2.1. Non-dimensional boundary-layer equations

The compressible boundary layers are studied in a Cartesian coordinate $\boldsymbol{x}^*=\left \{x^*,y^*\right \}$ that defines the streamwise and wall-normal directions, respectively. Hereafter, the asterisk denotes dimensional quantities. The velocities $(u^*,v^*)$ and temperature $T^*$ are normalised by ${u}_\infty ^*$ and ${T}_\infty ^*,$ with lengths scaled using the boundary-layer thickness $\delta ^*_o.$ The fluid properties, including the density $\rho ^*$ , the dynamic viscosity $\mu ^*$ and the thermal conductivity $\kappa ^*$ , are normalised by their respective free-stream values, $\rho _\infty ^*$ , $\mu ^*_\infty$ and $\kappa ^*_\infty$ . The pressure $p^*$ is normalised by $\rho ^*_\infty {u}^{*2}_\infty$ . The Reynolds number $\mathcal{R}$ is defined as $\mathcal{R}={\rho _\infty ^* {u}_\infty ^* \delta ^*_o} /{\mu _\infty ^*},$ and is taken to be asymptotically large. The oncoming flow is considered isentropic and air is assumed to be a perfect gas. The free-stream Mach number is defined as $\mathcal{M}_\infty ={u}_\infty ^*/a_\infty ^*=\mathcal{O}(1)$ , where $a_\infty ^*=(\gamma R^* {T}_\infty ^*)^{1/2}$ is the speed of sound in the free stream and $R^*=287.06$ $\textrm {J}$ $\textrm {kg}^{-1}$ ${\textrm {K}}^{-1}$ is the ideal gas constant for air. The plate is assumed to be infinitely thin so that shocks are weak and distant from the boundary layer. The effects of shocks on the boundary layer are hence neglected. The adiabatic wall temperature is estimated as $T_{ad}=1+(\gamma -1)\sqrt {\textit{Pr}}\mathcal{M}_{\infty }^2/2.$ It should be emphasised that the adiabatic wall temperature $T_{ad}$ depends on multiple physical parameters, including its sensitivity to viscosity laws; this dependence underscores that the simplified expression cannot serve as a universally valid prediction. Therefore, in this study, $T_{ad}$ is solely chosen as a reference scaling parameter to analyse the effects of wall temperature variations. Henceforth, all variables are non-dimensional.

In the boundary layers, the streamwise slow variable is introduced as $\bar x=x/\mathcal{R}.$ When $\bar x$ and $y$ are both of $\mathcal{O}(1)$ , the streamwise velocity has a magnitude greater than that of the normal by a factor of $\mathcal{O}(\mathcal{R}).$ Hence, the rescaled velocity and pressure fields are

(2.1) \begin{eqnarray} (U,V)=(u,\mathcal{R}^{-1}v), \quad P= p. \end{eqnarray}

Substitution of (2.1) and $\bar x$ into the Navier–Stokes equations gives, at leading order, the compressible boundary-layer equations:

(2.2) \begin{equation} \left .\begin{array}{c} \dfrac {\partial (\rho U)}{\partial \bar x} + \dfrac {\partial (\rho V)}{\partial y} = 0, \\ \rho U \dfrac {\partial U}{\partial \bar x} + \rho V \dfrac {\partial U}{\partial y} = \dfrac {\partial }{\partial y} \left ( \mu \dfrac {\partial U}{\partial y} \right )\!, \\ \rho U \dfrac {\partial T}{\partial \bar x} + \rho V \dfrac {\partial T}{\partial y} = \dfrac {1}{\textit{Pr}} \dfrac {\partial }{\partial y} \left ( \kappa \dfrac {\partial T}{\partial y} \right ) + (\gamma - 1) \mathcal{M}_\infty ^2 \mu \left ( \dfrac {\partial U}{\partial y} \right )^2. \end{array} \right \} \end{equation}

The streamwise pressure gradient is neglected in the present study. Similarity solutions are obtained as $ \{U,V,T\}=\big \{F'(\eta ), T(\eta _c F'-F)/\sqrt {2\bar x},T(\eta )\big \},$ where $\eta =\sqrt {{\mathcal{R}}/{2 x}}\int _0^y1/T(\bar x,\hat y) \textrm {d}{\hat y},$ $\eta _c= (1/T)\int _0^{\eta } T(\hat \eta )\textrm {d}{\hat \eta }$ and the prime denotes differentiation with respect to the similarity variable $\eta$ . The compressible Blasius functions $F(\eta )$ and $T(\eta )$ are solutions to the boundary-value problem:

(2.3) \begin{equation} \left .\begin{array}{c} \big(\mu F^{\prime\prime}/T\big)'+{\textit{FF}}^{\prime\prime} =0,\\ \big( \mu T'/T \big)'+{\textit{Pr}}{\textit{FT}}'+ \mu (\gamma -1){\textit{Pr}} \mathcal{M}^2_\infty (F^{\prime\prime})^2/T=0,\\ F=F'=0, \quad T=T_w, \quad \textrm {at } \quad \eta =0;\\ F'\rightarrow 1, \quad T \rightarrow 1,\quad \textrm {as } \quad \eta \rightarrow \infty , \end{array}\right \} \end{equation}

where ${\textit{Pr}}$ is assumed constant and the scaled thermal conductivity is $\kappa =\mu$ . Three distinct viscosity models are examined: the Chapman law, proven convenient for supersonic boundary-layer analysis (Stewartson Reference Stewartson1964; Zhu & Wu Reference Zhu and Wu2022); the power law, providing empirical accuracy at medium Mach numbers; and the Sutherland law, which delivers reliable viscosity predictions at low to high Mach numbers. Equations (2.3) serve as the fundamental governing equations for analysing the TV relation in the present study. These equations were previously employed to provide the base-flow solution for receptivity and stability analyses of compressible boundary layers (Xu, Ricco & Marensi Reference Xu, Ricco and Marensi2024b ). The numerical methodology for solving these equations follows the established procedures used in Xu, Ricco & Duan (Reference Xu, Ricco and Duan2023) and Xu, Ricco & Duan (Reference Xu, Ricco and Duan2024b ). The present framework, while formulated under a high-Reynolds-number assumption, remains valid for finite-Reynolds-number flows as (2.2) can be alternatively derived through empirical scaling arguments (Anderson Reference Anderson2006; White & Majdalani Reference White and Majdalani2006). However, for flows $\mathcal{R}\sim \mathcal{O}(1)$ or $\mathcal{R}\ll 1$ , the framework would require modification. In practice, these limitations are inconsequential for hypersonic applications, where $\mathcal{R}\gg 1$ usually holds (Schneider Reference Schneider1999; Xu et al. Reference Xu, Ricco and Marensi2024a ). For a detailed discussion of the differences between finite- and high-Reynolds-number approaches to flow receptivity and instability, we refer the interested reader to Wu (Reference Wu2019) and Dong, Liu & Wu (Reference Dong, Liu and Wu2020).

2.2. Temperature–velocity relation

The work of VO revised the CB relation by introducing an additional term to account for non-unity Prandtl number effects, yielding a successful TV relation for low-Mach-number boundary layers. Their asymptotic expansion relies on two small parameters: $\textit{Pr} - 1$ and  $\epsilon _M$ . However, this approach is not designed for hypersonic flows; e.g. $\epsilon _M$ reaches 5 at $\mathcal{M}_\infty = 5$ and $\gamma = 1.4$ , violating the small- $\epsilon _M$ assumption. While we retain the assumption that $\textit{Pr}-1$ is small (which holds reasonably well for air), the second small-parameter assumption involving $\epsilon _M$ is relaxed in the present work. Meanwhile, the method of VO also needs to solve a differential equation to obtain the first-order solution. We aim to derive a TV relation that eliminates the need for solving any differential equations.

For convenience, a new variable is defined as

(2.4) \begin{eqnarray} \theta (\eta )=\dfrac {1}{s}(T-T_w), \end{eqnarray}

with $s=1-T_w.$ This definition maps the temperature from $[T_w, 1]$ to $\theta$ at the unit interval $[0, 1],$ which facilitates a direct comparison between $\theta$ and velocity. To establish the relationship between temperature and velocity, we solve the temperature equation provided in (2.3). With the definition $\sigma =(\gamma -1)\mathcal{M}^2_\infty /s,$ (2.3) is reformulated as

(2.5) \begin{equation} \left .\begin{array}{c} \big( \mu \theta '/T \big)' +PrF\theta ' +Pr\mu \sigma (F^{\prime\prime})^2/T =0,\\ \theta =0 \quad \textrm {at } \quad \eta =0; \quad \theta \rightarrow 1 \quad \textrm {as} \quad \eta \rightarrow \infty . \end{array}\right \} \end{equation}

To study the effect of Prandtl number, the temperature is expressed as an asymptotic series:

(2.6) \begin{eqnarray} \theta (\eta ) = \theta _0(\eta ) + \epsilon \theta _1(\eta ) + \epsilon ^2 \theta _2(\eta ) + \cdots , \end{eqnarray}

where the small parameter is defined as $\epsilon ={\textit{Pr}}- 1$ . The current study adopts the $\epsilon$ expansion from the work of VO, which was originally derived under the assumption of $\textit{Pr} \approx 1$ . This asymptotic expansion remains mathematically valid for laminar hypersonic boundary layers. In practice, atmospheric air under hypersonic conditions exhibits this behaviour, with ${\textit{Pr}}$ generally staying close to 1 (Anderson Reference Anderson2006; CGF; Zhao & Fu Reference Zhao and Fu2025). Substitution of series (2.6) into (2.5) and collecting equal powers of $\epsilon$ yield the following leading-order equation:

(2.7) \begin{equation} \left .\begin{array}{c} \big( \mu \theta ^{\prime}_0/T \big)' +F\theta ^{\prime}_0 +\mu \sigma (F^{\prime\prime})^2/T =0,\\ \theta _0=0 \quad \textrm {at} \quad \eta =0; \quad \theta _0 \rightarrow 1\quad \textrm {as} \quad \eta \rightarrow \infty , \end{array}\right \} \end{equation}

the equation at $O(\epsilon )$ :

(2.8) \begin{equation} \left .\begin{array}{c} \big( \mu \theta ^{\prime}_1/T \big)' +F\theta ^{\prime}_1 =\big( \mu \theta ^{\prime}_0/T \big)',\\ \theta _1=0 \quad \textrm {at} \quad \eta =0; \quad \theta _1 \rightarrow 0\quad \textrm {as} \quad \eta \rightarrow \infty , \end{array}\right \} \end{equation}

and the equation at $O(\epsilon ^2)$ :

(2.9) \begin{equation} \left .\begin{array}{c} \big( \mu \theta ^{\prime}_2/T \big)' +F\theta ^{\prime}_2 =\big( \mu \theta ^{\prime}_{1}/T \big)'-\big( \mu \theta ^{\prime}_{0}/T \big)',\\ \theta _n=0 \quad \textrm {at} \quad \eta =0; \quad \theta _2 \rightarrow 0 \quad \textrm {as} \quad \eta \rightarrow \infty . \end{array}\right \} \end{equation}

While the Mach number explicitly appears only in the leading-order equations, compressibility effects still influence the first- and second-order terms. This influence occurs indirectly through the forcing terms on the right-hand side of (2.8) and (2.9).

The series (2.6) differs from the expansions (4.12)–(4.13) in VO, where both $ F$ and $ T$ are expanded as asymptotic series. In (2.4), our approach retains $ T$ in the denominators and all $ F$ , which are seen as known functions. Only $\theta$ is expanded in (2.6) to study the effects of non-unity Prandtl number on $\theta .$ This approach is motivated by three key considerations. First, our goal is to reconstruct $\theta$ as an explicit function of the experimentally measured velocity $U$ , where $ U = F'$ is the only input. An asymptotic expansion on $ F$ yields a TV relation that would require solving differential equations, as in VO. Another limitation of the asymptotic expansion for $ F$ is its reliance on the assumption that $\epsilon _M\ll 1$ for compressible flows. This limitation restricts the applicability of the VO method for hypersonic flows, whereas our approach overcomes this constraint. Second, analytical solutions are attainable for the zeroth-order equation (2.7), which naturally incorporates both viscosity variations and compressibility effects. The latter is neglected in the zeroth-order equation of VO. Third, $ T$ serves solely as an auxiliary variable to facilitate the derivation of an explicit expression for $ \theta$ ; while required in the derivation process, it can ultimately be replaced by the leading-order approximation.

The solution of the temperature equation for incompressible boundary layers can be decomposed into two distinct components (Schlichting & Gersten Reference Schlichting and Gersten2016). We find that the solution of (2.7) similarly admits a two-part decomposition $\theta _0=\theta _{00}+\theta _{01},$ where $\theta _{00}$ and $\theta _{01}$ satisfy the equations

(2.10) \begin{equation} \left .\begin{array}{c} \big( \mu \theta ^{\prime}_{00}/T \big)' +F\theta ^{\prime}_{00} =0,\\ \theta _{00}=0 \quad \textrm {at} \quad \eta =0; \quad \theta _{00} \rightarrow 1 \quad \textrm {as} \quad \eta \rightarrow \infty \end{array}\right \} \end{equation}

and

(2.11) \begin{equation} \left .\begin{array}{c} \big( \mu \theta ^{\prime}_{01}/T \big)' +F\theta ^{\prime}_{01} +\mu \sigma (F^{\prime\prime})^2/T =0,\\ \theta _{01}=0 \quad \textrm {at} \quad \eta =0; \quad \theta _{01} \rightarrow 0 \quad \textrm {as} \quad \eta \rightarrow \infty . \end{array}\right \} \end{equation}

Equation (2.10) is formally identical to the velocity equation in (2.3), and its solution is

(2.12) \begin{eqnarray} \theta _{00}=F'. \end{eqnarray}

Note that $\theta _{00}$ is solely the exact solution of (2.5) when $\mathcal{M}_\infty =0$ and $\textit{Pr}=1.$ The exact solution is obtained by the method of variation of parameters:

(2.13) \begin{eqnarray} \theta _{01}=-\sigma \dfrac {F'(F'-1)}{2}. \end{eqnarray}

The detailed derivation is shown in Appendix A. Obviously, the solution $\theta _{01}$ arises from the boundary-layer compressibility effect, vanishing in the incompressible flow limit ( $\mathcal{M}_\infty \to 0$ ). The solutions (2.12) and (2.13) both accommodate any viscosity laws.

The solution $\theta _0,$ which inherently accounts for both viscosity variations and compressibility effects, is the exact solution of (2.5) at $\textit{Pr}=1.$ Indeed, the present first-order solution $T=s\theta _0+T_w$ is identical to the CB relation (1.1). Meanwhile, when $\textit{Pr}\ne 1,$ $\theta _0,$ the CB relation, serves as the zeroth-order solution to (2.5). As a result, the CB relation is doomed to predict temperature inaccurately due to the non-unity Prandtl number.

For the first-order equation (2.8), the exact solution for boundary layers with an arbitrary viscosity law can be derived using the method of variation of parameters. The first-order solution is found as

(2.14) \begin{eqnarray} \theta _{1} =\theta _{10}+\theta _{11}, \end{eqnarray}

where

(2.15) \begin{eqnarray} \theta _{10} = -F^{\prime }\int _{0}^{\eta }\dfrac { T}{\mu }F\, \textrm {d}\eta +\dfrac {1}{2}\dfrac { T}{\mu }F^{2} -\dfrac {1}{2}\int _{0}^{\eta }\left (\dfrac { T}{\mu }\right )^{\prime }F^{2}\, \textrm {d}\eta -a_0 F^{\prime } \end{eqnarray}

and

(2.16) \begin{align} \theta _{11} & = \dfrac {\sigma }{2} \theta _{10} +\dfrac {\sigma }{2}\left [-{\textit{FF}}^{\prime \prime }-\dfrac {1}{2}F^{\prime 2} + \dfrac {1}{2}F' +\int _{0}^{\eta }\left (\dfrac { T}{\mu }\right )^{\prime }\Bigl {(}F^{\prime }F^{2} +\dfrac {\mu }{ T}F^{\prime \prime }F\Bigr {)} \textrm {d}\eta \right . \nonumber \\ &\quad \displaystyle - \left . F^{\prime }\int _{0}^{\eta }\left (\dfrac { T}{\mu }\right )^{\prime }F^{2} \textrm {d}\eta \right ] -\dfrac {\sigma }{2} b_0 F'. \end{align}

The constants $a_0$ and $b_0$ are defined as

(2.17) \begin{eqnarray} a_0&=& -\int _{0}^{\infty }\left (\dfrac { T}{\mu }F- \eta - c_{1}\right )\textrm {d} \eta +\dfrac {1}{2}c_{1}^{2}-\dfrac {1}{2}\int _{0}^{\infty }\left (\dfrac { T}{\mu }\right )^{\prime }F^{2} \textrm {d}\eta \end{eqnarray}

and

(2.18) \begin{eqnarray} b_0&=&\int _{0}^{\infty }\left (\dfrac { T}{\mu }\right )^{\prime }\Bigl {(}F^{\prime }F^{2} +\dfrac {\mu }{ T}F^{\prime \prime }F-F^{2}\Bigr {)} \textrm {d}\eta , \end{eqnarray}

with $c_{1} = \int _{0}^{\infty }(U - 1) \textrm {d}\eta$ . The detailed derivation is provided in Appendix A.

For the boundary layers with the Chapman law, one can find a simplified solution for $\theta _{10}$ and $\theta _{11}$ as

(2.19) \begin{align} \displaystyle \theta _{10}&=-F' \int _0^\eta F \,\textrm {d} \eta +\dfrac {1}{2}F^2-a_0F' \end{align}

(2.20) \begin{align} \displaystyle \textrm {and} \quad \theta _{11}&=\dfrac {\sigma }{2}\left [\theta _{10}+\dfrac {1}{2}\left (1-F^{\prime 2}\right )-{\textit{FF}}^{\prime\prime}-\dfrac {1-F'}{2}\right ]\!, \\[9pt] \nonumber \end{align}

where the constant $a_0$ simplifies to $a_0 = - \int _0^\infty (F - \eta - c_1) \textrm {d}\eta + (1/2) c_1^2$ and the constant $b_0$ vanishes identically. Here $\theta _{10}$ and $\theta _{11}$ are the first-order corrections of $\theta _{00}$ and $\theta _{01},$ respectively. Second-order solutions can be obtained by solving (2.9) with the method of variation of parameters. In the present paper, we only focus on the first-order solution as it is accurate enough to predict the temperature from the velocity. Thus, the final first-order approximate solution is

(2.21) \begin{eqnarray} \theta = \theta _{00}+\theta _{01}+\epsilon (\theta _{10}+{\theta _{11}}). \end{eqnarray}

The solution (2.21) is expected to have sufficient accuracy for temperature prediction, with detailed error analysis to be presented in the following section. This expected accuracy originates from two principal aspects: (i) the zeroth-order solution $\theta _0$ fully captures the dominant viscosity variation and compressibility effects and (ii) the first-order correction $\theta _1$ provides an exact solution to address non-unity Prandtl number effects. The TV relation, developed from the generalised Reynolds analogy (Zhang et al. Reference Zhang, Bi, Hussain and She2014; MG; CGF), aims to establish a universal reciprocal of effective Prandtl number. While this goal could be achievable, it remains an open question that warrants further investigation. A limitation of the present TV relation is that it is currently confined to laminar boundary layers; its applicability to laminar channel flows, three-dimensional boundary layers and turbulent flows remains unexplored.

We also observe that $\theta _0$ , the CB relation (1.1), reduces to solution (2.12) in VO when $\textit{Pr}=1,$ $\mathcal{M}_\infty =0$ and the wall is adiabatic. Consequently, the first-order solution $\theta _1$ also reduces to solution (2.14) in VO as (2.8) simplifies to their (2.13). As a rational TV relation, the solution (2.21) provides a useful equation for predicting the temperature profile. Meanwhile, it can serve as a valuable tool for analysing the underlying mechanism of temperature increase at the wall or in the near-wall region at different wall temperatures, Mach numbers, heat capacity ratios and Prandtl numbers. As demonstrated by VO, (2.21) can also be applied to analyse wall quantities without requiring significant additional effort. The relation between the skin-friction coefficient and the wall-heat flux, the recovery temperature and the recovery factor are given in Appendix B. The parameter analysis is beyond the scope of the present study.

Note that the similarity variable $\eta$ is tied to the temperature, making it impossible to reconstruct the natural coordinate $y$ using the measured velocity. However, we notice that the zeroth-order solution $T_0=s\theta _0+T_w$ provides a reasonable approximation to $T$ to reconstruct $y$ from $\eta$ . In the form of $y,$ the first-order corrections read

(2.22) \begin{align} \theta _{10}(y;\mathcal{R}) & = -\dfrac {\mathcal{R}U}{2 x} \int _0^y \dfrac {1}{\mu _0} \int _0^y \dfrac {U}{T_0} \textrm {d} y\, \textrm {d} y +\dfrac {\mathcal{R}}{4 x}\dfrac { T_0}{\mu _0}\left (\int _0^y \dfrac {U}{T_0} \textrm {d} y\right )^2 \nonumber \\ &\quad -\dfrac {\mathcal{R}}{4 x}\int _{0}^{y}\left (\dfrac { T_0}{\mu _0}\right )_y\left (\int _0^y \dfrac {U}{T_0} \textrm {d} y\right )^2 \textrm {d}y -a_1 U \end{align}

and

(2.23) \begin{align} \theta _{11}(y;\mathcal{R}) &= \dfrac {\sigma }{2} \theta _{10} +\dfrac {\sigma }{2}\Bigg {\{}-T_0 U_y\int _0^y \dfrac {U}{T_0} \textrm {d} y-\dfrac {1}{2}U^2 + \dfrac {1}{2}U \nonumber \\ &\quad +\left .\int _{0}^{y}\left (\dfrac { T_0}{\mu _0}\right )_y\left [\dfrac {\mathcal{R}U}{2 x} \left (\int _0^y \dfrac {U}{T_0} \textrm {d} y\right )^2 +\mu _0 U_y \int _0^y \dfrac {U}{T_0} \textrm {d} y \right ] \textrm {d}y \right . \nonumber \\ &\quad \displaystyle - \dfrac {\mathcal{R}U}{2 x} \int _{0}^y\left (\dfrac { T_0}{\mu _0}\right )_y\left (\int _0^y \dfrac {U}{T_0} \textrm {d} y \right )^{2} \textrm {d} y \Bigg {\}} -\dfrac {\sigma }{2}b_1 U .\end{align}

The unknown functions $ T$ and $ \mu$ in (2.15) and (2.16) are replaced by their first-order approximations $ T_0$ and $ \mu _0$ in (2.22) and (2.23), respectively. The constants $a_1$ and $b_1$ are defined as

(2.24) \begin{align} a_1&= -\dfrac {\mathcal{R}}{2 x} \int _0^\infty \dfrac {1}{\mu _0} \int _0^y \dfrac {U}{T_0} \textrm {d} y\, \textrm {d} y +\dfrac {\mathcal{R}}{4 x}\left (\int _0^\infty \dfrac {U}{T_0} \textrm {d} y\right )^2 \nonumber \\ &\quad -\dfrac {\mathcal{R}}{4 x}\int _{0}^{\infty }\left (\dfrac { T_0}{\mu _0}\right )_y\left (\int _0^y \dfrac {U}{T_0} \textrm {d} y\right )^2 \textrm {d}y \end{align}

and

(2.25) \begin{align} b_1=\int _{0}^{\infty }\left (\dfrac { T_0}{\mu _0}\right )_y\left [\dfrac {\mathcal{R}U}{2 x} \left (\int _0^y \dfrac {U}{T_0} \textrm {d} y \right )^{2} +\mu _0 U_y\left (\int _0^y \dfrac {U}{T_0} \textrm {d} y \right )- \dfrac {\mathcal{R}}{2 x} \left (\int _0^y \dfrac {U}{T_0} \textrm {d} y \right )^{2} \right ] \textrm {d} y. \end{align}

For the Chapman law, the first-order corrections read

(2.26) \begin{align} \displaystyle \theta _{10}(y;\mathcal{R}) &= \bigg [-\dfrac {\mathcal{R}U}{2 x} \int _0^y \dfrac {1}{T_0} \int _0^y \dfrac {U}{T_0} \textrm {d} y\, \textrm {d} y +\dfrac {\mathcal{R}}{4 x}\left (\int _0^y \dfrac {U}{T_0} \textrm {d} y\right )^2 -a_1U\bigg ], \end{align}

(2.27) \begin{align} \displaystyle \theta _{11}(y;\mathcal{R}) &=\dfrac {\sigma }{2}\left [\theta _{10}+\dfrac {1}{2}\left (1-U^2 \right )-T_0{U_y}\int _0^y\dfrac {U}{T_0} \textrm {d}y-\dfrac {1-U}{2}\right ]\!, \end{align}

with the constant $a_1=\lim \limits _{y_0\to \infty } [ -({\mathcal{R}}/{2 x}) \int _0^{y_0} ({1}/{T_0}) \int _0^{y} ({U}/{T_0}) \textrm {d} y \textrm {d} y + ({\mathcal{R}}/{4 x}) (\int _0^{y_0} ({U}/ {T_0}) {} \textrm {d} y )^2].$ The remaining analysis employs (2.12), (2.13), (2.22) and (2.23) without explicit restatement. The temperature field reconstruction requires the velocity profile in $y$ , along with the parameters $\gamma$ , ${\textit{Pr}}$ , $\mathcal{M}_\infty$ and $\mathcal{R}$ as inputs. The Reynolds number in (2.22) and (2.23) governs the determination of $y$ . Indeed, all terms in (2.21) are Reynolds-number-independent, as shown in (2.12)–(2.13) and (2.15)–(2.16). The observed Reynolds-number independence arises because the flat-plate boundary layer admits self-similar solutions that are inherently independent of the Reynolds number.

Regarding the solution for $\theta _1$ (2.15)–(2.16) and the resulting new TV relation, several key points should be noted. First, substituting $T$ with the zero-order relation in the evaluation does not compromise the overall accuracy, which remains $O(\epsilon )$ . Second, unlike the zero-order solution $\theta _0$ , the first-order solution $\theta _1$ is algebraic (requiring no differential equation solution) but non-local – meaning $\theta _1(y)$ is not directly tied to $U(y)$ at the same point. As a result, its evaluation necessitates numerical quadrature. Third, the non-locality suggests that pointwise empirical relations are likely inadequate for this scenario. This non-locality may explain the Mach-number-dependent variation of prediction errors in the empirical relation of CGF.

It is necessary here to make a few remarks concerning the present relation and the TV relations of MG and CGF. The present relation (2.21) offers three key advantages. First, unlike the methods of MG and CGF, which require the wall-heat flux as an a priori input to determine the temperature profile, our approach treats it as a concomitant result. Second, it accommodates three distinct viscosity laws: Chapman, power and Sutherland laws, while the relations of MG and CGF work for a single viscosity law. Third, while MG and CGF restrict their analyses to a fixed heat capacity ratio and Prandtl number, our framework allows investigation of arbitrary heat capacity ratios and variable Prandtl numbers close to unity.