding & Chi Reference Spalding and Chi1964) theories considering variations in density and viscosity, and can be expressed as

(2.2) \begin{equation} (F_C)_{{vD}} = (F_C)_{{SC}} = \left [\int _0^1 \sqrt {\frac {\overline {\rho }}{\rho _\infty }}\,\text{d}z\right ]^{-2}, \end{equation}

with $z=\overline {u}/u_\infty$ , density $\overline {\rho }$ and streamwise velocity $\overline {u}$ . Here, subscripts ‘vD’ and ‘SC’ refer to the transformation factors from the vD-II and SC theories, respectively. Using the fact that the pressure is nearly constant in TBLs and the velocity–temperature relation, the factor $F_C$ can be further written as

(2.3) \begin{equation} (F_C)_{{vD}} = (F_C)_{{SC}} = \dfrac {\dfrac{T_r}{T_\infty}-1}{\left (\sin ^{-1}\alpha +\sin ^{-1}\beta \right )^2}, \end{equation}

where $\alpha =(2A^2-B)/(B^2+4A^2)^{1/2}$ , $\beta =B/(B^2+4A^2)^{1/2}$ , $A=[r(\gamma -1)/2\times Ma_\infty ^2\,T_\infty /T_w]^{1/2}$ and $B=T_r/T_w-1$ . Here, $r$ is the recovery factor, $T_r$ is the recovery temperature, and $\gamma$ is the heat capacity ratio. However, the factor $F_\theta$ differs between the two theories and is given as

(2.4) \begin{equation} (F_\theta )_{{vD}} = \dfrac {\mu _\infty }{\mu _w}, \quad (F_\theta )_{{SC}} = \left (\dfrac {T_\infty }{T_w}\right )^{0.702}\left (\dfrac {T_r}{T_w}\right )^{0.772}. \end{equation}

These two transformations are used most commonly but fail to predict $C_f$ on a highly cooled wall (Hopkins & Inouye Reference Hopkins and Inouye1971; Bradshaw Reference Bradshaw1977; Huang et al. Reference Huang, Duan and Choudhari2022). The key issue with these two theories is the use of a linear transformation to eliminate the influences of Mach number and heat transfer on $Re_\theta$ . Indeed, the momentum thickness is defined as

(2.5) \begin{equation} \theta = \int _0^{\delta _e}\frac {\overline {\rho }}{\rho _\infty }\frac {\overline {u}}{u_\infty }\left (1-\frac {\overline {u}}{u_\infty }\right )\text{d}y, \end{equation}

where $\delta _e$ represents the TBL edge, typically chosen at the location where $\overline {u}=0.99u_\infty$ . It is evident that the integrand in the definition of $\theta$ is a quadratic function of the velocity profile. Hence the factor $F_\theta$ in the linear transformation of (2.1) is unavailable to include all effects of Mach number and heat transfer on $\overline {u}$ . To address this concern, we redefine a momentum thickness $\theta ^*$ as

(2.6) \begin{equation} \theta ^* = \int _0^{\delta _e}\frac {\overline {\rho }}{\rho _\infty }\frac {\overline {U}_I}{U_{I,\infty }}\left (1-\frac {\overline {U}_I}{U_{I,\infty }}\right )\text{d}(y^*\delta _v), \end{equation}

where $\delta _e$ is determined at $\overline {U}_I=0.99U_{I,\infty }$ , the semi-local wall-normal coordinate is $y^*=y\sqrt {\tau _w\overline {\rho }}/\overline {\mu }$ , the viscous length scale is $\delta _v=\mu _w/\sqrt {\tau _w\rho _w}$ , and $\overline {U}_I$ using the physics-based GFM velocity transformation (Griffin et al. Reference Griffin, Fu and Moin2021) is given as

(2.7) \begin{equation} \overline {U}_I = \int _0^{y^*}\frac {\dfrac{1}{\overline {\mu }^+} \dfrac{\text{d}\overline {U}^+}{\text{d}y^*}}{1 + \dfrac{1}{\overline {\mu }^+}\dfrac{\text{d}\overline {U}^+}{\text{d}y^*} - \overline {\mu }^+ \dfrac{\text{d}\overline {U}^+}{\text{d}y^+}}\,\text{d}y^*. \end{equation}

Throughout this paper, the superscript $+$ indicates a non-dimensionalisation by the friction velocity $u_\tau =\sqrt {\tau _w/\rho _w}$ , $\delta _v$ and $\mu _w$ . Note that $\overline {U}_I$ in (2.7) is based on constant-stress-layer GFM transformation. In fact, the performances of $\overline {U}_I$ based on total-stress-based GFM transformation without the constant-stress-layer assumption, and on constant-stress-layer GFM transformation, are nearly identical (Griffin et al. Reference Griffin, Fu and Moin2021). Therefore, the constant-stress-layer assumption in (2.7) does not impact the establishment of the skin-friction scaling law, which has also been validated in Appendix A. The profiles of the transformed $\overline {U}_I(y^*)$ in compressible TBLs, with and without heat transfer, collapse to the incompressible law of the wall, and are independent of Mach number and heat transfer. Clearly, $\theta ^*$ is similar to the traditional momentum thickness $\theta$ , except that $\theta ^*$ is based on $\overline {U}_I$ and $y^*$ . Since the profiles of $\overline {U}_I(y^*)$ are independent of Mach number and heat transfer, $\theta ^*$ can be physically interpreted as a momentum thickness unaffected by the effects of Mach number and heat transfer in the velocity profiles of compressible TBLs. Therefore, only $\overline {\rho }$ in the redefined $\theta ^*$ is affected by Mach number and heat transfer. These effects can be reasonably included in a linear transformation factor $F_{\theta ^*}$ with a redefined Reynolds number $Re_{\theta ^*}=\rho _\infty u_\infty \theta ^*/\mu _\infty$ . It is important to highlight that by multiplying $y^*$ by $\delta _v$ in (2.6), $Re_{\theta ^*}$ can be precisely reduced to the $Re_{\theta }$ of the incompressible case, where $\overline {\rho }$ and $\overline {\mu }$ are nearly constant.

Subsequently, we will theoretically derive the scaling law for $C_f$ based on $Re_{\theta ^*}$ . Given the fact that the contribution of the integrand to $\theta ^*$ in both the viscous sublayer ( $\overline {U}_I/U_{I,\infty} \rightarrow 0$ ) and outer layer ( $\overline {U}_I/U_{I,\infty} \rightarrow 1$ ) is negligible, the log-law behaviour of $\overline {U}_I$ , expressed as

(2.8) \begin{equation} y^* = \dfrac{\exp (\kappa \overline {U}_I)}{E}, \end{equation}

is suitably employed to approximate $\theta ^*$ . Here, $\kappa$ is the von Kármán constant, and $E$ is a constant. By substituting (2.8) into (2.6), one can obtain

(2.9) \begin{equation} \theta ^* = \frac {\kappa }{E}\delta _vU_{I,\infty }\int _0^1\frac {\overline {\rho }}{\rho _\infty }Z(1-Z)\exp (\kappa U_{I,\infty }Z)\,\text{d}Z, \end{equation}

where $Z=\overline {U}_I/U_{I,\infty }$ . The integral term in above equation, $\int _0^1(\overline {\rho }/\rho _\infty) Z(1-Z){} \!\exp(\kappa U_{I,\infty }Z)\,\text{d}Z = 1/(\kappa U_{I,\infty })\int _0^1(\overline {\rho }/\rho _\infty) Z(1-Z)\,\text{d}\exp (\kappa U_{I,\infty }Z)$ , can be integrated by parts and expressed as

(2.10) \begin{align} & \frac {1}{\kappa U_{I,\infty }}\int _0^1\frac {\overline {\rho }}{\rho _\infty } Z(1-Z)\,\text{d}\exp (\kappa U_{I,\infty }Z) \notag \\ & =\frac {1}{\kappa U_{I,\infty }}\left \lbrace \left [\frac {\overline {\rho }}{\rho _\infty } Z(1-Z)\exp (\kappa U_{I,\infty }Z)\right ]_0^1- \int _0^1\exp (\kappa U_{I,\infty }Z)\,\text{d}\frac {\overline {\rho }}{\rho _\infty } Z(1-Z)\right \rbrace \notag \\ & =-\frac {1}{\kappa U_{I,\infty }}\int _0^1\left [\frac {\overline {\rho }}{\rho _\infty } (1-2Z)-\frac {Z(1-Z)}{\rho _\infty }\frac {\text{d}\rho }{\text{d}Z} \right ] \exp (\kappa U_{I,\infty }Z)\,\text{d}Z. \end{align}

Here, the term $ \bigl[{(\overline {\rho }}/{\rho _\infty }) Z(1-Z)\exp (\kappa U_{I,\infty }Z) \bigr]_0^1$ represents the difference in values of $ [({\overline {\rho }}/{\rho _\infty }) Z(1-Z)\exp (\kappa U_{I,\infty }Z) ]$ at the locations $Z=1$ and $Z=0$ . Similarly, by repeatedly applying integration by parts to (2.10), (2.9) can be expressed as an asymptotic series of $1/\kappa U_{I,\infty }$ as

(2.11) \begin{align} \theta ^* &= \frac {\kappa }{E}\delta _vU_{I,\infty } \times\left \lbrace \frac {\exp (\kappa U_{I,\infty })+ \dfrac{1}{\rho _\infty ^+}}{(\kappa U_{I,\infty })^2} \vphantom{\frac {\left \lbrace \dfrac {\text{d}}{\text{d}Z}\left [\dfrac {\overline {\rho }}{\rho _\infty } (1-2Z)-\dfrac {Z(1-Z)}{\rho _\infty }\dfrac {\text{d}\rho }{\text{d}Z} \right ] \exp (\kappa U_{I,\infty }Z)\right \rbrace _0^1 }{\left \lbrace \dfrac {\text{d}}{\text{d}Z}\left [\dfrac {\overline {\rho }}{\rho _\infty } (1-2Z)-\dfrac {Z(1-Z)}{\rho _\infty }\dfrac {\text{d}\rho }{\text{d}Z} \right ] \exp (\kappa U_{I,\infty }Z)\right \rbrace _0^1 }} \right . \notag \\&\quad + \left . \frac {\left \lbrace \dfrac {\text{d}}{\text{d}Z}\left [\dfrac {\overline {\rho }}{\rho _\infty } (1-2Z)-\dfrac {Z(1-Z)}{\rho _\infty }\dfrac {\text{d}\rho }{\text{d}Z} \right ] \exp (\kappa U_{I,\infty }Z)\right \rbrace _0^1 }{(\kappa U_{I,\infty })^3} + O\left [\frac {1}{(\kappa U_{I,\infty })^4}\right ]\right \rbrace . \end{align}

According to the logarithmic law of TBLs, it can be estimated that $U_{I,\infty }\gtrsim 20$ , leading to $\kappa U_{I,\infty }$ being of the order of $O(10)$ for TBLs. The ratio of magnitude of the third-order term related to $1/(\kappa U_{I,\infty })^3$ to the magnitude of the second-order term related to $1/(\kappa U_{I,\infty })^2$ is of the order of $O(10^{-1})$ . Moreover, given the fact that pressure is nearly constant in TBLs, $1/\rho _\infty ^+\approx T_\infty /T_w$ . For common high-speed TBLs, $T_\infty /T_w \lt 1$ , which implies $1/\rho _\infty ^+\lt 1\ll \exp (\kappa U_{I,\infty })\sim O(10^4)$ . To this end, the term $1/\rho _\infty ^+$ , which is much smaller than $\exp (\kappa U_{I,\infty })$ , along with the higher-order terms that are smaller than the second-order term, can be neglected. Consequently, $Re_{\theta ^*}$ can be determined by

(2.12) \begin{equation} Re_{\theta ^*} = \frac {1}{E}\frac {\rho _\infty u_\infty \mu _w}{\mu _\infty \sqrt {\tau _w\rho _w}}\frac {\exp (\kappa U_{I,\infty })}{\kappa U_{I,\infty }}. \end{equation}

Moreover, according to (2.7), $U_{I,\infty }$ is determined by

(2.13) \begin{equation} U_{I,\infty } = \frac {u_\infty }{\sqrt {\dfrac{\tau _w}{\rho _w}}}\int _0^1\frac {1}{\overline {\mu }^+ + \dfrac{\text{d}\overline {U}^+}{\text{d}y^*} -(\overline {\mu }^+)^2 \dfrac{\text{d}\overline {U}^+}{\text{d}y^+}}\,\text{d}z. \end{equation}

Letting $F=\int _0^1[\overline {\mu }^++\text{d}\overline {U}^+/\text{d}y^*-(\overline {\mu }^+)^2\,\text{d}\overline {U}^+/\text{d}y^+]^{-1}\,\text{d}z$ , which is determined by given viscosity and velocity profiles in TBLs, the functional relation between $Re_{\theta ^*}$ and $C_f$ can be obtained by substituting (2.13) into (2.12) as

(2.14) \begin{equation} \sqrt {\frac {2}{C_f \left(\dfrac{\rho _\infty }{\rho _w} \right)F^{-2}}} = \frac {1}{\kappa }\ln \left ( \frac {\rho _w\mu _\infty }{\rho _\infty \mu _w}F\,Re_{\theta ^*}\right ) + C, \end{equation}

with a constant $C=\ln (E\kappa )/\kappa$ . Obviously, we can define a novel transformation for $C_f$ and $Re_{\theta ^*}$ as

(2.15) \begin{equation} C_{f,i}=F_{C^*} C_f,\quad Re_{\theta ,i}=F_{\theta ^*}\, Re_{\theta ^*}, \end{equation}

where the new transformation factors are expressed as

(2.16) \begin{equation} F_{C^*}=\frac {\rho _\infty }{\rho _w}F^{-2},\quad F_{\theta ^*}= \frac {\rho _w\mu _\infty }{\rho _\infty \mu _w}F. \end{equation}

Employing the present transformation, i.e. (2.15), the scaling for $C_f$ of a compressible TBL can be written as

(2.17) \begin{equation} \left (\dfrac{2}{C_{f,i}}\right )^{1/2} \propto \ln Re_{\theta ,i}. \end{equation}

The functional relation between the newly transformed $C_{f,i}$ and $Re_{\theta ,i}$ is exactly the same as the incompressible scaling for $C_f$ expressed by (1.1). Furthermore, in the incompressible TBL with constant $\overline {\rho }$ and $\overline {\mu }$ , it is clear that $F_{C^*}=1$ , $F_{\theta ^*}=1$ and $Re_{\theta ^*}=Re_{\theta }$ . Hence the newly transformed $C_{f,i}$ and $Re_{\theta ,i}$ can be precisely reduced to the incompressible counterparts when the effects of Mach number and heat transfer are negligible. The preceding discussions on the innovative transformation indicate that this novel approach theoretically maps the scaling for $C_f$ of compressible TBLs for air described by the ideal gas law to the incompressible relation for $C_f$ . Furthermore, by performing a linear fit of the data to determine the constants $\kappa _f$ and $C$ , the scaling for $C_f$ of a compressible TBL can be quantified as

(2.18) \begin{equation} \left (\dfrac{2}{C_{f,i}}\right )^{1/2} = \frac {1}{\kappa _f}\ln Re_{\theta ,i} + C. \end{equation}

Given the approximations involved in deriving the skin-friction scaling, the constant obtained by linearly fitting $ (2/C_{f,i} )^{1/2}$ and $\ln Re_{\theta ,i}$ in (2.18) will differ from the value of the von Kármán constant $\kappa$ obtained from the stream velocity profile, and is therefore denoted as $\kappa _f$ . Additionally, based on the definition of $Re_{\theta ^*}$ , the newly transformed $Re_{\theta ,i}$ can be further expressed as $Re_{\theta ,i}=F\rho _w u_\infty \theta ^*/\mu _w$ . It is evident that $\mu _\infty$ used in the present definition of $Re_{\theta ^*}$ does not appear in the final form of transformed $Re_{\theta ,i}$ . In other words, in the definition of $Re_{\theta ^*}$ , replacing $\mu _\infty$ with $\mu _w$ , shear stress-weighted average viscosity introduced by Kianfar et al. (Reference Kianfar, Di Renzo, Williams, Elnahhas and Johnson2023) to account for the relative influence of turbulence on the skin friction, or any other viscosity, does not change the final form of the transformed $Re_{\theta ,i}$ .

3. Validation of the newly proposed scaling law

To verify the scaling law for $C_f$ , we conduct direct numerical simulations (DNS) of compressible TBLs, and also collect as much published DNS data as possible on compressible TBLs with adiabatic (Li et al. Reference Li, Tong, Yu and Li2009; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Volpiani et al. Reference Volpiani, Iyer, Pirozzoli and Larsson2018; Zhang et al. Reference Zhang, Wan, Liu, Sun and Lu2018, Reference Zhang, Wan, Sun and Lu2024; Maeyama & Kawai Reference Maeyama and Kawai2023; Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023), cooled (Zhang et al. Reference Zhang, Wan, Liu, Sun and Lu2018, Reference Zhang, Wan, Sun and Lu2022; Li et al. Reference Li, Tong, Yu and Li2019; Volpiani et al. Reference Volpiani, Bernardini and Larsson2020a ; Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023), and heated (Volpiani et al. Reference Volpiani, Bernardini and Larsson2018, Reference Volpiani, Bernardini and Larsson2020a ) walls. The data cover a fairly wide range of flow conditions, with $Ma_\infty$ ranging from $0.5$ to $14$ , friction Reynolds number $Re_\tau$ ranging from $100$ to $2400$ , and $T_w/T_r$ ranging from $0.15$ to $1.9$ . A wall-to-recovery temperature ratio $T_w/T_r$ less than $1$ signifies a cooled wall, $T_w/T_r$ equal to $1$ denotes an adiabatic wall, and $T_w/T_r$ greater than $1$ indicates a heated wall. Detailed parameters regarding the DNS data of TBLs can be found in tables 1, 2 and 3.

Table 1.

The parameters for compressible TBLs self-simulated using the open-source code STREAmS (Bernardini et al. Reference Bernardini, Modesti, Salvadore and Pirozzoli2021, Reference Bernardini, Modesti, Salvadore, Sathyanarayana, Della Posta and Pirozzoli2023) in fully developed turbulent regions. Here, $Ma_\infty$ is the free-stream Mach number, $T_w/T_r$ is the wall-to-recovery temperature ratio, $Re_\tau$ is the friction Reynolds number, $Re_{\delta _e}$ is the Reynolds number based on boundary layer thickness, $Re_\theta$ is the Reynolds number based on momentum thickness, and $Re_{\theta ^*}$ is the redefined Reynolds number based on transformed momentum thickness.

Table 2.

The parameters for compressible TBLs of Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022, Reference Zhang, Wan, Sun and Lu2024), Li et al. (Reference Li, Fu, Ma and Gao2009, Reference Li, Tong, Yu and Li2019) and Volpiani et al. (Reference Volpiani, Bernardini and Larsson2018, Reference Volpiani, Bernardini and Larsson2020a ) in fully developed turbulent regions. The representations of the parameters are presented in table 1.

Table 3.

The parameters for compressible TBLs of Maeyama & Kawai (Reference Maeyama and Kawai2023), Zhang et al. (Reference Zhang, Duan and Choudhari2018), Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023) and Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) in fully developed turbulent regions. The representations of the parameters are presented in table 1.

Figure 1.

The transformed $(2/C_{f,i})^{1/2}$ versus transformed $Re_{\theta ,i}$ : (a) present theory (using (2.15)), (b) vD-II theory (using (2.1) with $(F_C)_{{vD}}$ and $(F_\theta )_{{vD}}$ ) and (c) SC theory (using (2.1) with $(F_C)_{{SC}}$ and $(F_\theta )_{{SC}}$ ). The coloured symbols represent DNS data from both adiabatic and diabatic compressible TBLs, with colours indicating the wall-to-recovery temperature ratios. The black symbols $\times$ denote DNS and experimental data for incompressible TBLs, with $Re_\theta \leqslant 3000$ from Schlatter & Örlü (Reference Schlatter and Örlü2010), $4000\leqslant Re_\theta \leqslant 6500$ from Sillero et al. (Reference Sillero, Jiménez and Moser2013), and $13\,000\lt Re_\theta \lt 52\,000$ (corresponding to $6000\lt Re_\tau \lt 20\,000$ ) from Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). The dashed and dash-dotted lines represent the incompressible correlations of Coles–Fernholz (modified by Nagib et al. (Reference Nagib, Chauhan and Monkewitz2007), i.e. $(2/C_{f,i})^{1/2}_{{CF}}=2.604\ln Re_{\theta ,i}+4.127$ ) and Smits et al. (Reference Smits, Matheson and Joubert1983) (i.e. $(C_{f,i})_{{SM}}=0.024\,Re_{\theta ,i}^{-1/4}$ ), respectively. The squared Pearson correlation coefficient $R^2$ between $(2/C_{f,i})^{1/2}$ and $\ln Re_{\theta ,i}$ for each transformation is provided in each plot. For a pair of variables $(X,Y)$ , $R^2$ is defined as $R^2 = \text{cov}^2(X,Y)/(\sigma _X^2\sigma _Y^2)$ , where $\text{cov}$ denotes the covariance, $\sigma _X$ is the standard deviation of $X$ , and $\sigma _Y$ is the standard deviation of $Y$ .

Figure 1(a) displays the correlation between the transformed $ (2/C_{f,i} )^{1/2}$ and $Re_{\theta ,i}$ according to the proposed theory, employing logarithmic coordinate for $Re_{\theta ,i}$ . It should be noted that a second-order difference scheme is uniformly employed to calculate the derivatives in $F_{C^*}$ and $F_{\theta ^*}$ for all data. With a squared Pearson correlation coefficient $R^2$ as high as $0.99$ between $ (2/C_{f,i} )^{1/2}$ and $\ln Re_{\theta ,i}$ , it is indicated that the transformed $C_{f,i}$ of compressible TBLs with and without heat transfer strictly satisfies the incompressible scaling for $C_f$ , i.e. $ (2/C_{f,i} )^{1/2}\propto \ln Re_{\theta ,i}$ , based on present theory. For comparison, the results of vD-II and SC theories are depicted in figures 1(b) and 1(c), respectively. No significant linear relationship is observed between $ (2/C_{f,i} )^{1/2}$ and $\ln Re_{\theta ,i}$ under these two theories, with $R^2$ values $0.85$ and $0.86$ , much less than $0.99$ of present theory.

To quantitatively confirm if the present theory effectively collapses the compressible scaling for $C_f$ to the incompressible relation, the constants $\kappa _f$ and $C$ are determined by linearly fitting present data for compressible TBLs, and (2.18) is plotted in figure 1. Two commonly used incompressible correlations for $C_f$ , namely the modified Coles–Fernholz (Nagib et al. Reference Nagib, Chauhan and Monkewitz2007) and Smits et al. (Reference Smits, Matheson and Joubert1983) relations, are also depicted. It is evident that the $C_{f,i}$ correlation of present theory lies between two incompressible relations at $Re_{\theta ,i}\gtrsim 500$ . However, the $C_{f,i}$ correlation of vD-II theory deviates from two incompressible relations at $Re_{\theta ,i}\lesssim 10\,000$ , and that of SC theory notably deviates from incompressible relations. Moreover, the $C_f$ of incompressible DNS data (Schlatter & Örlü Reference Schlatter and Örlü2010; Sillero et al. Reference Sillero, Jiménez and Moser2013) and experimental data (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) also falls between two incompressible relations, following the $C_{f,i}$ correlation of present theory. Hence comparing with vD-II and SC theories, the present theory elegantly maps the compressible scaling for $C_f$ to the incompressible relation. Additionally, the performance of present theory for TBLs at supercritical pressure is discussed in Appendix B.

Figure 2.

The error of the skin-friction coefficient, defined as $|(2/C_{f,i})^{1/2}_{{DNS}}-(2/C_{f,i})^{1/2}_{{CF}} |/(2/C_{f,i})^{1/2}_{{CF}}$ : (a) present theory, (b) vD-II theory, and (c) SC theory. The black dashed line in each plot represents the maximum error.

Error statistics of $C_{f,i}$ from DNS data, compared to the modified Coles–Fernholz relation (Nagib et al. Reference Nagib, Chauhan and Monkewitz2007), are provided in figure 2 to assess the theory’s performance in predicting $C_f$ for compressible TBLs. The maximum errors for the present, vD-II and SC theories are slightly below $5\, \%$ , slightly below $10\, \%$ , and surpassing $14\, \%$ , respectively. The data from compressible TBLs with heated and extensively cooled walls exhibit a significant error for vD-II theory, aligning with observations on the vD-II theory’s inadequacy in predicting $C_f$ on a highly cooled wall (Hopkins & Inouye Reference Hopkins and Inouye1971; Bradshaw Reference Bradshaw1977; Huang et al. Reference Huang, Duan and Choudhari2022). The significant error in the SC theory indicates a notable deviation from incompressible relations, leading to its failure in predicting $C_f$ of compressible TBLs. Therefore, the present theory provides the most reliable predictions of $C_f$ for compressible TBLs with and without heat transfer.

Figure 3.

The transformed $C_{f,i}$ versus transformed $Re_{\theta ,i}$ : (a) present theory, (b) vD-II theory, and (c) SC theory.

The distributions of $C_{f,i}$ versus $Re_{\theta ,i}$ are depicted directly in figure 3. Both compressible and incompressible data collapse to the $C_{f,i}$ correlation of the present theory, lying between two incompressible relations. Additionally, $C_{f,i}$ exhibits a typical decreasing relation with increasing $Re_{\theta ,i}$ . In the vD-II and SC theories, the data fail to collapse to their $C_{f,i}$ correlation. The $C_{f,i}$ of a highly cooled wall ( $T_w/T_r\lesssim 0.3$ ) is overestimated by the vD-II theory but underestimated by the SC theory. This phenomenon is also noted by Huang et al. (Reference Huang, Duan and Choudhari2022). These observations further suggest that the newly proposed theory effectively unifies the compressible and incompressible scaling of $C_f$ .