1.1. The law of the wall – velocity scaling

Advancements in the wall-scaling law in compressible boundary layers mainly stem from the foundational work of Van Driest (VD) (Reference Van Driest1951), who proposed the velocity transformation (namely, VD transformation or VDT) grounded in Morkovin’s scaling and Prandtl’s incompressible mixing length theory. However, the VDT does not perform well for diabatic boundary layers and channel flows. To address this issue, Zhang et al. (Reference Zhang, Bi, Hussain, Li and She2012) introduced a viscosity-weighted transformation of the mean-velocity profile based on a boundary-layer thickness length scale using Reynolds stress structure parameters to fulfil the Mach-number-invariant condition. The performance improvement was found to be still limited under diabatic conditions (Griffin et al. Reference Griffin, Fu and Moin2021; Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023). Huang et al. (Reference Huang, Coleman and Bradshaw1995) developed a semi-local scaling method to enhance the mapping of wall-normal turbulence statistics in the logarithmic region. Pecnik & Patel (Reference Pecnik and Patel2017) provided theoretical support for the semi-local scaling and identified the quasi-similarity between variable-property and constant-property flows. Ma, Yang & Ihme (Reference Ma, Yang and Ihme2018) examined this approach under transcritical conditions and reported reduced effectiveness near the heated wall, where density fluctuation is stronger than at the cooled wall. Trettel & Larsson (TL) (Reference Trettel and Larsson2016) incorporated the stress balance assumption and developed a new transformation (namely, TL transformation) to overcome the VDT’s insufficiency for non-adiabatic wall-bounded flows. The TL-transformed mean-velocity profiles collapse consistently at various Reynolds numbers, Mach numbers and wall heat transfer rates in supersonic channel flows. Wan et al. (Reference Wan, Zhao, Liu, Wang and Lei2020) and Bai, Griffin & Fu (Reference Bai, Griffin and Fu2022) demonstrated that the TL transformation works well for the inner region of the boundary layer in supercritical channel flows. More recent efforts by Griffin et al. (Reference Griffin, Fu and Moin2021) integrated the frameworks of Zhang et al. (Reference Zhang, Bi, Hussain, Li and She2012) and Trettel & Larsson (Reference Trettel and Larsson2016), and derived a new transformation based on the universality of the production-to-consumption ratio and a stress-based mixing function, yet their approach struggles with non-air-like viscosity laws and SCP conditions. Hasan et al. (Reference Hasan, Larsson, Pirozzoli and Pecnik2023) accounted for intrinsic compressibility effects and extended the capabilities of the TL transformation to a broader range of flow scenarios, covering wall-bounded flows of both ideal gases and SCP fluids. However, Zhu et al. (Reference Zhu, Song, Yang and Xia2024) revealed that the efficacy of both the Griffin’s and Hasan’s transformations is diminished in low Reynolds number channel and pipe flows. A more complex transformation that incorporates intrinsic compressibility effects and the multi-layer structure of mixing lengths was developed, whereas its performance in SCP regimes remains unconfirmed.

1.2. The law of the wall – temperature-/enthalpy–velocity relations

Regarding the studies of thermal scaling, one major line of the research is on the similarity between velocity and temperature fields to establish a temperature–velocity (T–V) relation. Unlike coordinate-based wall-scaling laws, this approach is generally regarded as a weaker form of scaling (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022). An early T–V relation was established by Busemann (Reference Busemann1931), who showed that, for compressible laminar boundary layers with a molecular Prandtl number ( $\mathit{Pr}$ ) of unity, velocity and temperature follow a quadratic relation. Walz (Reference Walz1969) later modified this relation to account for the deviation of $\mathit{Pr}$ from unity through the introduction of a recovery factor, $r = \mathit{Pr}^{1/3}$

(1.1) \begin{equation} \tilde {T}_{\textit{Walz}} = \tilde {T}_w + \big(\tilde {T}_r - \tilde {T}_w \big) \frac {\tilde {u}}{\tilde {u}_e} + \big(\tilde {T}_e - \tilde {T}_r \big)\left (\frac {\tilde {u}}{\tilde {u}_e}\right )^2\!, \end{equation}

where $T$ and $u$ denote temperature and velocity, with subscripts ` $w$ ’ and ` $e$ ’ referring to wall and boundary-layer-edge values, respectively. The recovery temperature is defined as $\tilde {T}_r = \tilde {T}_e + r\tilde {u}_e^2/(2c_{\!p})$ , and $c_{\!p}$ is specific heat capacity at constant pressure. The tilde $(\tilde {\ })$ denotes a Favre-averaged quantity. While (1.1) showed good performance for adiabatic compressible turbulent boundary layers, it deviates significantly from direct numerical simulation (DNS) data under strong cold-wall conditions (Duan & Martin Reference Duan and Martin2011; Zhang et al. Reference Zhang, Duan, Choudhari and Wu2014). To address these limitations, Zhang et al. (Reference Zhang, Duan, Choudhari and Wu2014) introduced a generalised recovery factor, $r_{\!g}$ to revise $\tilde {T}_{\textit{Walz}}$ by incorporating cold-wall and diabatic effects. A local generalised recovery temperature $\tilde {T}_{r_{\!g}}$ was defined to replace $\tilde {T}_r$

(1.2) \begin{equation} \tilde {T}_{\textit{Zhang}} = \tilde {T}_w + \big(\tilde {T}_{r_{\!g}} - \tilde {T}_w \big) \frac {\tilde {u}}{\tilde {u}_e} + \big(\tilde {T}_e - \tilde {T}_{r_{\!g}}\big) \left (\frac {\tilde {u}}{\tilde {u}_e}\right )^2\!, \end{equation}

where $\tilde {T}_{r_{\!g}} = \tilde {T}_e + r_{\!g} \tilde {u}_e^2 / (2 c_{\!p})$ and $r_{\!g} = (\tilde {T}_w - \tilde {T}_e) / (\tilde {u}_e^2 / (2 c_{\!p})) + (2 c_{\!p} / \tilde {u}_e) (\partial \tilde {T} / \partial \tilde {u})_w$ . Equation (1.2) was shown to perform well in hypersonic turbulent boundary layers of calorically perfect gases, where $c_{\!p}$ is constant (Zhang, Duan & Choudhari Reference Zhang, Duan and Choudhari2018; Mo et al. Reference Mo, Zhang, Duan and Choudhari2023). However, its applicability to wall-bounded SCP flows remains to be confirmed, as real-fluid effects caused by abrupt property variations render the perfect-gas assumption invalid. In particular, $c_{\!p}$ varies dramatically and nonlinearly across the pseudo-boiling line, complicating the relation between enthalpy and temperature in the energy transport.

To mitigate the explicit dependence of T–V relations on thermal and chemical models, Duan & Martin (Reference Duan and Martin2011) introduced the concept of recovery enthalpy, $\tilde {h}_r = \tilde {h} + r(\tilde {u}^2/2)$ , and established a semi-empirical enthalpy–velocity (H–V) relation

(1.3) \begin{equation} \tilde {h} = \tilde {h}_w + \big(\tilde {h}_r - \tilde {h}_w\big) f\!\left (\frac {\tilde {u}}{\tilde {u}_e}\right ) - \frac {r\tilde {u}^2}{2}, \end{equation}

where the recovery factor $r$ is assumed to be $0.9$ , and the empirical function $f(\tilde {u}/\tilde {u}_e)$ is given by fitting the DNS data of hypersonic reacting turbulent boundary layers

(1.4) \begin{equation} f\left (\frac {\tilde {u}}{\tilde {u}_e}\right ) = 0.8259\frac {\tilde {u}}{\tilde {u}_e} + 0.1741\left (\frac {\tilde {u}}{\tilde {u}_e}\right )^2\!. \end{equation}

For calorically perfect gases, (1.3) reduces to the classical Walz’s relation (1.1) with $\tilde {u}/\tilde {u}_e$ replaced by $f(\tilde {u}/\tilde {u}_e)$ . However, its empirical nature, including a constant recovery factor and data-fitted function $f(\tilde {u}/\tilde {u}_e)$ , may jeopardise its reliability for SCP conditions, where property variations are far more drastic. Inspired by Duan & Martin (Reference Duan and Martin2011) and Zhang et al. (Reference Zhang, Duan, Choudhari and Wu2014), Li et al. (Reference Li, Zhang, Duan and Choudhari2025) have recently developed a new H-V relation without empirical function $f(\tilde {u}/\tilde {u}_e)$ but introduced the generalised recovery enthalpy $\tilde {h}_{r_{\!g}^*}$

(1.5) \begin{equation} \tilde {h} = \tilde {h}_w + \big(\tilde {h}_{r_{\!g}^*} - \tilde {h}_w\big)\frac {\tilde {u}}{\tilde {u}_e} + \big(\tilde {h}_e - \tilde {h}_{r_{\!g}^*}\big)\left (\frac {\tilde {u}}{\tilde {u}_e}\right )^2\!. \end{equation}

The generalised recovery enthalpy $\tilde {h}_{r_{\!g}^*}$ is determined as

(1.6) \begin{equation} \tilde {h}_{r_{\!g}^*} = \tilde {h}_e + r_{\!g}^* \tilde {u}_e^2 / 2, \end{equation}

and the recovery factor $r_{\!g}^*$ defined as

(1.7) \begin{equation} r_{\!g}^* = \big(\tilde {h}_w - \tilde {h}_e\big)/\big(\tilde {u}_e^2/2\big) + 2\mathit{Pr}_w \, \bar {q}_w / (\tilde {u}_e \bar {\tau }_w). \end{equation}

Numerical studies (Li et al. Reference Li, Zhang, Duan and Choudhari2025) of hypersonic reacting turbulent boundary layers at atmospheric pressure confirmed the improved accuracy of this relation.

Despite these advances, existing T–V and H–V relations were developed primarily for compressible flows under near-atmospheric conditions. Their extension to SCP wall-bounded flows remains uncertain due to the strong influence of real-fluid effects. Moreover, recent studies (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022; Cheng & Fu Reference Cheng and Fu2024) argue that these relations are not closed by themselves, and limited by both the empirical recovery factor and outer boundary conditions, which prevents their application to complex flow configurations. In § 4.1, an assessment of H–V relations for SCP fluid flows will be conducted.

1.3. The law of the wall – thermal transformation

Another category for thermal scaling is to develop transformation-based thermal wall models that map the mean scalar profile of compressible flows into its `incompressible’ equivalent form. Recently, Chen, Gan & Fu (Reference Chen, Gan and Fu2025) have demonstrated that the temperature-transformation wall model for compressible laminar and turbulent flows at atmospheric pressure serves as a good alternative to the T–V wall model and successfully excludes the dependence on boundary-layer-edge quantities. However, compared with velocity transformation, thermal transformation faces additional complexity due to dependencies on heat flux, specific heat capacity and thermal conductivity (Bradshaw & Huang Reference Bradshaw and Huang1995). Furthermore, most studies on thermal boundary layers have primarily focused on flows under the ideal-gas assumption and cannot be directly extended to SCP fluid flows, where real-fluid effects play a critical role. Early efforts, such as Kader’s temperature transformation (Kader Reference Kader1981) based on the Reynolds analogy, were made for low-speed incompressible flows and assumed constant thermophysical properties

(1.8) \begin{equation} \overline {\theta }^+ = \frac {\overline {\theta }}{\overline {\theta }_\tau } = \begin{cases} \textit{Pr}_w y^+ & \text{sublayer region}, \\ \frac {1}{\kappa } \ln (y^+) + \beta (\textit{Pr}_w) & \text{logarithmic region}. \end{cases} \end{equation}

Here, overbar $\overline {(\boldsymbol{\cdot })}$ denotes the Reynolds-averaged physical quantity, and the mean temperature difference is defined as $\overline {\theta } = \overline {T} - \overline {T}_w$ ; $\kappa$ is a constant of 0.47; $\beta (\textit{Pr}_w)$ is another constant determined by the wall Prandtl number $\textit{Pr}_w$ and is given by $\beta (\textit{Pr}_w) = (3.85 \textit{Pr}_w^{1/3} - 1.3 )^2 + 2.12 \ln (\textit{Pr}_w)$ . The friction temperature is defined as $\overline {\theta }_\tau = \overline {q}_w / (\overline {\rho }_w \, \overline {c_{\!p}}_w \, \overline {u}_\tau )$ , where $q$ , $\rho$ and $u_\tau$ are the heat flux, density and wall friction velocity, respectively. Hereinafter, the wall-normal coordinates $y$ , $y^+$ and $y^*$ denote the physical coordinate, the coordinate scaled by the wall viscous length $\delta _v^+$ (Pope Reference Pope2001) and the coordinate scaled by the semi-local viscous length $\delta _v^*$ (Huang et al. Reference Huang, Coleman and Bradshaw1995), respectively. Specifically,

(1.9) \begin{align} y^+ &= \dfrac {y}{\delta _v^+},y^*= \dfrac {y}{\delta _v^*},\delta _v^+= \dfrac {\overline {\mu }_w}{\overline {\rho }_w\,\overline {u}_\tau },\delta _v^*= \dfrac {\overline {\mu }(y)}{\overline {\rho }(y)\,\overline {u}_\tau ^*}, \overline {u}_\tau = \sqrt {\dfrac {\overline {\tau }_w}{\overline {\rho }_w}},\overline {u}_\tau ^*= \sqrt {\dfrac {\overline {\tau }_w}{\overline {\rho }(y)}}, \end{align}

with $\mu$ and $\tau _w$ being dynamic viscosity and viscous shear stress.

Note that (1.8), which does not account for property variations, is generally inapplicable to compressible or other variable-property flows. Lee et al. (Reference Lee, Jung, Sung and Zaki2014) modified Kader’s formulation by incorporating semi-local Prandtl number $\textit{Pr}^*$ instead of the wall-based $\textit{Pr}_w$ . Nonetheless, their method was pointed out to be inadequate because of neglecting the impact of property variations on the log-law slope. Patel et al. (Reference Patel, Boersma and Pecnik2017) prescribed the relations for density $\rho$ , viscosity $\mu$ and thermal conductivity $\lambda$ as functions of temperature and derived a temperature scaling using the semi-local coordinate. This allows the influence of property variations to be parameterised by the semi-local Reynolds number $\textit{Re}_{\tau }^*$ and Prandtl number $\textit{Pr}^*$ . The transformed mean scalar is expressed as

(1.10) \begin{equation} \overline {\theta }^* = \int _0^{\overline {\theta }_{\textit {VD}}^+} \left [1 + \frac {y}{\textit{Re}_\tau ^*} \frac {{\rm d} \textit{Re}_\tau ^*}{{\rm d}y}\right ] {\rm d}\overline {\theta }_{\textit{VD}}^+ = \int _0^{y^*}\! \dfrac {1}{\alpha _t / \overline {\mu } + 1 / \textit{Pr}^*} {\rm d}y^*, \end{equation}

where

(1.11) \begin{align} {\rm d}\overline {\theta }_{\textit {VD}}^+ = \sqrt {\dfrac {\overline {\rho }(y)}{\overline {\rho }_w}} \, d\!\left (\dfrac {\overline {\theta }}{\overline {\theta }_\tau }\right )\!, \textit{Re}_\tau ^* = \textit{Re}_\tau \sqrt {\dfrac {\overline {\rho }(y)}{\overline {\rho }_w}} \, \dfrac {\overline {\mu }_w}{\overline {\mu }(y)},\textit{Pr}^* = \textit{Pr}_w \, \dfrac {\overline {\mu }(y)/\overline {\mu }_w}{\overline {\lambda }(y)/\overline {\lambda }_w}. \end{align}

Here, $\bar {\theta }_{\textit {VD}}^+$ denotes the VD-type temperature, and $\alpha _t = -\bar {\rho } \, \widetilde {v'' \theta ''} / ({\rm d}\bar {\theta }/{{\rm d}y} )$ represents the turbulent heat conductivity. The transformed scalar $\bar {\theta }^*$ is independent of $\textit{Re}_\tau ^*$ while exhibiting a $\textit{Pr}^*$ -dependent shift. Chen et al. (Reference Chen, Huang, Shi, Yang and Lv2022) extended these transformations to channel and boundary-layer flows with varying wall conditions. Huang et al. (Reference Huang, Coleman, Spalart and Yang2023) simplified the expression by leveraging analogies with incompressible transport equations. More recently, Cheng & Fu (Reference Cheng and Fu2024) introduced a Mach-independent thermal scaling law at a given constant Prandtl number. In SCP wall-bounded flows, however, $c_{\!p}$ and $\lambda$ feature sharp temperature-dependent variations, leading to varying local Prandtl number in energy transport, which must be taken into account.

Establishing a reliable thermal scaling law at SCP is more challenging due to the strongly nonlinear variations of thermophysical properties near the pseudo-critical temperature $T_{pc}$ , which is defined as the temperature at which $c_{\!p}$ peaks (Jiang et al. Reference Jiang, Zhang, Xu and Shi2008; Yoo Reference Yoo2013). The interplay of these abnormal property variations and the near-wall turbulent dynamics poses challenges to the validity of existing thermal transformation rules developed at atmospheric pressure. Guo, Yang & Ihme (Reference Guo, Yang and Ihme2022) adopted Patel’s transformation at SCP and reported poor collapse of transformed mean scalar profiles, likely attributed to its insufficient treatment of near-wall property variations. To address this issue, Wan et al. (Reference Wan, Zhao, Liu, Wang and Lei2020) modified Patel’s transformation by incorporating variable  $\bar {c}_{\!p}$ into the semi-local Prandtl number ( $\textit{Pr}^* = \textit{Pr}_w \bar {\mu }(y)\bar {c}_{\!p}(y)\bar {\lambda }_w / (\bar {\mu }_w \overline {c_{\!p}}_w \bar {\lambda }(y))$ ) and the VD-type temperature gradient ( ${\rm d}\bar {\theta }_{\textit{VD}}^+/{\rm d}y = \sqrt {\bar {\rho }(y)/\bar {\rho }_w}\bar {c}_{\!p}(y)/\overline {c_{\!p}}_w d(\bar {\theta }/\bar {\theta }_\tau )/{\rm d}y$ ). While the modified transformation improved the collapse of the slopes in logarithmic region at small wall-temperature differences, it still performs unsatisfactorily in capturing the overall transformed temperature profiles. Toki, Teramoto & Okamoto (Reference Toki, Teramoto and Okamoto2020) proposed a VD-based temperature transformation that incorporates the variation of $\bar {c}_{\!p}$ at SCP and the mixing length hypothesis for energy transport with a linear heat flux, yielding

(1.12) \begin{equation} \bar {\theta }_{\textit {Toki}}^* = \int _0^{\bar {\theta }^+}\! \sqrt {\frac {\bar {\rho }(y)}{\bar {\rho }_w}} \frac {\tilde {c}_{\!p}(y)}{\bar {c_{\!p}}_w} \, {\rm d}\bar {\theta }^+. \end{equation}

Although (1.12) shows a slight improvement over (1.10), significant deviations still persist, especially near the heated wall where large property variations occur (Guo et al. Reference Guo, Yang and Ihme2022). Guo et al. (Reference Guo, Yang and Ihme2018) introduced the TL scaling for thermal transformation by considering a range of thermophysical property variations

(1.13) \begin{align} \bar {\theta }_{\textit {TL}}^+ = \int _0^{\bar {\theta }^+}\! \left ( \frac {\overline {\rho (y) c_{\!p}^2(y)}}{\bar {\rho }_w \overline {c_{\!p}}_w^2} \right )^{1/2} \left ( 1 + \frac {1}{2} \frac {y}{\overline {\rho (y) c_{\!p}^2(y)}} \frac {{\rm d}\overline {\rho (y) c_{\!p}^2(y)}}{{\rm d}y} - \frac {y}{\bar {\mu }(y)} \frac {{\rm d}\bar {\mu }(y)}{{\rm d}y} \right ) {\rm d}\bar {\theta }^+. \\[-18pt]\nonumber \end{align}

Despite the modification on property variations, the improvement on T-law was shown to be limited against the VD-based transformation (Guo et al. Reference Guo, Yang and Ihme2018). This is likely due to large gradients in $c_{\!p}$ , which cause large variations and inaccuracies in the associated terms of the TL-like derivation (Guo et al. Reference Guo, Yang and Ihme2018). Furthermore, we observe that the TL transformation in (1.13) did not account for the variations in local thermal conductivity, which could be significant in the vicinity of the pseudo-critical region. More recently, Guo et al. (Reference Guo, Yang and Ihme2022) proposed an improved T-law that incorporates the contribution of viscous dissipation to the total heat flux, leading to a reduced uncertainty – by approximately 18 % – in the slope of the transformed mean temperature profiles within the logarithmic region. In spite of numerous efforts on T-law, the transformed scalar profiles still fail to collapse effectively at variable semi-local Prandtl numbers at SCP.

The primary objective of this study is to establish a theoretical framework to characterise the T-law for wall-bounded channel flows at SCP. Unlike existing approaches using temperature transformation, we introduce a novel enthalpy transformation that considers the effects of property variations sufficiently. The use of enthalpy can effectively circumvent the involvement of the isobaric specific heat gradient term, which approaches infinity across the pseudo-critical temperature, thereby remedying the large uncertainty encountered in temperature transformation. For demonstration, DNSs of fully developed SCP channel flows at different wall-temperature differences and bulk Reynolds numbers are performed for detailed examination. The remainder of this paper is structured as follows: the mathematical foundation of the enthalpy transformation is presented in § 2; the computational set-up is briefed in § 3; the performance of the new scaling law is assessed and compared with those of existing H–V relations and temperature-based scaling laws in § 4; conclusions are provided in § 5.