Revisiting the theory of van Driest: a general scaling law for the skin-friction coefficient of high-speed turbulent boundary layers

Abstract

The skin-friction coefficient is a dimensionless quantity defined by the wall shear stress exerted on an object moving in a fluid, and it decreases as the Reynolds number increases for wall-bounded turbulent flows over a flat plate. In this work, a novel transformation, based on physical and asymptotic analyses, is proposed to map the skin-friction relation of high-speed turbulent boundary layers (TBLs) for air described by the ideal gas law to the incompressible skin-friction relation. Through this proposed approach, it has been confirmed theoretically that the transformed skin-friction coefficient $C_{f,i}$, and the transformed momentum-thickness Reynolds number $Re_{\theta ,i}$ for compressible TBLs with and without heat transfer, follow a general scaling law that aligns precisely with the incompressible skin-friction scaling law, expressed as $ (2/C_{f,i} )^{1/2}\propto \ln Re_{\theta ,i}$. Furthermore, the reliability of the skin-friction scaling law is validated by compressible TBLs with free-stream Mach number ranging from $0.5$ to $14$, friction Reynolds number ranging from $100$ to $2400$, and the wall-to-recovery temperature ratio ranging from $0.15$ to $1.9$. In all of these data, $ (2/C_{f,i} )^{1/2}$ and $\ln Re_{\theta ,i}$ based on the present theory collapse to the incompressible relation, with a squared Pearson correlation coefficient reaching an impressive value $0.99$, significantly exceeding $0.85$ and $0.86$ based on the established van Driest II and the Spalding–Chi transformations, respectively.

1. Introduction

The turbulent boundary layer (TBL) is among the most intriguing and important turbulent flows, drawing numerous researchers to engage in both physical and mathematical modelling of this phenomenon (Bradshaw 1977; Duan et al. 2010; Pirozzoli 2011; Smits et al. 2011; Marusic & Monty 2019; Chen et al. 2023; Cheng & Fu 2024). It is well known that incompressible TBLs at high Reynolds numbers exhibit several nearly universal scaling laws (van Driest 1951; Nagib et al. 2007; Marusic et al. 2013; Chen & Sreenivasan 2021). For example, the mean streamwise velocity profiles versus the wall-normal coordinate can be unified into the law of the wall through the non-dimensionalisation with respect to the friction velocity and kinematic viscosity. The skin-friction coefficient $C_f$ decreases along the streamwise direction, and is widely recognised to exhibit a functional relation with the Reynolds number $Re_\theta$ based on the momentum thickness $\theta$ . The scaling law between $C_f$ and $Re_\theta$ for incompressible TBLs over a flat plate can be expressed as (Nagib et al. 2007)

(1.1) \begin{equation} \left (\frac{2}{C_f}\right )^{1/2}\propto \ln Re_{\theta }, \end{equation}

as a result of the logarithmic law of the mean streamwise velocity.

However, the compressible TBLs with high free-stream Mach number and non-negligible heat transfer do not directly obey the above scaling laws observed in incompressible cases. Consequently, significant efforts are dedicated to mapping compressible TBLs onto the incompressible counterparts by considering variations in mean properties such as density and viscosity inspired by Morkovin hypothesis (Bradshaw 1977). Such a transformation holds not only theoretical significance but also practical importance for reduced-order turbulence modelling, since it would enable the established incompressible wall models to be seamlessly applied to compressible flows (Chen et al. 2024). An exemplary instance of the mapping is the velocity transformation. Over the past decades, several variants have been proposed for the velocity transformation of compressible TBLs (Zhang et al. 2012; Trettel & Larsson 2016; Volpiani et al. 2020b ; Griffin et al. 2021; Hasan et al. 2023), building upon the pioneering work of van Driest (1951). Among these methods, the physics-based Griffin-Fu-Moin (GFM) transformation (Griffin et al. 2021) combining the modified transformation of Zhang et al. (2012) with the transformation of Trettel & Larsson (2016) successfully collapses mean streamwise velocity profiles of compressible TBLs with and without heat transfer into the law of the wall observed in incompressible scenarios.

In the field of aerospace engineering, it is essential to predict $C_f$ on a surface where high-speed airflow passes along with intense heat transfer. To this end, there are approximately 30 published theories for calculation of $C_f$ of compressible TBLs (Spalding & Chi 1963). The theories presented by van Driest (1951) and Spalding & Chi (1964) exhibit lowest root mean square error, and are two of the most commonly used models to estimate $C_f$ . Specifically, considering variations in density and viscosity, van Driest (1951) introduced a scaling of $C_f$ with Reynolds number for compressible TBLs, which can be reduced to the incompressible relation as the Mach number approaches zero and the heat transfer becomes negligible. According to Spalding & Chi (1964), the compressible scaling of $C_f$ can be mapped to the incompressible relation by multiplying $C_f$ and $Re_{\theta }$ by $F_C$ and $F_{\theta }$ , respectively. Here, $F_C$ and $F_{\theta }$ are functions of free-stream Mach number $Ma_\infty$ and temperature $T$ . Spalding & Chi (1964) formulated $F_C$ and $F_{\theta }$ based on the theory of van Driest (1951), called the van Driest II (vD-II) transformation. Moreover, an empirically modified $F_{\theta }$ using $C_f$ data in the presence of heat transfer is proposed as the Spalding–Chi (SC) transformation (Spalding & Chi 1964). Hopkins & Inouye (1971) compared the performance of the above two transformations, and concluded that neither theory provided accurate predictions of $C_f$ for problems with wall-to-recovery temperature ratios below $0.3$ . The review by Bradshaw (1977) further remarked that these two theories failed to predict $C_f$ on a very cold wall. In a more recent study, Huang et al. (2022) confirmed that neither of the two theories could map the compressible scaling of $C_f$ to incompressible relation for a highly cooled wall.

Based on the aforementioned discussions, none of the theories could consistently predict the $C_f$ of compressible TBLs with and without heat transfer. To this end, we revisit the theory of van Driest (1951), and introduce a novel transformation for $C_f$ in this work. This proposed approach effectively maps the scaling law of $C_f$ for high-speed TBLs in air described by the ideal gas law, particularly those involving highly cooled walls, to the incompressible relation (i.e. (1.1)).

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 (1964), 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 1951) and SC (Spalding & Chi 1964) 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 1971; Bradshaw 1977; Huang et al. 2022). 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. 2021) 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. 2021). 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. (2023) 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. 2009; Pirozzoli & Bernardini 2011; Volpiani et al. 2018; Zhang et al. 2018, 2024; Maeyama & Kawai 2023; Cogo et al. 2023), cooled (Zhang et al. 2018, 2022; Li et al. 2019; Volpiani et al. 2020a ; Cogo et al. 2023), and heated (Volpiani et al. 2018, 2020a ) 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. 2021, 2023) 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. (2022, 2024), Li et al. (2009, 2019) and Volpiani et al. (2018, 2020a ) 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 (2023), Zhang et al. (2018), Cogo et al. (2023) and Pirozzoli & Bernardini (2011) 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 (2010), $4000\leqslant Re_\theta \leqslant 6500$ from Sillero et al. (2013), and $13\,000\lt Re_\theta \lt 52\,000$ (corresponding to $6000\lt Re_\tau \lt 20\,000$ ) from Samie et al. (2018). The dashed and dash-dotted lines represent the incompressible correlations of Coles–Fernholz (modified by Nagib et al. (2007), i.e. $(2/C_{f,i})^{1/2}_{{CF}}=2.604\ln Re_{\theta ,i}+4.127$ ) and Smits et al. (1983) (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. 2007) and Smits et al. (1983) 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 2010; Sillero et al. 2013) and experimental data (Samie et al. 2018) 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. 2007), 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 1971; Bradshaw 1977; Huang et al. 2022). 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. (2022). These observations further suggest that the newly proposed theory effectively unifies the compressible and incompressible scaling of $C_f$ .

4. Conclusions

By redefining the Reynolds number, i.e. $Re_{\theta ^*}$ , the defects of vD-II and SC transformations of $C_f$ that do not completely absorb the effects of Mach number and heat transfer in high-speed TBLs for air described by the ideal gas law are overcome. Based on physical and asymptotic analyses, we derived a novel transformation utilising $Re_{\theta ^*}$ to precisely map the compressible scaling law for $C_f$ to the incompressible relation, expressed as $ (2/C_{f,i} )^{1/2}\propto \ln Re_{\theta ,i}$ . Moreover, the transformed $C_{f,i}$ and $Re_{\theta ,i}$ can be precisely reduced to the incompressible skin-friction coefficient and the Reynolds number when the effects of Mach number and heat transfer are negligible. By employing the novel theory, the transformed $C_{f,i}$ from the data of compressible TBLs over a flat plate with a fairly wide range of flow conditions elegantly collapses to the incompressible scaling law of $C_f$ . Therefore, the newly established theory effectively unifies the scaling law for $C_f$ in high-speed TBLs, both with and without heat transfer, and in incompressible TBLs.

Since the GFM transformation used to derive the scaling law is effective only for TBLs with air described by the ideal gas law over smooth flat plates with zero-pressure gradient, the present skin-friction scaling law is limited to these specific conditions. However, redefining $Re_{\theta ^*}$ to establish the skin-friction scaling law in the present study is enlightening. Future investigations can establish skin-friction scaling laws for TBLs with pressure gradients, surface roughness, supercritical pressure, or non-air-like viscosity law, utilising $Re_{\theta ^*}$ in conjunction with an appropriate velocity transformation. Moreover, the skin-friction scaling law established in the present work is of practical value. Specifically, the present skin-friction scaling law, validated by extensive DNS data, can serve as a reference for assessing the accuracy of methods that employ turbulence models, such as large eddy simulation and Reynolds-averaged Navier–Stokes methods, in simulating high-speed TBLs. Since the present method unifies the skin-friction scaling relations of compressible and incompressible TBLs, the $C_f$ of high-speed TBLs can be obtained using results from incompressible flows at the same $Re_{\theta ,i}$ .