2.1. The compressible RANS equations

The Reynolds-averaged governing equations for compressible flows, including the continuity, momentum and total energy equations, are given as (Gatski & Bonnet Reference Gatski and Bonnet2009; Raje et al. Reference Raje, Parish, Hickey, Cinnella and Duraisamy2025)

(2.1) \begin{equation} \frac {\partial \bar \rho }{\partial t} + \frac {\partial \bar \rho \tilde u_{\!j}}{\partial x_{\!j}} = 0, \end{equation}

(2.2) \begin{equation} \frac {\partial \bar \rho \tilde u_i}{\partial t} + \frac {\partial \bar \rho \tilde u_i \tilde u_{\!j}}{\partial x_{\!j}} = -\frac {\partial \bar p}{\partial x_i} + \frac {\partial }{\partial x_{\!j}} \left ( \bar \tau _{ij} - \bar \rho\, \widetilde {u_i^{\prime \prime }u_{\!j}^{\prime \prime }} \right) + \bar F\delta _{i1}, \end{equation}

(2.3) \begin{equation} \frac {\partial \bar \rho \tilde E}{\partial t} + \frac {\partial \bar \rho \tilde E \tilde u_{\!j}}{\partial x_{\!j}} = -\frac {\partial \overline {p u_{\!j}}}{\partial x_{\!j}} + \frac {\partial \overline {\tau _{ij} u_i}}{\partial x_{\!j}} + \frac {\partial }{\partial x_{\!j}} \left ( \overline {\lambda \frac {\partial T}{\partial x_{\!j}}} - \bar \rho\, \widetilde {E^{\prime \prime }u_{\!j}^{\prime \prime }} \right) + \overline {\textit{Fu}_1}. \end{equation}

Here, $i = 1, 2, 3$ correspond to the streamwise ( $x$ ), wall-normal ( $y$ ) and spanwise ( $z$ ) directions, respectively, and $\delta _{ij}$ is the Kronecker delta. The flow variables are defined as follows: $u_i$ is the $i$ th velocity component, and $\rho$ , $p$ and $T$ are the density, pressure and temperature, respectively. The viscous stress tensor is given by $\tau _{ij}=\mu (\partial u_i/\partial x_j+\partial u_j/\partial x_i-(2/3)(\partial u_k/\partial x_k)\delta _{ij})$ , $F$ denotes the streamwise body force ( $F = 0$ in ZPG CTBLFs), and $E = c_v T + u_i u_i / 2$ is the total energy per unit mass, where $c_v = c_{\!p} / \gamma = R / (\gamma - 1)$ is the specific heat at constant volume, $\gamma$ is the specific heat ratio, and $R$ is the specific gas constant. Both Reynolds averaging ( $\bar \varGamma$ ) and Favre averaging ( $\tilde \varGamma =\overline {\rho \varGamma }\!/\!\bar \rho$ ) are employed for any flow quantity $\varGamma$ , with $\varGamma ^{\prime }$ and $\varGamma ^{\prime \prime }$ denoting the corresponding fluctuating components, respectively.

In the following, the total shear stress and total heat flux are defined as

(2.4) \begin{equation} \bar \tau _t = \bar \mu \frac {\partial \tilde u}{\partial y} - \bar \rho\, \widetilde {u^{\prime \prime }v^{\prime \prime }\ } \quad \text{and} \quad \bar q_t = -\bar \lambda \frac {\partial \tilde T}{\partial y} + c_{\!p} \bar \rho\, \widetilde {T^{\prime \prime }v^{\prime \prime }}. \end{equation}

Two non-dimensionalisation approaches are adopted. The inner scaling is characterised by the wall friction velocity $u_\tau = \sqrt {\bar \tau _w / \bar \rho _w}$ , the wall viscous length scale $\delta _\nu = \bar \mu _w / (\bar \rho _w u_\tau)$ , and the wall friction temperature $T_\tau = \bar q_w / (\bar \rho _w c_{\!p} u_\tau)$ , and is denoted using a superscript ‘+’. The semi-local scaling is defined using the semi-local friction velocity $u_\tau ^* = \sqrt {\bar \tau _w / \bar \rho }$ , the semi-local viscous length scale $\delta _\nu ^* = \bar \mu / (\bar \rho u_\tau ^*)$ , and the semi-local friction temperature $T_\tau ^* = \bar q_w / (\bar \rho c_{\!p} u_\tau ^*)$ , and is indicated by a superscript ‘*’.

2.2. Velocity transformations and compressible eddy viscosity models

For CWBTFs, the total shear stress equation (2.4) can be expressed in non-dimensional form by normalising with the wall shear stress $\bar \tau _w$ :

(2.5) \begin{equation} \left ( \frac {\partial \tilde u}{\partial y} \right)^* - \widetilde {u^{\prime \prime }v^{\prime \prime }\ }^* = \bar \tau _{t,{\textit{com}}}^+, \end{equation}

where $(\partial \tilde u / \partial y)^* = (\partial \tilde u / \partial y) / (u_\tau ^* \,/ \delta _\nu ^*) = \bar \mu ^+\, \partial \tilde u^+ \!/ \partial y^+$ represents the viscous shear stress, with $\tilde u^+ = \tilde u / u_\tau$ . The term $-\widetilde {u^{\prime \prime }v^{\prime \prime }\ }^* = -\widetilde {u^{\prime \prime }v^{\prime \prime }\ } / u_\tau ^{*2} = -\bar \rho\, \widetilde {u^{\prime \prime }v^{\prime \prime }\ } / (\bar \rho _w u_\tau ^2) = -\bar \rho ^+\, \widetilde {u^{\prime \prime }v^{\prime \prime }\ }^+$ denotes the turbulent shear stress, and $\bar \tau _{t,{\textit{com}}}^+ = \bar \tau _{t,{\textit{com}}} / \bar \tau _w$ is the total shear stress. Here, $\bar \mu ^+ = \bar \mu / \bar \mu _w$ and $\bar \rho ^+ = \bar \rho / \bar \rho _w$ denote the inner-scaled mean viscosity and density, respectively. Throughout this paper, the subscripts ‘ $com$ ’ and ‘ $in$ ’ refer to compressible and ‘incompressible’ states, respectively.

The total shear stress equation for the corresponding ‘incompressible’ state is given by

(2.6) \begin{equation} \left ( \frac {\partial \bar U}{\partial Y} \right)^+ - \overline {U^{\prime }V^{\prime }}^+ = \bar \tau _{t,{\textit{in}}}^+, \end{equation}

where the ‘incompressible’ viscous shear stress is defined as $(\partial \bar U \!/ \partial Y)^+ = (\partial \bar U \!/ \partial Y) / (u_\tau / \delta _\nu) = \partial \bar U^+ \!/ \partial Y^+$ , with $\bar U^+ = \bar U \!/ u_\tau$ , and the turbulent shear stress is $-\overline {U^{\prime }V^{\prime }}^+ = -\overline {U^{\prime }V^{\prime }} / u_\tau ^2$ . The ‘incompressible’ total shear stress is denoted as $\bar \tau _{t,{\textit{in}}}^+ = \bar \tau _{t,{\textit{in}}} / \bar \tau _w$ . Here, $U$ , $V$ and $Y$ represent the transformed ‘incompressible’ streamwise velocity, wall-normal velocity and wall-normal distance, respectively.

It can be observed that the introduction of semi-local scaling leads to a similar form of the total shear stress equations for both compressible and ‘incompressible’ states. To further simplify the expressions, we define the compressible and ‘incompressible’ eddy viscosities as $\bar \mu _{t,{\textit{com}}} / \bar \mu = -\widetilde {u^{\prime \prime }v^{\prime \prime }\ }^* \!/ (\partial \tilde u / \partial y)^*$ and $\bar \mu _{t,{\textit{in}}} / \bar \mu _w = -\overline {U^{\prime }V^{\prime }}^+ \!/ (\partial \bar U \!/ \partial Y)^+$ , respectively. Then (2.5) and (2.6) can be rewritten as

(2.7) \begin{equation} \left ( 1 + \frac {\bar \mu _{t,{\textit{com}}}}{\bar \mu } \right) \left ( \frac {\partial \tilde u}{\partial y} \right)^* = \bar \tau _{t,{\textit{com}}}^+, \end{equation}

(2.8) \begin{equation} \left ( 1 + \frac {\bar \mu _{t,{\textit{in}}}}{\bar \mu _w} \right) \left ( \frac {\partial \bar U}{\partial Y} \right)^+ = \bar \tau _{t,{\textit{in}}}^+. \end{equation}

As noted, the semi-local scaling renders the compressible mean momentum equation structurally similar to its ‘incompressible’ counterpart, which indirectly supports its validity for extending incompressible models to high-speed flows. Dividing (2.8) by (2.7) then leads to a general framework for mean velocity transformations, given by

(2.9) \begin{equation} \bar U^+ = \int _{0}^{\tilde u^+}{\frac {\bar \tau _{t,{\textit{in}}}^+}{\bar \tau _{t,{\textit{com}}}^+} \frac {1 + \bar \mu _{t,{\textit{com}}} / \bar \mu }{1 + \bar \mu _{t,{\textit{in}}} / \bar \mu _w}\, \bar \mu ^+\, \frac {{\partial } Y^+}{{\partial } y^+}\, \mathrm{d}\tilde u^+}. \end{equation}

It should be noted that an expression similar to (2.9) was first proposed by Hasan et al. (Reference Hasan, Larsson, Pirozzoli and Pecnik2023) for CWBTFs. However, the authors neglected the differences in total shear stresses before and after the transformation. A comparable framework, based on the mixing length concept, was introduced by Zhu et al. (Reference Zhu, Song, Yang and Xia2024). In fact, all transformations must satisfy (2.9), and we mainly consider six transformations in this paper: TL (Patel et al. Reference Patel, Boersma and Pecnik2016; Trettel & Larsson Reference Trettel and Larsson2016), GFM (Griffin et al. Reference Griffin, Fu and Moin2021b ), HLPP (Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023), ZSYX (Zhu et al. Reference Zhu, Song, Yang and Xia2024), SLVD (the semi-local van Driest transformation for the logarithmic layer, proposed by Younes & Hickey Reference Younes and Hickey2023), and VIPL (Volpiani et al. Reference Volpiani, Iyer, Pirozzoli and Larsson2020) transformations, as detailed in Appendix A.

On the one hand, (2.9) can be directly employed to develop new velocity transformations, provided that the Mach-invariance of eddy viscosity under a certain transformed coordinate $Y^+$ is assumed and validated, i.e.

(2.10) \begin{equation} \bar \mu _{t,{\textit{com}}} / \bar \mu = \bar \mu _{t,{\textit{in}}} / \bar \mu _w = f\!(Y^+, {\textit{Re}}_T), \end{equation}

where $f\!(Y^+, {\textit{Re}}_T)$ is a universal function describing the ‘incompressible’ eddy viscosity profile with respect to the inner coordinate $Y^+$ , parametrised by the friction Reynolds number ${\textit{Re}}_T$ . For instance, in the inner layer of high Reynolds number IWBTFs, it is typically adopted as $f\!(Y^+, {\textit{Re}}_T) = \kappa Y^+ [1 - \exp(-Y^+ \!/ A^+) ]^2$ , where $\kappa = 0.41$ and $A^+ = 17$ (Johnson & King Reference Johnson and King1985). The original TL and VIPL transformations, given in (A1) and (A6), serve as representative examples. They can be derived by substituting the transformed coordinates $Y^+ = y^*$ and $Y^+ = y_V^+$ into (2.10), while further neglecting the differences in total shear stresses before and after transformation, i.e. $\bar \tau _{t,{\textit{com}}}^+ = \bar \tau _{t,{\textit{in}}}^+$ , which was originally motivated by the constant-stress layer assumption (Trettel & Larsson Reference Trettel and Larsson2016; Volpiani et al. Reference Volpiani, Iyer, Pirozzoli and Larsson2020).

On the other hand, (2.9) can also be used to develop compressible eddy viscosity models by incorporating advanced mean velocity scalings (Yang & Lv Reference Yang and Lv2018; Hendrickson et al. Reference Hendrickson, Subbareddy and Candler2022, Reference Hendrickson, Subbareddy, Candler and Macdonald2023; Chen et al. Reference Chen, Gan and Fu2024), which may be derived through various approaches. For clarity and convenience, we first introduce the modified TL and VIPL transformations,

(2.11) \begin{equation} \bar U_{{\textit{TL}},m}^+ = \int _{0}^{\bar U_{\textit{TL}}^+}{\frac {\bar \tau _{t,{\textit{in}}}^+}{\bar \tau _{t,{\textit{com}}}^+}\, \mathrm{d}\bar U_{\textit{TL}}^+} \quad \text{and} \quad \bar U_{\textit{VIPL},m}^+ = \int _{0}^{\bar U_{\textit{VIPL}}^+}{\frac {\bar \tau _{t,{\textit{in}}}^+}{\bar \tau _{t,{\textit{com}}}^+}\, \mathrm{d}\bar U_{\textit{VIPL}}^+}, \end{equation}

which extend the original formulations (A1) and (A6) by additionally accounting for differences in $\bar \tau _{t,{\textit{com}}}^+$ and $\bar \tau _{t,{\textit{in}}}^+$ . As presented in Appendix A, the TL, GFM, HLPP, ZSYX and SLVD transformations all acknowledge the semi-local transformed coordinate $Y^+ = y^*$ , implying a consistent form of the ‘incompressible’ eddy viscosity, $\bar \mu _{t,{\textit{in}}} / \bar \mu _w = f\!(y^*, {\textit{Re}}_{\tau ,e}^*)$ . Based on (2.9), these mean velocity scalings can thus be interpreted as different modelling approaches for the compressible eddy viscosity $\bar \mu _{t,{\textit{com}}} / \bar \mu$ . Substituting $Y^+ = y^*$ into (2.9) and combining with the modified TL transformation in (2.11), the compressible eddy viscosity corresponding to any $y^*$ -based transformation $\bar U_{\textit{trans}}^+ (y^*)$ can be expressed as

(2.12) \begin{equation} 1 + \left ( \frac {\bar \mu _{t,{\textit{com}}}}{\bar \mu } \right)_{\textit{model}} = \frac {\partial \bar U_{\textit{trans}}^+ / \partial \tilde u^+}{\partial \bar U_{{\textit{TL}},m}^+ / \partial \tilde u^+} \left [ 1 + f\!(y^*, {\textit{Re}}_{\tau ,e}^*) \right ]. \end{equation}

Equation (2.12) offers physical insight into the extension of incompressible eddy viscosity models to high-speed flows through the inclusion of compressibility effects. Clearly, a more accurate transformation yields a compressible eddy viscosity model $(\bar \mu _{t,{\textit{com}}} / \bar \mu)_{\textit{model}}$ that agrees better with compressible DNS data $(\bar \mu _{t,{\textit{com}}} / \bar \mu)_{\textit{DNS}}$ . It is noteworthy that (2.12) differs slightly from the formulations proposed in previous studies (Hendrickson et al. Reference Hendrickson, Subbareddy and Candler2022, Reference Hendrickson, Subbareddy, Candler and Macdonald2023; Chen et al. Reference Chen, Gan and Fu2024), where the authors refined the semi-local model implied by the TL transformation using the GFM transformation. Their model can be expressed as

(2.13) \begin{equation} \left ( \frac {\bar \mu _{t,{\textit{com}}}}{\bar \mu } \right)_{\textit{model}} = \frac {\partial \bar U_{\textit{GFM}}^+ / \partial \tilde u^+}{\partial \bar U_{\textit{TL}}^+ / \partial \tilde u^+} f\!(y^*, {\textit{Re}}_{\tau ,e}^*). \end{equation}

A detailed examination of their derivation indicates that (2.13) represents a special case of (2.12) within the logarithmic layer, without accounting for the difference between $\bar \tau _{t,{\textit{com}}}^+$ and $\bar \tau _{t,{\textit{in}}}^+$ . Nevertheless, as the GFM transformation largely retains the TL scaling below the buffer layer, this simplification does not significantly impact the overall model accuracy.

2.3. Temperature transformations and compressible eddy thermal conductivity models

As previously noted by Huang et al. (Reference Huang, Coleman, Spalart and Yang2023) and Modesti & Pirozzoli (Reference Modesti and Pirozzoli2024), the formulation of the mean temperature scalings may proceed in a manner analogous to that used for mean velocity transformations. We begin by considering the case with wall heat transfer, i.e. $\bar q_w \neq 0$ . The compressible total heat flux equation in (2.4) is non-dimensionalised using the wall heat flux $\bar q_w$ , yielding

(2.14) \begin{equation} \frac {1}{Pr} \left ( \frac {\partial \tilde T}{\partial y} \right)^* - \widetilde {T^{\prime \prime }v^{\prime \prime }}^* = \bar q_{t,{\textit{com}}}^+, \end{equation}

where ${\textit{Pr}}^{-1}\,(\partial \tilde T \!/ \partial y)^* = - Pr^{-1}\, (\partial \tilde T \!/ \partial y) / (T_\tau ^* \!/ \delta _\nu ^*) = {\textit{Pr}}_w^{-1}\, \bar \lambda ^+\, \partial \tilde T^+ \!/ \partial y^+$ represents the molecular heat flux, with $\tilde T^+ = (\tilde T_w - \tilde T) / T_\tau$ denoting the inner-scaled mean temperature. The term $-\widetilde {T^{\prime \prime }v^{\prime \prime }}^* = \widetilde {T^{\prime \prime }v^{\prime \prime }} / (T_\tau ^* u_\tau ^*) = \bar \rho\, \widetilde {T^{\prime \prime }v^{\prime \prime }} / (\bar \rho _w T_\tau u_\tau) = -\bar \rho ^+\, \widetilde {T^{\prime \prime }v^{\prime \prime }}^+$ is the turbulent heat flux, and $\bar q_{t,{\textit{com}}}^+ = \bar q_{t,{\textit{com}}} / \bar q_w$ the total heat flux. Here, $\bar \lambda ^+ = \bar \lambda / \bar \lambda _w$ denotes the inner-scaled mean thermal conductivity, and ${\textit{Pr}} = c_{\!p} \bar \mu / \bar \lambda$ is the local molecular Prandtl number.

For the transformed ‘incompressible’ counterpart, the total heat flux equation is given by

(2.15) \begin{equation} \frac {1}{{\textit{Pr}}_w} \left ( \frac {\partial \bar {\mathcal{T}}}{\partial Y} \right)^+ - \overline {\mathcal{T}^{\prime }V^{\prime }}^+ = \bar q_{t,{\textit{in}}}^+, \end{equation}

where the ‘incompressible’ molecular heat flux is defined as ${\textit{Pr}}_w^{-1}\,(\partial \bar {\mathcal{T}} / \partial Y)^+ = -{\textit{Pr}}_w^{-1}\,(\partial \bar {\mathcal{T}} / \partial Y) / (T_\tau / \delta _\nu) = {\textit{Pr}}_w^{-1}\,\partial \bar {\mathcal{T}}^+ \!/ \partial Y^+$ , with $\bar {\mathcal{T}}^+ = (\bar {\mathcal{T}}_w - \bar {\mathcal{T}}) / T_\tau$ , and the turbulent heat flux is $-\overline {\mathcal{T}^{\prime }V^{\prime }}^+ = \overline {\mathcal{T}^{\prime }V^{\prime }} / (T_\tau u_\tau)$ . The ‘incompressible’ total heat flux is denoted as $\bar q_{t,{\textit{in}}}^+ = \bar q_{t,{\textit{in}}} / \bar q_w$ .

Consistent with the approach in § 2.2, we introduce compressible and ‘incompressible’ eddy thermal conductivities, defined respectively as $\bar \lambda _{t,{\textit{com}}} / \bar \lambda = - Pr\, \widetilde {T^{\prime \prime }v^{\prime \prime }}^* \!/ ( {\partial \tilde T} / {\partial y})^*$ and $\bar \lambda _{t,{\textit{in}}} / \bar \lambda _w = - {\textit{Pr}}_w\, \overline {\mathcal{T}^{\prime }V^{\prime }}^+ \!/ ( {\partial \bar {\mathcal{T}}} / {\partial Y})^+$ , for further simplifications. Equations (2.14) and (2.15) can then be rewritten as

(2.16) \begin{equation} \left ( 1 + \frac {\bar \lambda _{t,{\textit{com}}}}{\bar \lambda } \right) \frac {1}{Pr} \left ( \frac {\partial \tilde T}{\partial y} \right)^* = \bar q_{t,{\textit{com}}}^+, \end{equation}

(2.17) \begin{equation} \left ( 1 + \frac {\bar \lambda _{t,{\textit{in}}}}{\bar \lambda _w} \right) \frac {1}{{\textit{Pr}}_w} \left ( \frac {\partial \bar {\mathcal{T}}}{\partial Y} \right)^+ = \bar q_{t,{\textit{in}}}^+. \end{equation}

It is worth noting that (2.16) becomes ill-defined in the presence of adiabatic walls, because $\bar q_w = 0$ . To address this issue, we reformulate (2.16) into a more unified expression, following the methodology proposed in previous studies (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022a ; Huang et al. Reference Huang, Coleman, Spalart and Yang2023; Cheng & Fu Reference Cheng and Fu2024a ):

(2.18) \begin{equation} \left ( 1 + \frac {\bar \lambda _{t,{\textit{com}}}}{\bar \lambda } \right) \frac {\bar \lambda ^+}{{\textit{Pr}}_w} \frac {\partial \theta }{\partial y^+} = B_q \bar q_{t,{\textit{com}}}^+, \end{equation}

where $\theta = 1 - \tilde T \!/ \tilde T_w$ , $B_q = \bar q_w / (\bar \rho _w c_{\!p} u_\tau \tilde T_w)$ and $\bar q_{t,{\textit{com}}}^+ = \bar q_{t,{\textit{com}}} / \bar q_w$ . Dividing (2.17) by (2.18) then yields a general framework for the mean temperature transformation, given by

(2.19) \begin{equation} \bar {\mathcal{T}}^+ = \int _{0}^{\theta }{\frac {{\textit{Pr}}_w}{Pr} \frac {1 + \bar \lambda _{t,{\textit{com}}} / \bar \lambda }{1 + \bar \lambda _{t,{\textit{in}}} / \bar \lambda _w} \frac {\bar q_{t,{\textit{in}}}^+}{B_q \bar q_{t,{\textit{com}}}^+} \bar \mu ^+ \frac {\partial Y^+}{\partial y^+}\, \mathrm{d} \theta }. \end{equation}

An expression similar to (2.19) was derived by Modesti & Pirozzoli (Reference Modesti and Pirozzoli2024) for variable-property forced air convection, where compressibility effects are negligible. In their context, neither the wall-normal variation of the local molecular Prandtl number nor the dimensionless treatment employed in (2.18) was required, as these aspects were not relevant to their flow configuration. Moreover, they considered only the special case $Y^+ = y^*$ , yielding a temperature transformation analogous to the HLPP method (Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023), whereas the present framework offers a more general and rigorous formulation by explicitly defining the ‘incompressible’ reference state and examining the corresponding total heat flux equation.

As discussed in § 2.2, (2.19) can be similarly applied to develop novel temperature transformations, provided that the eddy thermal conductivity is Mach-invariant under a suitable transformed coordinate $Y^+$ , i.e.

(2.20) \begin{equation} \bar \lambda _{t,{\textit{com}}} / \bar \lambda = \bar \lambda _{t,{\textit{in}}} / \bar \lambda _w = g(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w), \end{equation}

where the universal function characterising the ‘incompressible’ eddy thermal conductivity profile may take the form $g(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_w\, \kappa _T Y^+ [1 - \exp(-Y^+ \!/ A_T^+) ]^2$ in the inner layer of high Reynolds number IWBTFs, with $\kappa _T = 0.48$ and $A_T^+ = 20$ (Chen et al. Reference Chen, Lv, Xu, Shi and Yang2022b ). To establish a connection between (2.9) and (2.19), we further introduce the concept of the mixed (or total) Prandtl number ${\textit{Pr}}_m$ (Walz Reference Walz1969; Lusher & Coleman Reference Lusher and Coleman2022; Huang et al. Reference Huang, Coleman, Spalart and Yang2023), which characterises the ratio of total momentum to total heat transport. Accordingly, (2.19) is reformulated as

(2.21) \begin{equation} \bar {\mathcal{T}}^+ = \int _{0}^{\theta }{\frac {{\textit{Pr}}_{m,{\textit{in}}}}{{\textit{Pr}}_{m,{\textit{com}}}} \frac {\bar \tau _{t,{\textit{com}}}^+}{\bar \tau _{t,{\textit{in}}}^+} \frac {\bar q_{t,{\textit{in}}}^+}{B_q \bar q_{t,{\textit{com}}}^+} \frac {\partial \bar U^+}{\partial \tilde u^+}\, \mathrm{d}\theta }, \end{equation}

where ${\textit{Pr}}_{m,{\textit{com}}} = c_{\!p} (\bar \mu + \bar \mu _{t,{\textit{com}}}) / (\bar \lambda + \bar \lambda _{t,{\textit{com}}})$ and ${\textit{Pr}}_{m,{\textit{in}}} = c_{\!p} (\bar \mu _w + \bar \mu _{t,{\textit{in}}}) / (\bar \lambda _w + \bar \lambda _{t,{\textit{in}}})$ . Equation (2.21) was originally proposed by Zhu et al. (Reference Zhu, Zhang, Yang, Ji and Xia2025b ) for CTCFs. Compared to (2.19), it offers a straightforward and effective approach for developing and improving temperature transformations by coupling with any advanced mean velocity scalings. Nevertheless, it is worth noting that (2.19) is more advantageous in cases where the local molecular Prandtl number exhibits significant wall-normal variations, as it explicitly retains the ${\textit{Pr}}_w / Pr$ term. In contrast, (2.21) accounts for such effects implicitly through the modelling of ${\textit{Pr}}_{m,{\textit{com}}}$ and ${\textit{Pr}}_{m,{\textit{in}}}$ , which can be challenging in practice.

In this study, (2.21) is employed to derive the TL- (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022a ; Huang et al. Reference Huang, Coleman, Spalart and Yang2023), GFM-, HLPP-, ZSYX-, SLVD- (Cheng & Fu Reference Cheng and Fu2024a ) and VIPL-type temperature transformations by substituting the corresponding velocity transformation kernels $\partial \bar U^+ \!/ \partial \tilde u^+$ presented in Appendix A, under the assumption that the mixed Prandtl number is Mach-invariant under the transformed coordinate, i.e. ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ , which is validated in detail in § 4.

By substituting $\partial \bar U^+ \!/ \partial \tilde u^+$ from (2.9) into (2.21), inserting the resulting $\partial \bar {\mathcal{T}}^+ \!/ \partial \theta$ into (2.19), and assuming ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ , a general framework for extending incompressible eddy thermal conductivity models to high-speed flows is thereby established:

(2.22) \begin{equation} 1 + \left ( \frac {\bar \lambda _{t,{\textit{com}}}}{\bar \lambda } \right)_{\textit{model}} = \frac {Pr}{{\textit{Pr}}_w} \frac {1 + (\bar \mu _{t,{\textit{com}}} / \bar \mu)_{\textit{model}}}{1 + f\!(Y^+, {\textit{Re}}_T)} \left [ 1 + g(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) \right ]. \end{equation}

As noted in § 2.2, the modified TL- and VIPL-type transformations imply that $(\bar \mu _{t,{\textit{com}}} / \bar \mu)_{\textit{model}} = f\!(Y^+, {\textit{Re}}_T)$ , leading to

(2.23) \begin{equation} 1 + \left (\frac {\bar \lambda _{t,{\textit{com}}}}{\bar \lambda } \right)_{\textit{model}} = \frac {Pr}{{\textit{Pr}}_w} \left [ 1 + g(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) \right ], \end{equation}

where $Y^+ = y^*$ and $Y^+ = y_V^+$ , respectively. Additionally, by substituting (2.12) into (2.22), the modelled eddy thermal conductivity associated with the GFM-, HLPP-, ZSYX- and SLVD-type transformations can similarly be interpreted as further refinements of the semi-local model implied by $\bar U_{{\textit{TL}},m}^+$ , i.e.

(2.24) \begin{equation} 1 + \left (\frac {\bar \lambda _{t,{\textit{com}}}}{\bar \lambda } \right)_{\textit{model}} = \frac {Pr}{{\textit{Pr}}_w} \frac {\partial \bar U_{\textit{trans}}^+ / \partial \tilde u^+}{\partial \bar U_{{\textit{TL}},m}^+ / \partial \tilde u^+} \left [ 1 + g(y^*, {\textit{Re}}_{\tau ,e}^*,{\textit{Pr}}_w) \right ]. \end{equation}

In fact, constructing temperature transformations by drawing analogies between the total heat flux equations in the compressible and ‘incompressible’ states has been a common practice in previous studies. Specifically, Chen et al. (Reference Chen, Huang, Shi, Yang and Lv2022a ) and Huang et al. (Reference Huang, Coleman, Spalart and Yang2023) demonstrated that the semi-local-scaled mean momentum and energy equations exhibit the same structure as their ‘incompressible’ counterparts, and accordingly derived the VD-type and TL-type temperature transformations. By contrast, the recent study of Cheng & Fu (Reference Cheng and Fu2024a ) departed from this conventional path, employing instead a semi-local empirical modification of the VD-type transformation, inspired by the SLVD transformation (Younes & Hickey Reference Younes and Hickey2023). Compared to these prior works, the present study makes two main theoretical advancements. First, while our analysis remains grounded in the total heat flux equation, its novelty lies in clarifying the explicit definition of the ‘incompressible’ reference state, and providing a direct and rigorous derivation of the general framework (2.19). All temperature transformations must satisfy this framework, as it essentially reflects the fundamental physics of mean energy transport. Moreover, it demonstrates how temperature transformations can be used to extend incompressible eddy thermal conductivity models to high-speed compressible flows. Second, we introduce the mixed Prandtl number to explicitly link mean momentum and energy transport, leading to the derivation of (2.21). This equation provides a systematic pathway for constructing more accurate and robust temperature transformations by incorporating state-of-the-art mean velocity scalings. Indeed, all transformation methods proposed in the aforementioned studies can be recovered as special cases of this equation. In summary, the present study establishes a more rigorous and systematic theoretical framework for analysing and extending temperature transformations.

4.1. The CTCFs with symmetric isothermal $w$ alls

In this subsection, DNS data of CTCFs with symmetric isothermal walls from Zhang & Xia (Reference Zhang and Xia2020), Yao & Hussain (Reference Yao and Hussain2020), Zhang, Song & Xia (Reference Zhang, Song and Xia2023), Li et al. (Reference Li, Chen, Xiao, Mao and Yao2025b ) and Gerolymos & Vallet (Reference Gerolymos and Vallet2024a ) are employed to assess the performance of various transformation methods. The datasets span a range of bulk Mach numbers $0.5 \leqslant M_b \leqslant 3.0$ and semi-local Reynolds numbers $177 \leqslant {\textit{Re}}_{\tau ,e}^* \leqslant 1483$ , as summarised in table 2.

Table 2.

Summary of DNS datasets for CTCFs with symmetric isothermal walls. Data sources are abbreviated as follows: ZX for Zhang & Xia (Reference Zhang and Xia2020) and Zhang et al. (Reference Zhang, Song and Xia2023), YH for Yao & Hussain (Reference Yao and Hussain2020), LCXMY for Li et al. (Reference Li, Chen, Xiao, Mao and Yao2025b ), and GV for Gerolymos & Vallet (Reference Gerolymos and Vallet2023, Reference Gerolymos and Vallet2024a ,Reference Gerolymos and Vallet b ). Here, the bulk Mach number is defined as $M_b = u_b / \sqrt {\gamma R T_{\textit{ref}}}$ , where $u_b$ is the bulk velocity, $T_{\textit{ref}}$ is the reference temperature, and $\tilde T_w = T_{\textit{ref}}$ is the wall temperature. The characteristic Reynolds numbers are ${\textit{Re}}_{\tau ,e}^* = y^* |_{y = \delta _e}$ and ${\textit{Re}}_{V,e} = y_V^+ |_{y = \delta _e}$ , where $\delta _e = h$ is the channel half-height. ‘Incom. ref.’ refers to the transformed ‘incompressible’ DNS data used for reference, as listed in table 1. Note that the molecular Prandtl number is not constant in cases M07R0990T10I and M16R0970T10I (with consistent ${\textit{Pr}}_w \approx 0.71$ at the wall), and the final character ‘I’ in each dataset label denotes the isothermal wall boundary condition.

The Mach-independence assumption of the mixed Prandtl number is first examined and validated. As shown in figure 2, the ${\textit{Pr}}_{m,{\textit{com}}}$ profiles exhibit a similar collapse to ${\textit{Pr}}_{m,{\textit{in}}}$ for $Y^+ \lesssim 50$ , regardless of whether $Y^+ = y^*$ or $Y^+ = y_V^+$ is used. This behaviour is also well captured by the proposed model in (3.7). Beyond the peak region, some deviations are observed for those cases with relatively low Reynolds numbers, as expected. Nevertheless, figure 3 demonstrates that the assumption ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ remains an effective simplification, with the ratio ${\textit{Pr}}_{m,{\textit{in}}} / {\textit{Pr}}_{m,{\textit{com}}}$ confined within 0.95–1.05 for $Y^+ \leqslant 0.1\, {\textit{Re}}_T$ , thereby covering the entire inner layer (Pope Reference Pope2000).

Figure 2.

Compressible mixed Prandtl number ${\textit{Pr}}_{m,{\textit{com}}}$ plotted as a function of (a) $Y^+ = y^*$ and (b) $Y^+ = y_V^+$ in CTCFs with symmetric isothermal walls, based on the DNS datasets listed in table 2. The grey solid and dashed lines indicate the predictions of (3.7) with wall Prandtl numbers ${\textit{Pr}}_w = 0.72$ and ${\textit{Pr}}_w = 0.71$ , respectively. Hereafter, the colour bar represents the magnitude of the characteristic Reynolds numbers ${\textit{Re}}_T$ , providing a unique identifier for each DNS case within a given flow configuration.

Figure 3.

Wall-normal distributions of the mixed Prandtl number ratio ${\textit{Pr}}_{m,{\textit{in}}} / {\textit{Pr}}_{m,{\textit{com}}}$ in CTCFs with symmetric isothermal walls, plotted as functions of (a) $Y^+ = y^*$ and (b) $Y^+ = y_V^+$ . Here, ${\textit{Pr}}_{m,{\textit{com}}}(Y^+)$ denotes the mixed Prandtl number profiles for the cases listed in table 2, while ${\textit{Pr}}_{m,{\textit{in}}}(Y^+)$ corresponds to the ‘incompressible’ reference states with comparable ${\textit{Re}}_T$ , as listed in table 1. The theoretical expectation is unity, with the grey shaded region bounded by black dashed lines indicating the $\pm$ 5 % tolerance band. Filled circles mark the positions $Y^+ = 0.1\, {\textit{Re}}_T$ for each case.

To apply the temperature transformation defined in (2.21), it remains to determine $\bar \tau _{t,{\textit{com}}}^+$ and $B_q \bar q_{t,{\textit{com}}}^+$ . The integral forms of the Reynolds-averaged momentum and total energy equations can be expressed as (Huang, Coleman & Bradshaw Reference Huang, Coleman and Bradshaw1995)

(4.4) \begin{equation} \bar {\mu }\frac {\mathrm{d}\tilde {u}}{\mathrm{d}y}+\bar {\mu }\frac {\mathrm{d}\overline {u^{\prime \prime }}}{\mathrm{d}y}+\overline {\mu ^{\prime }\frac {\mathrm{d} u ^{\prime }}{\mathrm{d}y}}-\bar {\rho }\,\widetilde {u^{\prime \prime }v^{\prime \prime }\ } = \bar \tau _w -\int _0^y {\bar F\, \mathrm{d}y}, \end{equation}

(4.5) \begin{equation} \overline {\tau _{i2}u_i}-\overline {\rho \tilde {u}_iu_{i}^{\prime \prime }v^{\prime \prime }}-\frac {1}{2}\overline {\rho u_{i}^{\prime \prime }u_{i}^{\prime \prime }v^{\prime \prime }}+\bar {\lambda }\frac {\mathrm{d}\tilde {T}}{\mathrm{d}y}+\bar {\lambda }\frac {\mathrm{d}\overline {T^{\prime \prime }}}{\mathrm{d}y}+\overline {\lambda ^{\prime }\frac {\mathrm{d}T^{\prime }}{\mathrm{d}y}}-c_{\!p}\bar {\rho }\widetilde {T^{\prime \prime }v^{\prime \prime }}=-\bar {q}_w-\!\!\int _0^y{\overline {\textit{Fu}}\mathrm{d}y}, \end{equation}

where $\bar F = \bar \tau _w / h$ for flows driven by a uniformly distributed body force. Following the modelling strategies introduced by Zhang et al. (Reference Zhang, Bi, Hussain and She2014) and Huang et al. (Reference Huang, Coleman, Spalart and Yang2023), we adopt the quasi-parallel flow approximation $u_i u_i \approx u^2$ and the small turbulence intensity approximation $u^2 \approx \tilde u^2 + 2\tilde u u^{\prime \prime }$ . Additionally, the molecular diffusion flux of a quantity $\phi$ , $\overline {\mu\, \mathrm{d}\phi / \mathrm{d}y}$ , is approximated by its dominant term $\bar \mu\, \mathrm{d} \tilde \phi / \mathrm{d}y$ (Huang et al. Reference Huang, Coleman and Bradshaw1995). By combining (4.4) and (4.5), and assuming $\overline {\textit{Fu}} \approx \bar F \tilde u$ , the total shear stress and total heat flux can be approximated as (Zhu et al. Reference Zhu, Zhang, Yang, Ji and Xia2025b )

(4.6) \begin{equation} \bar \tau _{t,{\textit{com}}}^+ \approx 1 - y / h \quad \text{and} \quad B_q \bar q_{t,{\textit{com}}}^+ \approx B_q + (\gamma - 1) M_\tau ^2 \int _0^{\tilde u^+}{\bar \tau _{t,{\textit{com}}}^+\, \mathrm{d} \tilde u^+}, \end{equation}

where $B_q = \bar q_w / (\bar \rho _w c_{\!p} u_\tau \tilde T_w)$ and $M_\tau = u_\tau / \sqrt {\gamma R \tilde T_w}$ . It can be seen that these simplifications intentionally omit higher-order terms that are typically unavailable in open-source DNS datasets, thereby maintaining the simplicity of the formulation, and enhancing its applicability in turbulence modelling (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022a ; Cheng & Fu Reference Cheng and Fu2024a ). A detailed derivation of (4.6), along with the associated approximations, is provided in Zhu et al. (Reference Zhu, Zhang, Yang, Ji and Xia2025b ). The physical significance of the higher-order terms and the implications of their omission have been thoroughly examined by Huang et al. (Reference Huang, Coleman and Bradshaw1995, Reference Huang, Coleman, Spalart and Yang2023), Lee et al. (Reference Lee, Helm, Martín and Williams2023) and Cheng & Fu (Reference Cheng and Fu2024a ). Furthermore, the performance of the total heat flux approximation is directly evaluated in Appendix C.

Figure 4(a–f) present the transformed mean velocity profiles using the TL, GFM, HLPP, ZSYX, SLVD and VIPL methods, respectively. The corresponding integrated percentage errors, defined by (4.1), are shown in the insets. Here, the integration bounds $\alpha = 0$ and $\beta = \max \{ 40, 0.1\,{\textit{Re}}_T \}$ are chosen to cover the viscous sublayer, the buffer layer, and at least one-third of the logarithmic region. Theoretically, if a mean velocity scaling is truly Mach-independent, then the transformed profiles should collapse onto their corresponding ‘incompressible’ references at comparable ${\textit{Re}}_T$ , thereby minimising the integrated percentage error. Consistent with the results of Zhu et al. (Reference Zhu, Song, Yang and Xia2024), the TL, HLPP and ZSYX transformations demonstrate satisfactory performance, with integrated percentage errors typically below 2 %. Among them, the TL transformation tends to overpredict the mean shear around the buffer layer, which is more pronounced in the case of the GFM transformation. The HLPP transformation shows a downward shift in the log-law intercept due to overcompensation for intrinsic compressibility effects. The ZSYX transformation achieves the highest accuracy, with the transformed profiles nearly collapsing onto the ‘incompressible’ reference, and integrated percentage errors all below 1 %. In contrast, the VIPL and SLVD transformations are found to be unsuitable for symmetric CTCFs, with maximum errors reaching approximately 9 % and 34 %, respectively. This is not surprising, as the former was developed using a data-driven approach based on CTBLF datasets, while the latter was originally proposed as a scaling for the logarithmic region of ZPG CTBLFs.

Figure 4.

Transformed mean velocity profiles $\bar U_{\textit{trans}}^+ (Y^+)$ in CTCFs with symmetric isothermal walls, based on the DNS datasets listed in table 2. Plots in (a–f) correspond to the TL, GFM, HLPP, ZSYX, SLVD and VIPL transformations, respectively. The grey dashed and solid lines denote the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 180$ and ${\textit{Re}}_T \approx 1000$ , sourced from Lee & Moser (Reference Lee and Moser2015) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016), respectively. Insets show the errors of all eight cases, with the shaded grey area indicating integrated percentage errors within 2 %. The characteristic Reynolds number ${\textit{Re}}_T$ is used to identify the different cases.

Equation (2.19) is first applied with the compressible total heat flux $B_q \bar {q}_{t,{\textit{com}}}^+$ extracted directly from the DNS data. For the three cases from Yao & Hussain (Reference Yao and Hussain2020) and Li et al. (Reference Li, Chen, Xiao, Mao and Yao2025b ), however, $B_q \bar {q}_{t,{\textit{com}}}^+$ must be modelled using (4.6) due to the lack of the necessary statistical quantities. The corresponding results are presented in figure 5. Ideally, (2.21) preserves the accuracy of the velocity transformation if $\bar \tau _{t,{\textit{com}}}^+ = 1 - y/h$ and ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ hold. Specifically, a strictly Mach-independent mean velocity scaling enables an accurate temperature transformation, effectively collapsing the mean temperature profiles onto their corresponding ‘incompressible’ references at similar ${\textit{Re}}_T$ . Conversely, if the Mach number independence is not satisfactorily achieved, then the transformed profiles display pronounced scatter and systematic deviations. As observed, the HLPP- and ZSYX-type transformations are the most accurate, with mean integrated percentage errors below 2 %, and minimal scatter. The TL- and GFM-type transformations perform slightly worse due to overprediction in the buffer layer, but they can still be considered effective. In contrast, the SLVD- and VIPL-type transformations continue to be inapplicable. These behaviours are consistent with the corresponding velocity transformations shown in figure 4, suggesting that both the total shear stress approximation and the Mach independence of the mixed Prandtl number are valid, although the overall accuracy is slightly affected.

Figure 5.

Transformed mean temperature profiles $\bar {\mathcal{T}}_{\textit{trans}}^+ (Y^+)$ in CTCFs with symmetric isothermal walls, where the compressible total heat flux $B_q \bar q_{t,{\textit{com}}}^+$ is directly extracted from the DNS data whenever available. Plots in (a–f) correspond to the TL-, GFM-, HLPP-, ZSYX-, SLVD- and VIPL-type transformations, respectively. The grey dashed and solid lines denote the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 180$ and ${\textit{Re}}_T \approx 1000$ , sourced from Abe et al. (Reference Abe, Kawamura and Matsuo2004) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016), respectively. The characteristic Reynolds number ${\textit{Re}}_T$ is used to identify the different cases.

An effective and concise model for $B_q \bar {q}_{t,{\textit{com}}}^+$ is highly desirable, as it is essential for eliminating reliance on higher-order statistics and facilitating the application of temperature transformations in near-wall modelling. Figure 6 presents the transformed mean temperature profiles obtained by further employing the approximation (4.6). In this context, the TL-, GFM-, HLPP- and ZSYX-type transformations remain effective, with errors typically remaining below 4 %. The VIPL- and SLVD-type transformations continue to be unsuitable, although the approximation (4.6) partially mitigates their inaccuracies. Notably, this approximation induces visible deviations in the outer region, as $B_q \bar {q}_{t,{\textit{com}}}^+$ increases more rapidly from its negative near-wall value toward zero, with these deviations being more significant there, as demonstrated in Appendix C. Specifically, case M30R0180T10I shows a sudden rise in the logarithmic region, while cases M07R0990T10I and M16R0970T10I exhibit singular behaviour. Nevertheless, the performance and accuracy of the transformations in the inner layer are largely unaffected, and the results remain consistent with those in figure 5. Therefore, (4.6) is confirmed to be applicable for this type of flow (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022a ).

Figure 6.

Transformed mean temperature profiles $\bar {\mathcal{T}}_{\textit{trans}}^+ (Y^+)$ in CTCFs with symmetric isothermal walls, where the compressible total heat flux $B_q \bar q_{t,{\textit{com}}}^+$ is approximated by (4.6) instead. The figure follows the same format and layout as figure 5.

Figure 7.

Transformed eddy thermal conductivity profiles $( \bar \lambda _{t,{\textit{in}}} / \bar \lambda _w)_{\textit{trans}} (Y^+)$ in CTCFs with symmetric isothermal walls. Plots in (a–f) correspond to the TL-, GFM-, HLPP-, ZSYX-, SLVD- and VIPL-type transformations, respectively. The grey solid lines indicate the damping function model proposed by Chen et al. (Reference Chen, Lv, Xu, Shi and Yang2022b ), while the grey dashed lines represent the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 180, 395, 550, 1000$ from Abe et al. (Reference Abe, Kawamura and Matsuo2004) and Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016). Insets show the near-wall region ( $Y^+ \in [1, 20]$ ), encompassing the viscous sublayer and part of the buffer layer.

Figure 7 presents the transformed eddy thermal conductivity profiles corresponding to the aforementioned transformation methods, by employing (4.2) and (4.3). Note that the three cases from Yao & Hussain (Reference Yao and Hussain2020) and Li et al. (Reference Li, Chen, Xiao, Mao and Yao2025b ) are excluded due to the absence of reported turbulent heat flux data, $-c_{\!p} \bar \rho\, \widetilde {T^{\prime \prime }v^{\prime \prime }}$ . As expected from the theoretical analysis in § 2.3, the results closely align with those shown in figure 5. Overall, the TL-, ZSYX- and HLPP-type transformations exhibit excellent performance, with their transformed profiles collapsing onto the corresponding ‘incompressible’ reference across the entire inner layer. As highlighted in the insets, the SLVD-type transformation is the only method that fails in the conductive sublayer ( $Y^+ \lesssim 5$ ), leading to its breakdown in both figures 5 and 6, due to cumulative error propagation. For the high-Mach-number case M30R0180T10I, the GFM- and VIPL-type transformations fall noticeably below the ‘incompressible’ reference in the buffer layer, consistent with their overprediction of $\bar {\mathcal{T}}_{\textit{trans}}^+(Y^+)$ observed earlier. Interestingly, despite its failure near the wall, the SLVD-type transformation performs remarkably well in the logarithmic region, whereas the GFM-type transformation exhibits comparatively poorer performance.

4.2. The CTCFs with asymmetric thermal boundary conditions

In this subsection, DNS data of asymmetric CTCFs from Lusher & Coleman (Reference Lusher and Coleman2022) are analysed. The channel configuration consists of an isothermal cold wall and an adiabatic hot wall, and the flows near each wall are considered separately to investigate the effects of wall thermal boundary conditions (Lusher & Coleman Reference Lusher and Coleman2022; Huang et al. Reference Huang, Coleman, Spalart and Yang2023; Cheng & Fu Reference Cheng and Fu2024a ; Zhang, Song & Xia Reference Zhang, Song and Xia2024). The location of the maximum mean streamwise velocity is defined as the effective boundary layer edge, i.e. $y = \delta _e,\ \mathrm{d}\tilde u / \mathrm{d}y = 0$ . Here, $\delta _e$ represents the effective boundary layer thickness on either the isothermal or adiabatic wall side. The datasets cover a range of bulk Mach numbers $1.6 \leqslant M_b \leqslant 3.4$ , with corresponding semi-local Reynolds numbers $228 \leqslant {\textit{Re}}_{\tau ,e}^* \leqslant 613$ on the isothermal cold wall side, and $182 \leqslant {\textit{Re}}_{\tau ,e}^* \leqslant 823$ on the adiabatic hot wall side, as summarised in table 3.

Table 3.

Summary of DNS datasets for CTCFs with asymmetric thermal boundary conditions. The open-source DNS data from Lusher & Coleman (Reference Lusher and Coleman2022), specifically the cases labelled ‘aiB’, ‘aiC’, ‘aiD’, ‘aiE’, ‘aiD2’ and ‘aiF2’ in their original work, are referred to as LC. The characteristic Reynolds numbers are ${\textit{Re}}_{\tau ,e}^* = y^* |_{y = \delta _e}$ and ${\textit{Re}}_{V,e} = y_V^+ |_{y = \delta _e}$ , where $\delta _e$ denotes the effective boundary layer edge, identified by the condition $\mathrm{d}\tilde u / \mathrm{d}y = 0$ (Lusher & Coleman Reference Lusher and Coleman2022; Huang et al. Reference Huang, Coleman, Spalart and Yang2023). Other definitions are consistent with those provided in table 2. The final character in each dataset label indicates the wall thermal boundary condition: ‘I’ for isothermal, and ‘A’ for adiabatic.

The Mach-invariance assumption for the mixed Prandtl number on each wall side is first examined. As shown in figure 8, ${\textit{Pr}}_{m,{\textit{com}}}$ demonstrates a satisfactory collapse for $Y^+ \lesssim 50$ , which is well captured by (3.7), exhibiting only weak sensitivity to Mach number, Reynolds number, and the wall temperature parameter. On the isothermal cold wall side, $Y^+ = y^*$ proves to be a more suitable choice than $Y^+ = y_V^+$ , as the latter shows noticeable scatter around $Y^+ \approx 10$ . Additionally, the ${\textit{Pr}}_{m,{\textit{com}}}(Y^+)$ profiles on the isothermal cold wall side exhibit greater scatter beyond the peak region than those on the adiabatic hot-wall side, which may be attributed to the Reynolds number effects observed in figure 1(d). Nevertheless, the assumption ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ remains valid on both wall sides, as is further confirmed by figure 9, where deviations are confined within $\pm$ 5 % for $Y^+ \leqslant 0.1\, {\textit{Re}}_T$ .

Figure 8.

Compressible mixed Prandtl number ${\textit{Pr}}_{m,{\textit{com}}}$ plotted as a function of (a,c) $Y^+ = y^*$ and (b,d) $Y^+ = y_V^+$ , on the (a,b) isothermal and (c,d) adiabatic wall sides of asymmetric CTCFs. The DNS datasets correspond to those summarised in table 3. The grey solid lines represent predictions of (3.7) using wall Prandtl number ${\textit{Pr}}_w = 0.7$ .

Figure 9.

Wall-normal distributions of the mixed Prandtl number ratio ${\textit{Pr}}_{m,{\textit{in}}} / {\textit{Pr}}_{m,{\textit{com}}}$ , plotted as functions of (a,c) $Y^+ = y^*$ and (b,d) $Y^+ = y_V^+$ on the (a,b) isothermal and (c,d) adiabatic wall sides of asymmetric CTCFs. The DNS datasets correspond to those summarised in table 1 and table 3. Symbols and shading follow the conventions of figure 3.

The mean momentum and total energy equations remain governed by (4.4) and (4.5), except that the mean body force is redefined as $\bar F = ( \bar \tau _{w1} + \bar \tau _{w2}) / 2h$ , where $\bar \tau _{w1}$ and $\bar \tau _{w2}$ denote the absolute values of the mean wall shear stress on the two respective walls, and $h$ represents the channel half-height. By adopting the same approximations as in § 4.1, the total shear stress and total heat flux are simplified as (Zhu et al. Reference Zhu, Zhang, Yang, Ji and Xia2025b )

(4.7) \begin{equation} \bar \tau _{t,{\textit{com}}}^+ \approx 1 - y / \delta _e \quad \text{and} \quad B_q \bar q_{t,{\textit{com}}}^+ \approx B_q + (\gamma - 1) M_\tau ^2 \int _0^{\tilde u^+}{\bar \tau _{t,{\textit{com}}}^+\, \mathrm{d} \tilde u^+}, \end{equation}

where $y$ and $\delta _e$ denote the wall-normal distance and the effective boundary layer thickness adjacent to each wall, respectively.

Figure 10(a–f) present the transformed mean velocity profiles on the isothermal cold wall side, corresponding to the TL, GFM, HLPP, ZSYX, SLVD and VIPL transformations. In this subsection, the integration bounds in (4.1) are also chosen as $\alpha = 0$ and $\beta = \max \{ 40, 0.1\,{\textit{Re}}_T \}$ . The associated integrated percentage errors are displayed in the insets, with the grey shaded region indicating errors below 2 %. Overall, the performance of these transformations closely resembles that observed in symmetric CTCFs. Among them, the TL and ZSYX transformations are the most accurate, typically yielding errors under 2 %. The GFM and HLPP transformations perform slightly worse, showing overprediction and underprediction around the buffer layer, respectively, though their average errors remain below 4 %. In contrast, the SLVD and VIPL transformations are not applicable on the isothermal cold wall side, and perform worse than in symmetric CTCFs, as the adiabatic wall on the opposite side forces more heat through the isothermal wall. It is worth noting that the TL, HLPP and ZSYX transformations nearly collapse all profiles, aligning closely with the ‘incompressible’ reference at ${\textit{Re}}_T \approx 1000$ . While Huang et al. (Reference Huang, Coleman, Spalart and Yang2023) and Cheng & Fu (Reference Cheng and Fu2024a ) regarded such performance to be perfect, the transformed velocity profiles are in fact expected to exhibit a log-law shift due to Reynolds number effects, as discussed in § 3. Our recent study (Zhu et al. Reference Zhu, Song, Zhang, Yang, Ji and Xia2025a ) suggests that this behaviour may be attributed to the modulation effect of the spatially uniform body force $F = f_I$ on the flow field, which enhances turbulence intensity on the isothermal cold wall side beyond what is implied by the characteristic Reynolds number ${\textit{Re}}_T$ . Incorporating a density-weighted body force $F = \rho f_{II}$ (where $f_{II}$ is spatially uniform) may yield transformed profiles that more accurately match the corresponding ‘incompressible’ reference.

Figure 10.

Transformed mean velocity profiles $\bar U_{\textit{trans}}^+ (Y^+)$ on the isothermal wall side of asymmetric CTCFs, based on the DNS datasets listed in table 3. The figure follows the same format and layout as figure 4.

Figure 11 presents the transformed mean temperature profiles corresponding to the six transformation methods considered. Here, $\bar \tau _{t,{\textit{com}}}^+$ is approximated using (4.7), while $B_q \bar q_{t,{\textit{com}}}^+$ is directly obtained from the DNS data. Notably, the TL-, HLPP- and ZSYX-type transformations display increased scatter in the logarithmic region, which is mainly attributable to the neglect of fluctuating components in the viscous shear stress approximation. In contrast, the GFM- and VIPL-type transformations benefit from a mutual cancellation of errors, resulting in improved accuracy compared to their velocity counterparts. To further enhance the practicality of the mean temperature transformation, the compressible total heat flux is further approximated using (4.7), and the corresponding results are shown in figure 12. As demonstrated in Appendix C, this approximation is essentially valid under the present flow configuration. Accordingly, the accuracy of the temperature transformations remains largely unaffected. In this context, the GFM-type transformation demonstrates the highest accuracy, with integrated percentage errors typically below 2 %, and minimal scatter. The TL- and ZSYX-type transformations perform slightly less accurately but still yield reliable results, maintaining mean errors below 4 %. The HLPP-type transformation, however, is no longer recommended due to a pronounced downward shift in the logarithmic intercept, with maximum errors reaching up to 7 %.

Figure 11.

Transformed mean temperature profiles $\bar {\mathcal{T}}_{\textit{trans}}^+ (Y^+)$ on the isothermal wall side of asymmetric CTCFs, where the compressible total heat flux $B_q \bar q_{t,{\textit{com}}}^+$ is directly extracted from the DNS data. The figure follows the same format and layout as figure 5.

Figure 12.

Transformed mean temperature profiles $\bar {\mathcal{T}}_{\textit{trans}}^+ (Y^+)$ on the isothermal wall side of asymmetric CTCFs, where the compressible total heat flux $B_q \bar q_{t,{\textit{com}}}^+$ is approximated by (4.7) instead. The figure follows the same format and layout as figure 5.

Figure 13 illustrates the transformed mean velocity profiles on the adiabatic wall side for the six transformation methods under consideration. The performance of these methods aligns well with previously reported behaviours near adiabatic walls (Griffin et al. Reference Griffin, Fu and Moin2021b ; Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023; Huang et al. Reference Huang, Coleman, Spalart and Yang2023; Zhu et al. Reference Zhu, Song, Yang and Xia2024). Specifically, the GFM, HLPP, ZSYX and VIPL transformations demonstrate excellent accuracy, with integrated percentage errors typically below 1 %. The TL transformation, while maintaining errors approximately 2 %, exhibits a noticeable upward deviation from the ‘incompressible’ reference profile. It is particularly noteworthy that the SLVD transformation nearly collapses all profiles onto the ‘incompressible’ reference at ${\textit{Re}}_T \approx 1000$ , a result previously regarded as favourable by Cheng & Fu (Reference Cheng and Fu2024a ). However, as discussed in § 3, an ideal transformation is in fact expected to reproduce Reynolds-number-dependent shifts in the transformed profiles.

Figure 13.

Transformed mean velocity profiles $\bar U_{\textit{trans}}^+ (Y^+)$ on the adiabatic wall side of asymmetric CTCFs, based on the DNS datasets listed in table 3. The figure follows the same format and layout as figure 4.

Figure 14 presents the transformed mean temperature profiles, where $B_q \bar q_{t,{\textit{com}}}^+$ is directly obtained from the DNS data. Interestingly, the TL-type transformation now proves to be the most accurate, with all errors remaining below 2 % and exhibiting the most concentrated distribution. In contrast, although the GFM, HLPP, ZSYX and VIPL transformations exhibit near-perfect performance for mean velocity, their corresponding temperature transformations perform less favourably at low Reynolds numbers. Comparison with figure 13 suggests that these discrepancies are primarily attributed to the approximation of $\bar \tau _{t,{\textit{com}}}^+$ in (4.7) and the assumption ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ . Nevertheless, these methods remain highly effective, with mean errors still below 3 %. The SLVD-type method continues to collapse all profiles onto the high-Reynolds-number ‘incompressible’ reference, which is not expected for an ideal transformation. Figure 15 presents results obtained by further approximating $B_q \bar q_{t,{\textit{com}}}^+$ using (4.7). As shown, all transformed profiles lie noticeably above the ‘incompressible’ reference due to overpredictions in the conductive sublayer and buffer layer. This trend aligns with the results of Huang et al. (Reference Huang, Coleman, Spalart and Yang2023), where the authors attributed such discrepancy to low-Reynolds-number effects induced by high temperatures on the adiabatic wall side. However, consistent with the interpretation of Cheng & Fu (Reference Cheng and Fu2024a ), we demonstrate that the deviation primarily arises from the neglect of higher-order terms in the approximation of the compressible total heat flux via (4.7), through comparison with the results shown in figure 14, where $B_q \bar q_{t,{\textit{com}}}^+$ is obtained without approximation. Compared to previous results, this effect becomes more pronounced near the adiabatic wall, where the total heat flux approaches zero and the previously neglected terms grow in relative importance, as discussed in Appendix C. Most recently, Zhu et al. (Reference Zhu, Zhang, Yang, Ji and Xia2025b ) validated their proposed integral mean TV relation in asymmetric CTCFs, and found that the model remains highly effective under the same approximation (4.7). In fact, temperature transformations based on strictly Mach-independent mean velocity scalings are essentially equivalent to their integral relation, while avoiding the explicit modelling of ${\textit{Pr}}_{m,{\textit{com}}}$ . Therefore, it can be inferred that the inverse form of (2.21), and its further application in near-wall temperature modelling, remain promising.

Figure 14.

Transformed mean temperature profiles $\bar {\mathcal{T}}_{\textit{trans}}^+ (Y^+)$ on the adiabatic wall side of asymmetric CTCFs, where the compressible total heat flux $B_q \bar q_{t,{\textit{com}}}^+$ is directly extracted from the DNS data. The figure follows the same format and layout as figure 5.

Figure 15.

Transformed mean temperature profiles $\bar {\mathcal{T}}_{\textit{trans}}^+ (Y^+)$ on the adiabatic wall side of asymmetric CTCFs, where the compressible total heat flux $B_q \bar q_{t,{\textit{com}}}^+$ is approximated by (4.7) instead. The figure follows the same format and layout as figure 5.

Figure 16.

Transformed eddy thermal conductivity profiles $( \bar \lambda _{t,{\textit{in}}} / \bar \lambda _w)_{\textit{trans}} (Y^+)$ on the isothermal wall side of asymmetric CTCFs. The figure follows the same format and layout as figure 7.

Figure 17.

Transformed eddy thermal conductivity profiles $( \bar \lambda _{t,{\textit{in}}} / \bar \lambda _w)_{\textit{trans}} (Y^+)$ on the adiabatic wall side of asymmetric CTCFs. The figure follows the same format and layout as figure 7.

Figures 16 and 17 present the transformed eddy thermal conductivity profiles on the isothermal cold wall and adiabatic hot wall sides, respectively, corresponding to the six transformations. On the isothermal cold wall side, only the SLVD-type transformation fails in the conductive sublayer, while the GFM- and VIPL-type methods exhibit noticeable inaccuracies in the buffer layer. Interestingly, the SLVD-type transformation performs best in the logarithmic region, whereas the other five transformations consistently overestimate the eddy thermal conductivity compared to the ‘incompressible’ reference at a similar characteristic Reynolds number ${\textit{Re}}_T$ . This trend is consistent with the earlier observation that the transformed mean velocity and temperature profiles nearly collapse onto ‘incompressible’ profiles with ${\textit{Re}}_T \approx 1000$ , and is attributed to turbulence enhancement induced by the spatially uniform body force (Zhu et al. Reference Zhu, Song, Zhang, Yang, Ji and Xia2025a ). On the adiabatic hot wall side, all transformed eddy thermal conductivity profiles show satisfactory agreement with their corresponding ‘incompressible’ references. While the ZSYX- and VIPL-type transformations perform the best, the SLVD-type method slightly overpredicts the eddy thermal conductivity in the buffer layer and logarithmic region, consistent with the profile shifts observed in figures 13 and 14.

4.3. The ZPG CTBLFs with and without wall heat transfer

In this subsection, DNS data for ZPG CTBLFs with adiabatic and diabatic walls from Zhang, Duan & Choudhari (Reference Zhang, Duan and Choudhari2018) and Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023) are employed to evaluate the performance of various transformation methods. The datasets cover a wide range of freestream Mach numbers $2.0 \leqslant M_\infty \leqslant 13.6$ , semi-local Reynolds numbers $541 \leqslant {\textit{Re}}_{\tau ,e}^* \leqslant 4952$ , and diabatic parameters $0.13 \leqslant \Theta \leqslant 1.00$ , as summarised in table 4. The corresponding DNS profiles of ITCFs with passive scalar transport at similar characteristic Reynolds number ${\textit{Re}}_T$ are still considered as the ‘incompressible’ reference.

Table 4.

Summary of DNS datasets for ZPG CTBLFs with and without wall heat transfer. Data sources are abbreviated as follows: ZDC refers to Zhang et al. (Reference Zhang, Duan and Choudhari2018), and CBCBP to Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023). The freestream Mach number is defined as $M_{\infty } = u_{\infty } / \sqrt {\gamma R T_{\infty }}$ , where $u_{\infty }$ and $T_{\infty }$ denote the freestream velocity and temperature, respectively. The characteristic Reynolds numbers are given by ${\textit{Re}}_{\tau ,e}^* = y^* |_{y = \delta _e}$ and ${\textit{Re}}_{V,e} = y_V^+ |_{y = \delta _e}$ , where $\delta _e = \delta _{99}$ represents the boundary layer thickness at which $\bar u \approx 0.99 u_{\infty }$ . The recovery temperature is defined as $\tilde T_r = [1 + r (\gamma - 1) M_{\infty }^2 / 2] T_{\infty }$ , and the diabatic parameter $\Theta = (\tilde T_w - T_{\infty }) / (\tilde T_r - T_{\infty })$ is used to characterise wall heat transfer effects (Zhang et al. Reference Zhang, Bi, Hussain and She2014; Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023, Reference Cogo, Chinappi, Bernardini and Picano2024; Gibis et al. Reference Gibis, Sciacovelli, Kloker and Wenzel2024).

Figure 18.

Wall-normal distributions of the mixed Prandtl number ratio ${\textit{Pr}}_{m,{\textit{in}}} / {\textit{Pr}}_{m,{\textit{com}}}$ in ZPG CTBLFs, plotted as functions of (a) $Y^+ = y^*$ and (b) $Y^+ = y_V^+$ , based on the DNS datasets listed in table 4. Symbols and shading follow the conventions of figure 3.

Figure 18 presents the distributions of the mixed Prandtl number ratio ${\textit{Pr}}_{m,{\textit{in}}} / {\textit{Pr}}_{m,{\textit{com}}}$ in ZPG CTBLFs, based on the open-source DNS dataset of Zhang et al. (Reference Zhang, Duan and Choudhari2018). It should be noted that the computed ${\textit{Pr}}_{m,{\textit{com}}}$ is subject to approximation errors, as the simplifications $-\bar \rho\, \widetilde {u^{\prime \prime }v^{\prime \prime }\ } \approx -\bar \rho\, \overline {u^{\prime }v^{\prime }}$ and ${\textit{Pr}}_t \approx (\overline {\rho u^{\prime }v^{\prime }}\, \partial \bar T \!/ \partial y) / (\overline {\rho v^{\prime }T^{\prime }}\, \partial \bar u / \partial y)$ were employed due to unavailable turbulence statistics. For most of the region with $Y^+ \leqslant 0.1\, {\textit{Re}}_T$ , the ratio remains nearly within 0.95–1.05. However, using $Y^+=y^*$ produces noticeably larger deviations than $Y^+=y_V^+$ , with relative errors in parts of the buffer layer exceeding the $\pm$ 5 % tolerance band. This indicates that the transformed coordinate $Y^+=y_V^+$ associated with the VIPL transformation may more effectively eliminate Mach number effects in this flow configuration. Notably, a pronounced singularity occurs near the mean temperature peak (where $\partial \tilde T \!/ \partial y = 0$ ) for cases with cold walls, suggesting a local breakdown of the assumption ${\textit{Pr}}_{m,{\textit{com}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w) = {\textit{Pr}}_{m,{\textit{in}}}(Y^+,{\textit{Re}}_T,{\textit{Pr}}_w)$ . The origin of this singularity and the validity of the current model (3.7) will be discussed and examined in detail below.

Figure 19(a–f) present the transformed mean velocity profiles using the TL, GFM, HLPP, ZSYX, SLVD and VIPL methods, respectively. The integration bounds in (4.1) are set as $\alpha = 0$ and $\beta = \max \{ 80, 0.1\,{\textit{Re}}_T \}$ , ensuring coverage of the viscous sublayer, the buffer layer, and at least one-third of the logarithmic region. The results are generally consistent with previous studies (Volpiani et al. Reference Volpiani, Iyer, Pirozzoli and Larsson2020; Griffin et al. Reference Griffin, Fu and Moin2021b ; Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023; Zhu et al. Reference Zhu, Song, Yang and Xia2024). Specifically, the TL and SLVD transformations exhibit significant scatter around the buffer layer and are therefore not considered applicable. The HLPP transformation clearly improves upon the TL method by accounting for the upward shift of the log-law induced by intrinsic compressibility effects, characterised by the wall friction Mach number $M_\tau$ . Nevertheless, the transformed profiles still show noticeable scatter, and the extent of the logarithmic region is generally reduced compared to the ‘incompressible’ references. Accordingly, the integrated percentage errors tend to increase with ${\textit{Re}}_{\tau ,e}^*$ . In contrast, the GFM, ZSYX and VIPL transformations demonstrate satisfactory agreement with the corresponding ‘incompressible’ profiles across the entire inner layer, with all errors remaining below 6 %, and mean errors nearly below 2 %.

Figure 19.

Transformed mean velocity profiles $\bar U_{\textit{trans}}^+ (Y^+)$ in ZPG CTBLFs with and without wall heat transfer, based on the DNS datasets listed in table 4. Plots in (a–f) correspond to the TL, GFM, HLPP, ZSYX, SLVD and VIPL transformations, respectively. The grey dashed and solid lines denote the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 1000$ and ${\textit{Re}}_T \approx 5000$ , sourced from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) and Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021), respectively. Insets show the errors of all 11 cases, where closed ‘ $\square$ ’ symbols denote the five cases from Zhang et al. (Reference Zhang, Duan and Choudhari2018), and open ‘ $\square$ ’ symbols denote the six cases from Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023). The shaded grey area indicates integrated percentage errors below 2 %. The characteristic Reynolds number ${\textit{Re}}_T$ is used to identify the different cases.

To apply the temperature transformations defined in (2.21), we derive a simplified expression for the compressible total heat flux by adopting approximations similar to those in § 4.1 and § 4.2 (see Appendix B for details):

(4.8) \begin{equation} B_q \bar q_{t,{\textit{com}}}^+ \approx B_q + (\gamma -1) M_\tau ^2 \int _{0}^{\tilde u^+}{\bar \tau _{t,{\textit{com}}}^+\, \mathrm{d}\tilde u^+} + \int _1^{\bar \tau _{t,{\textit{com}}}^+}{\frac {\partial \theta }{\partial \tilde u^+}\, \mathrm{d}\bar \tau _{t,{\textit{com}}}^+}. \end{equation}

Compared to (4.6), the third term on the right-hand side accounts for the convective transport of mean enthalpy. While this term can be neglected for further simplification by assuming $\bar \tau _{t,{\textit{com}}}^+ \approx 1$ across the entire boundary layer, we retain it in this subsection due to its notable influence in the outer part of the logarithmic region. In the absence of an analytical expression for the total shear stress, we adopt a defect power-law model (Sun & Childs Reference Sun and Childs1973) to approximate its distribution, i.e. $\bar \tau _{t,{\textit{com}}}^+ \approx 1 - (y / \delta _e)^{3/2}$ , following the approach of previous studies (Chen & She Reference Chen and She2016; Wu et al. Reference Wu, Bi, Hussain and She2017; Zhu et al. Reference Zhu, Song, Yang and Xia2024).

Before proceeding further, it is important to highlight some challenges in dealing with ZPG CTBLFs over cold walls (i.e. $\Theta \lt 1$ or $\tilde T_w / \tilde T_r \lt 1$ ). In these flows, the mean temperature profiles exhibit non-monotonic variations in the wall-normal direction, primarily due to the combined effects of wall heat transfer and viscous heating. Near the mean temperature peak, the turbulent heat flux also approaches zero. However, a non-negligible discrepancy exists between the wall-normal locations where $\bar \lambda\, \partial \tilde T \!/ \partial y$ and $-c_{\!p} \bar \rho\, \widetilde {T^{\prime \prime }v^{\prime \prime }}$ vanish. This mismatch becomes increasingly pronounced with rising friction Mach number $M_\tau$ , reaching up to a 25 % offset when $M_\tau \approx 0.23$ , as reported by Gibis et al. (Reference Gibis, Sciacovelli, Kloker and Wenzel2024). Yet it remains unclear whether this behaviour arises from an underlying physical mechanism or from numerical artefacts. A particularly challenging issue is that the eddy thermal conductivity, $\bar \lambda _t = -c_{\!p} \bar \rho\, \widetilde {T^{\prime \prime }v^{\prime \prime }} / (\partial \tilde T \!/ \partial y)$ , becomes ill-defined near the temperature peak, further leading to singularities in both the turbulent and mixed Prandtl numbers, as observed in figure 18 (Chen et al. Reference Chen, Gan and Fu2024; Gibis et al. Reference Gibis, Sciacovelli, Kloker and Wenzel2024). Furthermore, according to (2.21), the transformed mean temperature profiles exhibit abrupt jumps near the zero-crossing point of the total heat flux (Cheng & Fu Reference Cheng and Fu2024a ; Chen et al. Reference Chen, Gan and Fu2025), which lies between the zero points of the molecular and turbulent heat fluxes. Consequently, the methods for directly evaluating $\bar {\mathcal{T}}_{\textit{trans}}^+(Y^+)$ presented in § 4.1 and § 4.2 cannot be reliably applied to ZPG CTBLFs.

To mitigate the impact of the discontinuities, we define the mean temperature gradient of the ‘incompressible’ state $\varPhi (Y^+) = \partial \bar {\mathcal{T}}^+ \!/ \partial Y^+$ as $\xi$ in (4.1), as suggested by Cheng & Fu (Reference Cheng and Fu2024a ). This enables separate quantitative assessments for various transformation methods in the conductive sublayer ( $\alpha = 0$ , $\beta = 5$ ) and logarithmic region ( $\alpha = 30$ , $\beta = \max \{ 80, 0.1\,{\textit{Re}}_T \}$ ). The results are illustrated in figures 20 and 21, where $\varPsi _{\textit{trans}} = Y^+ \varPhi _{\textit{trans}} = Y^+\, \partial \bar {\mathcal{T}}_{\textit{trans}}^+ / \partial Y^+$ is the transformed indicator function. Meanwhile, figure 22 shows the corresponding integrated percentage errors as functions of ${\textit{Re}}_T$ . It should be noted that the adiabatic case ( $\Theta = 1$ ), along with the five moderately cooled cases ( $0 \leqslant \Theta \lt 0.43$ ), is used to assess accuracy in the conductive sublayer, as any discontinuity (if present) lies within the buffer layer. Conversely, the adiabatic case, together with the five weakly cooled cases ( $0.43 \leqslant \Theta \lt 1$ ), is used to evaluate performance in the logarithmic region, where any singularity (if present) is confined to the conductive sublayer (Gibis et al. Reference Gibis, Sciacovelli, Kloker and Wenzel2024). To evaluate the intrinsic accuracy of various transformation methods, the compressible total heat flux $B_q \bar {q}_{t,{\textit{com}}}^+$ is directly computed from the available turbulence statistics in the five cases reported by Zhang et al. (Reference Zhang, Duan and Choudhari2018). In contrast, for the six cases presented by Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023), $B_q \bar {q}_{t,{\textit{com}}}^+$ is modelled by (4.8), due to the unavailability of the required statistical data. Overall, the performance of these transformation methods is highly consistent with that shown in figure 19. Specifically, most transformations yield mean errors typically below 5 % in the conductive sublayer, indicating generally good performance. However, the SLVD-type transformation fails in the two coldest-wall cases ( $\Theta \lt 0.2$ ), with maximum integrated errors reaching approximately 23 %. Although relative errors are not explicitly quantified, the TL-type transformation exhibits significant overprediction near the end of the buffer layer ( $20 \lesssim Y^+ \lesssim 30$ ). The HLPP- and ZSYX-type transformations offer partial improvements by incorporating intrinsic compressibility effects within their respective mean velocity scalings, but still exhibit noticeable scatter, particularly for cases with ${\textit{Re}}_{\tau ,e}^* \gt 4000$ . In contrast, the GFM-, SLVD- and VIPL-type transformations show minimal scatter. In the logarithmic region, the GFM-, ZSYX- and VIPL-type transformations demonstrate superior accuracy, with transformed profiles collapsing closely onto the ‘incompressible’ references at comparable ${\textit{Re}}_T$ , and the corresponding mean errors remaining below 5 %. It is worth noting that the SLVD transformation, previously regarded as effective within the logarithmic region, fails to exhibit satisfactory performance in the present study. This discrepancy is primarily attributed to the broader evaluation range adopted here ( $\alpha = 30$ , $\beta = \max \{80,0.1\,{\textit{Re}}_T\}$ ), in contrast to that ( $\alpha = 30$ , $\beta = 80$ ) used by Cheng & Fu (Reference Cheng and Fu2024a ). The present criterion ensures that at least one-third of the logarithmic region is included for flows with ${\textit{Re}}_T \gt 800$ , thereby providing a more rigorous and representative assessment.

Figure 20.

Transformed indicator function $\varPsi _{\textit{trans}}(Y^+) = Y^+\, \partial \bar {\mathcal{T}}_{\textit{trans}}^+ / \partial Y^+$ in the conductive sublayer of ZPG CTBLFs with and without wall heat transfer. The grey dashed and solid lines represent the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 1000$ and ${\textit{Re}}_T \approx 5000$ , obtained from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) and Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021), respectively.

Figure 21.

Transformed indicator function $\varPsi _{\textit{trans}}(Y^+) = Y^+\, \partial \bar {\mathcal{T}}_{\textit{trans}}^+ / \partial Y^+$ in the logarithmic region of ZPG CTBLFs with and without wall heat transfer. The grey dashed and solid lines represent the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 1000$ and ${\textit{Re}}_T \approx 4000$ , obtained from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016).

Figure 22.

Integrated percentage errors of the ‘incompressible’ mean temperature gradient, $\varPhi (Y^+) = \partial \bar {\mathcal{T}}^+ \!/ \partial Y^+$ , as functions of the characteristic Reynolds number ${\textit{Re}}_T$ in ZPG CTBLFs, evaluated within (a) the conductive sublayer and (b) the logarithmic region. Colour and symbol conventions follow those in figure 19.

Figure 23.

Transformed eddy thermal conductivity profiles $(\bar {\lambda }_{t,{\textit{in}}} / \bar {\lambda }_w)_{\textit{trans}}(Y^+)$ in ZPG CTBLFs with and without wall heat transfer. Plots in (a–f) correspond to the TL-, GFM-, HLPP-, ZSYX-, SLVD- and VIPL-type transformations, respectively. The grey solid lines indicate the damping function model proposed by Chen et al. (Reference Chen, Lv, Xu, Shi and Yang2022b ), while the grey dashed lines represent the ‘incompressible’ reference profiles at ${\textit{Re}}_T \approx 1000$ and ${\textit{Re}}_T \approx 5000$ , obtained from Pirozzoli et al. (Reference Pirozzoli, Bernardini and Orlandi2016) and Alcántara-Ávila et al. (Reference Alcántara-Ávila, Hoyas and Pérez-Quiles2021).

Figure 23 presents the transformed eddy thermal conductivity profiles associated with the aforementioned transformation methods, computed using (4.2) and (4.3). Only the five cases from Zhang et al. (Reference Zhang, Duan and Choudhari2018) are included, due to the availability of the necessary turbulence statistics. As previously noted, the cases with cold walls exhibit pronounced discontinuities below the buffer layer ( $Y^+ \lesssim 30$ ), attributed to the misalignment in the wall-normal positions of the zero crossings between $\bar \lambda\, \partial \tilde {T} \!/ \partial y$ and $-c_{\!p} \bar {\rho }\, \widetilde {T^{\prime \prime }v^{\prime \prime }}$ . Within the viscous sublayer ( $Y^+ \lesssim 5$ ), most transformation methods perform well, whereas the SLVD-type transformation shows notable scatter in the two coldest-wall cases, even when the discontinuity effects are excluded. In the logarithmic region, profiles associated with the TL- and HLPP-type transformations fall significantly below the corresponding ‘incompressible’ references for $y^* \gtrsim 200$ , leading to deviations from the classical logarithmic law in the outer layer of $\bar {\mathcal{T}}_{\textit{trans}}^+(Y^+)$ . Conversely, the SLVD-type transformation yields considerably higher values than the ‘incompressible’ references. The GFM-, ZSYX- and VIPL-type transformations, however, demonstrate excellent performance, with the transformed profiles closely collapsing onto the ‘incompressible’ reference profiles at comparable ${\textit{Re}}_T$ . As anticipated from the theoretical analysis in § 2.3, these results are in close agreement with those observed in figure 19, confirming that advanced mean velocity scalings offer strong potential for the development of more accurate compressible eddy thermal conductivity models.