Temperature–velocity relation for laminar adiabatic and diabatic hypersonic boundary layers

Abstract

We study the temperature–velocity (TV) relation for laminar adiabatic and diabatic hypersonic boundary layers. By applying an asymptotic expansion to the compressible boundary-layer temperature equation, we derive a first-order equation for the TV relation, where the zeroth-order solution is found to be the classical Crocco–Busemann quadratic relation. The ensuing relation predicts accurately the temperature profile by using the velocity for hypersonic boundary layers with Chapman, power and Sutherland viscosity laws, arbitrary heat capacity ratios, variable Prandtl numbers close to unity and Mach number of up to 10. The Mach-number- and wall-temperature-independent quantities in our relation are also investigated. The present relation has the potential to function as a temperature wall model for laminar hypersonic boundary layers, especially for cold-wall cases.

1. Introduction

The compressible velocity transformation and temperature–velocity (TV) relation have emerged as a prominent focus in hypersonic boundary-layer studies due to scientific interest and engineering applications (Griffin, Fu & Moin 2021; Cheng et al. 2024; Chen, Gan & Fu 2025). While both relations pose theoretical challenges, the TV relation is more complex due to the thermal–kinetic energy conversion and thermodynamic effects, especially under wall-cooling conditions. In this paper, we seek to develop an analytical TV relation for laminar adiabatic and diabatic hypersonic boundary layers to accurately predict the temperature profile.

High velocity and temperature gradients near the wall in hypersonic boundary layers raise difficulties for numerical predictions of skin friction and wall-heat flux (Hoffmann, Siddiqui & Chiang 1991; Mo & Gao 2024). Accurate predictions demand refined near-wall grid resolution, leading to extra computational costs. This grid sensitivity issue escalates for complex three-dimensional flows. Meanwhile, experimental measurements face parallel challenges, as direct measurement of the temperature profile (hence wall-heat flux) proves exceptionally difficult in hypersonic flows (Byrne, Danehy & Houwing 2003; Smotzer et al. 2024). A TV relation offers an alternative solution by providing an accurate temperature field from computed or measured velocity, while also offering an interpretation for the temperature increase/decrease from a viewpoint of velocity.

Busemann (1931) and Crocco (1932) first introduced a quadratic function of the streamwise velocity for compressible boundary layers, known as the Crocco–Busemann (CB) relation. The temperature $ T$ is relevant to the streamwise velocity $ u$ through

(1.1) \begin{equation} {T} = {T}_w + ({T}_r - {T}_w) {u} + (1 - {T}_{r}){{u}}^2, \end{equation}

where the temperature and velocity are normalised by the free-stream temperature and velocity, respectively. The symbol $ T_w$ represents the wall temperature. Hereafter, the subscript $ w$ denotes the value at the wall. The reference temperature is defined as $ {T}_r = 1 + 1/(2c_p)$ , where $c_p$ is the non-dimensional isobaric heat capacity. Relation (1.1) is valid for arbitrary viscosity laws. However, the CB relation, being based on the assumption of the Prandtl number $\textit{Pr}=1$ , does not provide an accurate temperature prediction for air, as the Prandtl number of air deviates from unity (van Driest 1952).

Van Driest (1952) extended the CB relation to account for the effects of non-unity Prandtl numbers. They showed numerically that the recovery temperature is given by ${T}_{r} = 1 + r /(2c_p),$ where the recovery factor $r\approx \textit{Pr}^{1/2}$ is a constant. Mo & Gao (2024) (MG) analysed the generalised recovery factor $r_g$ , revealing it as a constant yet smaller than $\textit{Pr}^{1/2}$ in severely cold-wall conditions. Chen et al. (2025) (CGF) further conducted a detailed investigation of $r_g$ across a broader range of parameters – including varying Mach numbers and wall temperatures – and revealed that $r_g$ is not a constant but instead varies along the wall-normal direction. By identifying a universal distribution of the effective Prandtl number $\textit{Pr}_e,$ originally defined by Zhang et al. (2014), their work culminated in an empirical TV relation. Focusing on wall-bounded turbulent flows, Zhang et al. (2014) developed an effective TV relation (Cheng & Fu 2024; CGF). Their approach additionally provides theoretical support for the empirical relation proposed by Duan & Martin (2011) in turbulent boundary layers.

However, the TV relations of MG and CGF rely on the simulation results as they are inherently constrained by the specification of viscosity law, heat capacity ratio $\gamma$ and Prandtl number ${\textit{Pr}}$ . Hence, the applicability of their TV relations is restricted to certain simulation datasets or experimental conditions that match the set-up used in MG and CGF. In practice, different numerical implementations employ varying viscosity laws (e.g. power law in MG and Sutherland law in CGF). Additionally, $\gamma$ and ${\textit{Pr}}$ vary across flight environments and wind tunnel conditions (Anderson 2006). Typically, $\gamma$ ranges from $1.29$ to $1.67$ , while ${\textit{Pr}}$ varies between 0.7 and 0.8, potentially reaching higher values under ultralow-temperature conditions. A generalised TV relation that transcends such restrictions is required to address these limitations. Based on the assumption of an adiabatic wall and a small parameter $\epsilon _M=(\gamma -1)\mathcal{M}_\infty ^2/2,$ where $\mathcal{M}_\infty$ is the free-stream Mach number, van Oudheusden (1997) (VO) proposed a rational TV relation for low-Mach-number boundary layers. However, these assumptions restrict its applications to hypersonic boundary layers, where $\epsilon _M\gg 1$ and the wall is always cold (Choudhari et al. 2009).

In this study, we derive an analytical TV relation that eliminates the constraints of MG and CGF by avoiding numerical data dependence while generalising the framework of VO through relaxation of assumptions of the adiabatic wall and small parameter $\epsilon _M.$ The proposed relation is aimed to be valid for hypersonic boundary layers with Chapman, power and Sutherland viscosity laws, while accommodating arbitrary heat capacity ratios and variable Prandtl numbers close to unity, multiple Mach numbers ( $0 \leqslant \mathcal{M}_\infty \leqslant 10$ ) and both adiabatic and diabatic wall conditions. The rest of the paper is organised as follows. In § 2, we describe the formulation of the problem and propose a new TV relation. In § 3, the numerical results of the new TV relation are presented. The Mach-number- or wall-temperature-independent quantities are also investigated. A conclusion is given in § 4.

2. Problem formulation

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 1964; Zhu & Wu 2022); 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 2024b ). The numerical methodology for solving these equations follows the established procedures used in Xu, Ricco & Duan (2023) and Xu, Ricco & Duan (2024b ). 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 2006; White & Majdalani 2006). 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 1999; Xu et al. 2024a ). 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 (2019) and Dong, Liu & Wu (2020).

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 2006; CGF; Zhao & Fu 2025). 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 2016). 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. 2014; 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.

3. Numerical results

In this section, we first validate the proposed relation by systematically comparing its predictions with the numerical results of the coupled boundary-value problem (2.3). The prediction errors are also investigated. Then we show the contribution of each term in temperature decompositions to analyse the Mach-number- and wall-temperature-independent quantities. While CGF illustrates the TV relation in a $T$ – $V$ contour, we present the results in a $T$ – $y$ contour to better emphasise parametric effects on the temperature profile. Fundamentally, the two representations are interchangeable since $y$ and velocity are both known quantities. To measure the prediction accuracy, the relative error is defined as $ \mathcal{E}=\int _0^{U} \left |1-T_{p}/{T_c} \right | \textrm {d} U,$ where $T_{p}$ and $T_{c}$ are the predicted temperature and the numerical solution of (2.3), respectively. The Reynolds number is chosen as $\mathcal{R}= 2\times 10^{5}.$ We adopt the same assumption as in VO, treating $\epsilon$ as a small parameter. The efficacy of this assumption, as demonstrated in VO for low-Mach-number cases, suggests it is well founded for ideal gases at $\textit{Pr}=0.71.$ To quantify the validity of this assumption in our approach, we examine how the prediction error $\mathcal{E}$ depends on ${\textit{Pr}}$ in Appendix C. The results clearly demonstrate that the present first-order solution achieves sufficient accuracy in capturing the temperature profile of the studied ideal gas resulting from non-unity Prandtl number effects.

Figure 1. Comparison of the present TV relation (2.21) (symbols) with the numerical results (lines): ( $a{-}c$ ) different viscosity laws at $\mathcal{M}_\infty =6.0$ ; ( $d{-}f$ ) different Mach numbers with the Sutherland law; ( $g{-}i$ ) different heat capacity ratios at $\mathcal{M}_\infty =8.0$ with the Sutherland law; ( $j{-}l$ ) different Prandtl numbers at $\mathcal{M}_\infty =8.0$ with the Sutherland law.

Figure 2. Prediction errors of the present TV relation (2.21) (symbols) and the CB relation (solid lines with symbols). ( $a$ ) Different viscosity laws at $\mathcal{M}_\infty =6.0.$ The dashed lines with symbols denote the prediction errors of the Duan & Martin (2011) (DM) relation. ( $b$ ) Different Mach numbers with the Sutherland law. The open diamonds and right triangles denote the prediction errors of the CGF relation at $T_w=1.00T_{ad}$ and at $T_w=0.20T_{ad}$ , respectively. ( $c$ ) Different heat capacity ratios at $\mathcal{M}_\infty =8.0$ with the Sutherland law. ( $d$ ) Different Prandtl numbers at $\mathcal{M}_\infty =8.0$ with the Sutherland law.

Figure 1( $a$ – $c$ ) compares the results from three viscosity laws (Chapman, power and Sutherland) under five wall temperature conditions at $\mathcal{M}_\infty = 6.0$ . Unless otherwise stated, we use $\gamma = 1.4$ and $\textit{Pr} = 0.71$ as default values. The predicted temperature profiles for all viscosity laws agree well with the computational results, which indicates that (2.21) removes the Sutherland law restriction in the approach of CGF. Figure 1( $d$ – $f$ ) displays the predicted temperature profiles across Mach numbers ranging from 0 to 10 for three wall temperature conditions: heated ( $T_w=1.25T_{ad}$ ), adiabatic ( $T_w=1.00T_{ad}$ ) and cold ( $T_w=0.50T_{ad}$ ) walls. In figure 1( $e$ ), at $\mathcal{M}_{\infty }=0,$ the wall temperature is adopted as $T_w=0.98$ to avoid a singularity in (2.4). The present relation accurately captures the profiles across all studied Mach numbers and wall temperatures.

The heat capacity ratio $\gamma$ plays a pivotal role in boundary-layer dynamics (Byrne et al. 2003). At high-enthalpy conditions, thermal and chemical energy excitation alters $\gamma$ in equilibrium and non-equilibrium states. They modify $\gamma$ , causing it to vary across the entire boundary layer rather than remaining constant. Vibrational excitation tends to decrease $\gamma$ , while chemical dissociation typically increases it (Anderson 2006). Moreover, when the incoming gas is CO $_{2}$ instead of air, the specific heat ratio $\gamma$ also exhibits a variation due to its thermodynamic properties (Anyoji et al. 2015; Ren et al. 2025). This effect substantially modifies both the thermal and velocity profiles in the boundary layer. A fixed value of $\gamma = 1.3$ is commonly adopted for CO $_{2}$ flows to take the thermodynamic properties into account, as shown in Anyoji et al. (2015). Nevertheless, the present TV relation accommodates the adjustment of $\gamma$ as a constant, meaning it does not depend on the flow field. Capturing global variations in $\gamma$ requires simulations that account for more complicated gas effects (Candler 2019; Ren et al. 2025).

Figure 1( $g$ – $i$ ) compares our TV relation across different heat capacity ratios under the Sutherland law. The Mach number is increased to 8, where heat capacity effects become significant (Byrne et al. 2003). The predictions compare well with those of computations. It is also observed that the temperature profile is sensitive to $\gamma$ , highlighting the importance of the TV relation in accounting for variations in $\gamma$ . The variation of the Prandtl number is studied in figure 1( $j$ – $l$ ). Three different Prandtl numbers are chosen as $0.8,$ $0.9$ and $1.2,$ as the case at $\textit{Pr}=0.71$ is shown in figure 2( $h$ ). Again, the predictions compare well with those of computations.

Figure 2 compares the prediction errors between the present TV and CB relations. Figure 2( $a$ ) shows that all cases yield small prediction errors ( $\mathcal{E}\lt 2\,\%$ ), as expected, since $\theta _1$ is an exact solution to (2.8) for any viscosity laws. The errors for both power and Sutherland laws are comparable to those of the Chapman law. The CB relation exhibits significant prediction inaccuracies, particularly for cold-wall cases. The prediction errors from Duan & Martin (2011) are also included to assess performance in laminar flows, showing smaller errors than the CB relation in severely cold-wall cases. Figure 2( $b$ ) compares the prediction errors between the present TV and CB relations at different Mach numbers. The prediction errors of CGF are also included for comparison at the selected Mach numbers. The prediction errors of the present TV and the CGF relations are both smaller than $3\,\%.$ The prediction errors of CGF exhibit a Mach-number dependence, whereas our method maintains nearly constant prediction errors across different Mach numbers. The CB relation is effective for low Mach numbers with adiabatic and heated walls. However, again, the errors of the CB relation are high for cooled-wall cases, even at $\mathcal{M}_\infty =0.$ Figure 2( $c$ ) reveals that both the present TV and CB relations yield the minimum error at $\gamma =1.3$ but the CB relation does not perform well for all cases. Figure 2( $d$ ) shows that the present TV relations substantially improve accuracy across different Prandtl numbers. At $\textit{Pr}=0.9$ , the CB relation shows minimal errors, consistent with its $\textit{Pr}=1$ theoretical basis. Overall, the TV relations maintain acceptable accuracy ( $\mathcal{E}\lt 3\,\%$ ) for hypersonic boundary layers under various conditions, including different viscosity laws (Chapman, power and Sutherland), variable $\gamma$ and ${\textit{Pr}}$ , arbitrary Mach numbers ( $0 \leqslant \mathcal{M}_\infty \leqslant 10$ ) and both adiabatic and diabatic wall conditions.

Figure 3. Contributions of the decomposed terms for temperature: $(a)$ Mach-number effect at $\mathcal{M}_\infty =0$ (black lines), $4$ (blue lines) and $8$ (red lines); $(b)$ wall-temperature effect at $T_w=0.25T_{ad}$ (black lines), $0.50T_{ad}$ (blue lines) and $0.75T_{ad}$ (red lines).

The variables $\theta _{00}$ , $\theta _{01}$ , $\theta _{10}$ and $\theta _{11}$ can be recovered to four temperature components: $T_{00},$ $T_{01},$ $T_{10}$ and $T_{11},$ respectively. The first term, $T_{00} = s\theta _{00} + T_w$ , represents the contribution from the velocity distribution. In the limit of $\textit{Pr} = 1$ and $\mathcal{M}_\infty \rightarrow 0$ , $T_{00}$ behaves as a passive scalar and is the only non-zero term, while the other three components vanish. The second term, $T_{01}=s\theta _{01}$ , reflects the compressibility effect as $T_{01}$ is null at $\mathcal{M}_\infty \rightarrow 0$ . The third and fourth terms, $T_{10} = \epsilon s\theta _{10}$ and $T_{11} = \epsilon s\theta _{11}$ , emerge due to deviations from unity Prandtl number ( ${\textit{Pr}} \neq 1$ ) and are forced by the $T_{10}$ and $T_{11}$ components, respectively.

Figure 3 displays the contributions of decomposed terms in (2.21), with $\theta _{10}$ and $\theta _{11}$ computed using (2.15) and (2.16), respectively. Figure 3( $a$ ) shows the Mach-number effect on decomposed terms at $T_w=2.8.$ The Mach-number increase significantly enhances $T_{01}$ , leading to a temperature increase. Notably, both $T_{00}$ and $T_{10}$ (approaching zero) exhibit Mach-number independence. Component $T_{11}$ contributes negatively to the temperature rise. Figure 3( $b$ ) shows the wall-temperature effect on the components at $\mathcal{M}_\infty =8.$ The temperature increases with wall temperature solely through $T_{00},$ while other terms remain unchanged. As the wall temperature decreases, $ T_{01}$ becomes increasingly dominant due to the reduction in $ T_{00}$ , underscoring the heightened importance of the compressibility effect in cold-wall conditions. The observed independent quantities indicate that (2.21) is a potentially valuable tool for thermodynamic analysis at different flow parameters.

4. Conclusions

In this paper, we have utilised the asymptotic expansion to investigate the TV relation for laminar hypersonic boundary layers with adiabatic and diabatic conditions. The asymptotic expansion has been applied to the temperature equation to account for a non-unity Prandtl number. We have derived a first-order asymptotic solution for temperature as an explicit function of velocity, where the zeroth-order solution is found to be the classical CB relation. By inputting the velocity profile, the asymptotic solution can accurately capture the temperature profiles for high-speed boundary layers with Chapman, power and Sutherland viscosity laws and Mach number of up to 10. The relation remains valid for boundary-layer flows with arbitrary heat capacity ratios and variable Prandtl numbers close to unity. Numerical results have highlighted the Mach-number- and wall-temperature-independent terms in our relation.

The present relation is particularly valuable for determining temperature profiles in the near-wall region, where experimental measurements are exceptionally challenging in cold-wall hypersonic boundary layers. Moreover, this analytical approach establishes a theoretical framework for further investigation of three-dimensional boundary layers through cross-flow analysis. Potential applications include Falkner–Skan–Cooke flows admitting similarity solutions (Liu 2022). The effects of wall temperature and Mach number on the turbulent mean flow, both laminar and turbulent stress contributions, warrant a further study via reliable Reynolds stress modelling approaches (Chen, Gan & Fu 2024). In addition, the present relation has the potential to function as a temperature wall model for laminar hypersonic boundary layers over cold walls. Once the velocity transformation is established, as shown in Griffin et al. (2021) and MG, the proposed wall temperature model can be fully implemented.