2.1. Basic relations of GRA

For CPG turbulent boundary layers, Zhang et al. (Reference Zhang, Bi, Hussain and She2014) define $r_g$ from a GRA relation on the total enthalpy,

(2.1) \begin{equation} \bar {H}_g - H_w = c_p \frac {\partial \bar {T}}{\partial \bar {u}}|_w \bar {u}, \end{equation}

where $\bar {H}$ is the total enthalpy and $\bar {H}_g=c_p\bar {T} + r_g \bar {u}^2/2$ is the general recovery enthalpy. Equation (2.1) is explicitly expressed as

(2.2a,b) \begin{align} \bar {T} + \frac {r_g}{2c_p}\bar {u}^2 - T_w = \frac {\partial \bar {T}}{\partial \bar {u}}|_w \bar {u}, \quad \text {then} \quad r_g = \frac {2c_p}{\bar {u}^2} \left ( T_w - \bar {T} + \frac {\partial \bar {T}}{\partial \bar {u}}|_w \bar {u}\right ). \end{align}

By definition, $r_g$ varies with the wall-normal height $y$ , and can be obtained a posteriori using $\bar {T}$ and $\bar {u}$ . Zhang et al. (Reference Zhang, Bi, Hussain and She2014) further introduce an effective turbulent Prandtl number $Pr_e$ , which relates to $r_g$ as

(2.3a) ${\displaystyle r_g = \frac {c_p}{\bar {u}} \left ( \frac {\partial \bar {T}}{\partial \bar {u}}|_w - \frac {1}{Pr_e} \frac {\partial \bar {T}}{\partial \bar {u}} \right ) \quad }$

(2.3b) ${\quad \text {or conversely, } \frac {1}{Pr_e} = 1 + \frac {\bar {u}^2}{2c_p} \frac {\partial r_g}{\partial \bar {u}}/\frac {\partial \bar {T}}{\partial \bar {u}}.}$

The combination of (2.2) and (2.3) finally leads to an important ODE for the TV relation,

(2.4) \begin{equation} \bar {T} - \frac {\bar {u}}{2} \left ( \frac {\partial \bar {T}}{\partial \bar {u}}|_w + \frac {1}{Pr_e} \frac {\partial \bar {T}}{\partial \bar {u}} \right ) = T_w, \end{equation}

which can be solved provided a prescribed distribution of $Pr_e$ , and under the outer boundary condition, $\bar {T}|_{\bar {u}=\bar {u}_e}=\bar {T}_e$ . Zhang et al. (Reference Zhang, Bi, Hussain and She2014) further assume $Pr_e\approx 1$ , which is equivalent to assuming $r_g$ a constant, as seen from (2.3b ). As a result, the integration of (2.4) leads to the quadratic relation (1.1). The remaining unknown $({\partial \bar {T}}/{\partial \bar {u}})_w$ can be computed from (2.3a ) at the boundary-layer edge using $r_{g,e}$ , and $r_{g,e}$ is related to the analogy factor $s$ as

(2.5a,b) \begin{align} r_g = r_{g,e} = r \left [ s Pr + (1-s Pr) {\Theta } \right ], \quad \ \ s \equiv \frac {2C_h}{C_f} = \frac {\bar {q}_w \bar {u}_e}{\bar {\tau }_w c_p (\bar {T}_r - T_w)}, \end{align}

where the non-dimensional wall temperature $\Theta =({T_w-\bar {T}_e})/({\bar {T}_r - \bar {T}_e})$ is also called the diabatic parameter; $C_f$ and $C_h$ are the coefficients of skin friction and heat transfer, and $\tau _w$ and $q_w$ are the wall friction and heat flux. Since $s$ is found to be a robust parameter (Chi & Spalding Reference Chi and Spalding1966; Bons Reference Bons2005; Fu et al. Reference Fu, Karp, Bose, Moin and Urzay2021), $r_{g,e}$ can be computed a priori, so (1.1) can be fully determined. Notably, the original definition of $Pr_e$ by Zhang et al. (Reference Zhang, Bi, Hussain and She2014) is based on turbulent fluctuations, which gives $Pr_e$ physical interpretations. The resulting (2.4) provides another way to evaluate $Pr_e$ a posteriori, using only $\bar {T}$ and $\bar {u}$ without fluctuation statistics. As will be shown below, such an evaluation from the mean flow is critical to interpret the TV relations and improve modelling.

Interestingly, Wenzel, Gibis & Kloker (Reference Wenzel, Gibis and Kloker2022) (and also Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023)) showed that a form of Eckert number, ${Ec}=\bar {u}_e^2/[c_p(\bar {T}_r-T_w)]$ for a CPG, is an insightful parameter to account for heat-transfer effects. In fact, we note that $Ec$ can be related to $r_{g,e}$ as

(2.6a,b) \begin{align} {Ec}^{-1} = \frac {r}{2}(1-\Theta ) = \frac {r-r_{g,e}}{2(1-s Pr)} \quad \text {or}\quad r_{g,e} = r - 2 (1-s Pr) {Ec}^{-1}. \end{align}

Since the parameters $r$ , $Pr$ and $s$ are nearly independent of the Mach number $Ma$ and $T_w/\bar {T}_r$ , ${Ec}^{-1}$ and $r_{g,e}$ are essentially equivalent in accommodating heat-transfer effects; ${Ec}^{-1}$ measures the departure of $r_{g,e}$ from $r$ .

The above GRA relations are also generally applicable to turbulent channel and pipe flows, if all the edge quantities are replaced by the channel/pipe centreline quantities ( $\bar {u}_c$ , $\bar {T}_c$ , $r_{g,c}$ , etc.; Modesti & Pirozzoli (Reference Modesti and Pirozzoli2019); Song et al. (Reference Song, Zhang, Liu and Xia2022)). Note that (1.1) has been expressed using both the Reynolds-averaged and Favre-averaged variables among literatures. The differences between these two forms are usually minor (e.g. Zhang, Duan & Choudhari (Reference Zhang, Duan and Choudhari2018)), so the Reynolds average is used here throughout, denoted by overbars. In addition to turbulent flows, Mo & Gao (Reference Mo and Gao2024) reformulate the GRA relations for laminar boundary layers. They define $r_g$ the same as in (2.2), and assume $r_g$ a constant, which is mathematically equivalent to $Pr_e=1$ and leads to exactly the same ODE (2.4) (after enforcing $Pr_e=1$ ) for laminar flows. The constant $r_g$ can be obtained from (2.5) and the same quadratic relation (1.1b ) is arrived. Therefore, the relations (2.2) to (2.5) turn out to be universal for both turbulent and laminar flows, in both boundary layer and channel (and pipe) configurations.

In short, two components are required to solve (2.4) for the TV relation: the first is the wall-normal distribution of $Pr_e$ , which determines the shape of $\bar {T}$ ; the second is $r_{g,e}$ , or equivalently $s$ , which determines the wall boundary condition $({\partial \bar {T}}/{\partial \bar {u}})_w$ . Previous models assume constancy of $r_g$ (and $Pr_e$ ) for both turbulent and laminar flows, leading to the quadratic TV relation and temperature wall models. In the following, we carefully examine these assumptions for laminar and turbulent flows, respectively. Both channel flow and flat-plate boundary layers are considered. Although laminar flows are much easier to be resolved than turbulence, we still present the laminar TV results for three reasons. First, as mentioned in § 1, the laminar TV relation can be used to efficiently improve the prediction of $C_f$ and $C_h$ by largely relaxing the near-wall grid density; this strategy for flat plates can be directly applied to curved-wall cases which inhibits simple self-similar solutions (Mo & Gao Reference Mo and Gao2024). Therefore, a comprehensive evaluation on the accuracy of the laminar TV relation is required. Second, Mo & Gao (Reference Mo and Gao2024) only studied three severely cold-wall boundary layer cases ( $T_w/\bar {T}_r\lt 0.2$ ), where $r_g$ has a mild wall-normal variation. Here, we explore a wide parameter space ( $Ma$ from 2 to 10 and $T_w/\bar {T}_r$ from 0.1 to 1.6 for boundary layers), and will show that $r_g$ severely differs at different $T_w/\bar {T}_r$ but a universal distribution of $Pr_e$ can be identified. Third, a number of similarities of the TV relation between laminar and turbulent flows will be reported, suggesting some fundamental resemblance between the two flows, as already partly exhibited from their common GRA relations above.

Only CPG air flows are considered. The laminar flow is solved from the Navier–Stokes (NS) equations (detailed later), where the viscosity $\mu$ is from Sutherland’s law (reference temperature set to 270 K) and a constant $Pr=0.71$ is adopted throughout. For turbulent cases, elaborated DNS datasets of wide parameter ranges are employed. The factor $r$ for all cases is set to $Pr^{1/3}$ .

2.2. Laminar channel flows

The compressible NS equations for laminar channel flow degenerate into ODEs,

(2.7) \begin{equation} \left (\bar {\mu } \bar {u}_y\right )_y = \bar {p}_x = -\frac {\bar {\tau }_w}{h}, \qquad \bar {\mu } \bar {u}_y^2 + \frac {c_p}{Pr} \left (\bar {\mu } \bar {T}_y\right )_y = 0, \end{equation}

where the subscript $y$ denotes wall-normal derivatives, $\bar {p}_x$ is the streamwise pressure gradient and $h$ is the channel half-height ( $y\in [0,2h]$ ). The overbars are retained for consistency in notation. After non-dimensionalisation, the only parameter controlling the shape of the flow is the bulk Mach number $Ma_b$ , measuring the ratio of bulk velocity $\bar {u}_b$ and the wall speed of sound. The Chebyshev collocation point method is adopted to solve (2.7) using $N_y=201$ grid points, which achieves grid independence.

The temperature profiles at different $Ma_b$ are shown in figure 1(a) as a function of the streamwise velocity. The TV relation (1.1) is also displayed with $s$ computed from (2.5b ), as labelled in the legend. It is shown that even with $Ma_b$ up to 5, (1.1) agrees with the $\bar {T}$ from NS equations very well, which is not widely expected before for laminar channel flows. After attaining the mean flow, $r_g$ is determined by (2.2), as displayed in figure 1(b). An immediate observation is that $r_g$ increasingly varies with $\bar {u}$ as $Ma_b$ rises; it changes by over 15 % in the wall-normal direction at $Ma_b=5$ . Also, $r_g$ decreases more rapidly near the wall than in the outer region. Therefore, it seems difficult to trust the results of constancy $r_g$ in § 2.1, so a natural question is why does the TV relation still work?

Figure 1. Laminar channel results: (a) temperature from NS equations and (1.1), (b) general recovery factor, (c) effective Prandtl number and (d) the sensitivity factor for flows at different $Ma_b$ . The legends for panels (b)–(d) are the same.

As noted in § 2.1, $Pr_e$ is more significant than $r_g$ in determining the shape of $\bar {T}$ , so we define the counterpart of $Pr_e$ in laminar flows from (2.3) and (2.4), whose physical meaning will be discussed in § 4.1. For convenience of writing, we still use the notation $Pr_e$ , so (2.2) to (2.5) take the same form for both turbulent and laminar flows. From (2.4), $Pr_e$ can be expressed as

(2.8) \begin{equation} Pr_e^{-1} = \frac {2(\bar {T} - T_w)/\bar {u} - ({\partial \bar {T}}/{\partial \bar {u}})_w}{{\partial \bar {T}}/{\partial \bar {u}}}. \end{equation}

The distributions of $Pr_e^{-1}$ are plotted in figure 1(c) at $0\lt Ma_b\leqslant 5$ . Compared with $r_g$ , $Pr_e^{-1}$ is indeed less sensitive to $Ma_b$ and is approximately unity at $\bar {u}/\bar {u}_c\lesssim 0.6$ . In fact, its theoretical wall limit is just $Pr_{e,w}=1$ , obtained from (2.8). Further away from the wall, $Pr_e^{-1}$ rapidly diminishes and can be negative. To better quantify the effects of $Pr_e^{-1}$ variations, a sensitivity study on (2.4) is designed. Specifically, we compute the global variation of $\bar {T}$ when the $Pr_e^{-1}$ at a specific $y_0$ is perturbed, i.e.

(2.9) \begin{equation} \mathcal {E}(y_0) = \dfrac {1}{\int _{0}^{\bar {u}_c} {\bar {T}\,\mathrm {d} \bar {u}}} \int _{0}^{\bar {u}_c} {\left | \frac {\partial \bar {T}(Pr_e^{-1};\bar {u})}{\partial Pr_e^{-1}(y_0)}\right | \mathrm {d} \bar {u}}, \end{equation}

where $\bar {T}(Pr_e^{-1};\bar {u})$ represents the solution of (2.4) on the $\bar {u}$ grid, under the outer boundary condition $\bar {T}|_{\bar {u}=\bar {u}_c}=\bar {T}_c$ . The partial differentiation is computed after the numerical discretisation. The $\mathcal {E}$ in different cases are plotted in figure 1(d), where the spikes of $\mathcal {E}$ near the centreline are due to the singularity of $Pr_e$ there ( $Pr_e^{-1}=0$ ). A general trend is that $\mathcal {E}$ peaks in the middle regime ( $\bar {u}\approx 0.5\bar {u}_c$ ), and diminishes towards the wall and the centreline; meanwhile, $\mathcal {E}$ is amplified with the rise of $Ma_b$ . These features are consistent with the trend in figure 1(a) that the TV relation slightly degrades in the middle region and with $Ma_b$ increased.

Therefore, the success of the TV relation in figure 1(a) can be explained. On the one hand, the temperature is sensitive to the middle-regime $Pr_e^{-1}$ , but $Pr_e^{-1}$ is close to unity there. On the other hand, $Pr_e^{-1}$ deviates from unity further away towards the centreline, but $\bar {T}$ is already less sensitive to $Pr_e^{-1}$ . More physical explanations on the above trends will be presented in § 2.3, through comparisons with the boundary layer cases. For modelling usage, we note that $s$ varies mildly with $Ma_b$ . As listed in table 1, $s$ changes from 0.98 to 1.04 when $Ma_b$ is ranged from 0.01 to 6. Thereby, the TV relation (1.1) with $s\approx 1$ is a highly accurate model for laminar channel flows.

Table 1. Reynolds analogy factor (2.5b ) at different bulk Mach numbers for laminar channel flows.

2.3. Laminar boundary layers

The results for laminar boundary layers are rather different from § 2.2. The mean flow on a flat plate can be efficiently obtained by solving the self-similar equations (White Reference White1974),

(2.10) \begin{equation} \left \{\begin{array}{l} C_1 \bar {u}_{\eta \eta } + \left ( C_{1,\eta } + \dfrac {\Pi }{2} \right ) \bar {u}_\eta = 0, \\ C_2 \bar {T}_{\eta \eta } + \left ( C_{2,\eta } + c_p \dfrac {\Pi }{2} \right ) \bar {T}_\eta + C_1 \bar {u}_\eta ^2 = 0, \end{array}\right . \end{equation}

where $\eta$ is the transformed wall-normal coordinate, and $C_1=\bar {\rho }\bar {\mu }/\bar {\rho }_e\bar {\mu }_e$ , $C_2=c_pC_1/Pr$ and $\Pi =\int \bar {u}/\bar {u}_e\,\mathrm {d} \eta$ . Equation (2.10) is independent of streamwise locations, and we consider cases at different free steam Mach numbers $Ma_\infty$ (2–10) and $T_w/\bar {T}_r$ (0.1–1.6). The Chebyshev collocation point method with $N_y=201$ is also adopted, which leads to grid-independent mean flow and GRA parameters.

Figure 2. Laminar boundary layer results: general recovery factor for (a) adiabatic and (b) $Ma_\infty =6$ isothermal cases, and (c) effective Prandtl number and (d) the sensitivity factor for 20 cases with different $Ma_\infty$ and $T_w/\bar {T}_r$ , as labelled. The jumps in panel (c) are plotted as dotted lines for conciseness, and the red dashed line denotes (2.13).

The adiabatic wall cases ( $T_w=\bar {T}_r$ ) are investigated first and the $r_g$ at different $Ma_\infty$ is shown in figure 2(a). It is found that $r_g$ is not sensitive to $Ma_\infty$ and exhibits a universal wall-normal distribution for adiabatic walls. Its theoretical wall limit, obtained from (2.10), is $r_{g,w}=Pr=0.71$ , which represents a balance between viscous terms and molecular heat flux (see Appendix A). Nevertheless, $r_g$ is not a wall-normal constant, but keeps rising away from the wall due to stronger convection terms. The effects of wall cooling on $r_g$ are displayed in figure 2(b) at $Ma_\infty =6$ . A clear trend is that as $T_w/\bar {T}_r$ drops, $r_g$ exhibits weaker wall-normal variations, so the assumption of $r_g$ constancy roughly holds only for severely cooled walls, which was shown by Mo & Gao (Reference Mo and Gao2024). Nevertheless, the above trend does not mean that the TV relation becomes more accurate for cooler walls, as will be shown later.

As learned from § 2.2, $Pr_e$ is more significant than $r_g$ in interpreting the shape of $\bar {T}$ , so the $Pr_e^{-1}$ at different $Ma_\infty$ and $T_w/\bar {T}_r$ are plotted collectively in figure 2(c). A surprising observation is that the 20 curves (and also other $Ma_\infty$ and $T_w/\bar {T}_r$ cases not displayed) almost collapse with each other, except for abrupt jumps near the temperature peaks in cold-wall cases. In other words, $Pr_e^{-1}$ is nearly independent of $Ma_\infty$ and $T_w$ , beneficial for a robust modelling. The sensitivity examination of $\bar {T}$ on $Pr_e^{-1}$ is also performed as in (2.9). As shown in figure 2(d), the outer region $\bar {T}$ is mostly affected by the deviation of $Pr_e^{-1}$ , where $Pr_e^{-1}$ systematically departs from unity. Consequently, different from the channel flows, the TV relation (1.1) for laminar boundary layers has systematic deviations in the outer region, irrespective of $Ma_\infty$ and $T_w$ . This deviation is reflected by Mo & Gao (Reference Mo and Gao2024) (their figure 1b), and will be improved later.

Considering the collapse of $Pr_e^{-1}$ among cases in terms of $\bar {u}/\bar {u}_e$ , we seek for analytical relations from (2.10) for further interpretation. After eliminating the viscous parameters, an equation with separated variables is obtained (see Appendix A for derivations):

(2.11) \begin{equation} (1-Pr) \frac {\Psi _{\bar {u}}}{\Psi _e-\Psi } = \frac {\bar {T}_{\bar {u}\bar {u}} - \bar {T}_{\bar {u}\bar {u},w}}{\bar {T}_{\bar {u}}} = \frac {\bar {Q}_{\bar {u}\bar {u}}}{\bar {T}_{\bar {u}}}, \qquad \Psi = \int _0^{\bar {u}} \Pi \,\mathrm {d} \bar {u}, \end{equation}

where the left-hand side relates to the velocity, and the right-hand sides correspond to the TV relation; the subscript $\bar {u}$ denotes partial derivatives. If $\bar {T}$ is a perfect quadratic function of $\bar {u}$ , which is denoted as $\bar {T}_{qd}(\bar {u})$ , then the second-order derivative $\bar {T}_{\bar {u}\bar {u}}$ is a constant and the right-hand sides should remain zero ( $\bar {T}_{\bar {u}\bar {u}}=\bar {T}_{\bar {u}\bar {u},w}$ ). This is definitely invalid since the left-hand side monotonically increases from zero as $\bar {u}$ rises. In other words, a quadratic $\bar {T}(\bar {u})$ is not supported by the governing equation. Consequently, the non-zero right-hand side continuously contributes to the deviation of $\bar {T}$ from a quadratic function, though the accumulating error may be finally restricted by the enforced outer boundary condition $\bar {T}_e$ . For clarity, we further define a residual temperature $\bar {Q}=\bar {T}-\bar {T}_{qd}$ , then $\bar {Q}=0$ means a perfect quadratic $\bar {T}(\bar {u})$ (see Appendix A for details). From (2.11), the second-order derivative of $\bar {Q}$ relates to $\Psi$ , so $\bar {Q}_{\bar {u}\bar {u}}$ is non-zero except at the wall. However, substituting $\bar {Q}$ into (2.3) and (2.4) results in expressions of $Pr_e$ and $r_g$ as

(2.12) \begin{equation} Pr_e^{-1} = 1 - \left (\frac {\bar {Q}}{\bar {u}^2}\right )_{\bar {u}}\frac {\bar {u}^2}{\bar {T}_{\bar {u}}}, \qquad r_g = Pr - 2c_p\frac {\bar {Q}}{\bar {u}^2}. \end{equation}

Therefore, the introduction of $\bar {Q}$ connects the deviation of $\bar {T}$ from $\bar {T}_{qd}$ with the departure of ${Pr}_e$ from unity and of $r_g$ from $Pr$ .

Model improvement is required for laminar boundary layers considering the systematic deviation of ${Pr}_e$ and thus (1.1). The robust distribution of ${Pr}_e^{-1}$ in figure 2(c) can be modelled. For convenience of model usage, we provide a curve fitting

(2.13) \begin{equation} Pr_e^{-1} = 2 - [1-(\bar {u}/\bar {u}_e)^3]^{-0.1}, \end{equation}

then (2.4) can be numerically solved. Another source of error besides $Pr_e^{-1}$ is the estimation of $s$ , required for the wall boundary condition. Unlike in § 2.2, $s$ exhibits a moderate dependence on $Ma_\infty$ and $T_w/\bar {T}_r$ , as depicted in figure 3(a). Note that the $s$ around $T_w\approx \bar {T}_r$ is not displayed because both the numerator and denominator tend to zero (see (2.5 b)); the $s$ there is also of little significance because $r_{g,e}\approx r$ when $T_w\approx \bar {T}_r$ . The cold-wall boundary is more frequently encountered than the heated one in realistic high-speed applications (e.g. Bradshaw (Reference Bradshaw1977); Chen et al. (Reference Chen, Xi, Ren and Fu2022b )). In the former case, $s$ is generally 1.0–1.2, close to the turbulent counterparts. The temperature predictions using $Pr_e=1$ (i.e. (1.1)) and the fitted $Pr_e$ (2.13) are compared in figure 3(b), where $s=1.14$ are used for both. As pointed out above, (1.1) systematically underestimates the temperature in the outer region ( $\bar {u}\gtrsim 0.3\bar {u}_e$ ) at different $Ma_\infty$ and $T_w/\bar {T}_r$ . In comparison, solving (2.4) with (2.13) exhibits a notable improvement as expected, which attains a close agreement with the NS results.

Figure 3. Laminar boundary layer results: (a) contours of the Reynolds analogy factor at different $Ma_\infty$ and $T_w/\bar {T}_r$ ; (b) temperature from NS equations, (1.1) and (2.13) for three cases, and (c) the prediction error of $\bar {T}$ at different $Ma_\infty$ and $T_w/\bar {T}_r$ . The notation M4Tw02 in panel (b) denotes $Ma_\infty =4$ , $T_w=0.2\bar {T}_r$ and the same for the others.

To quantify the prediction accuracy of the TV relation, the relative error to the NS results,

(2.14) \begin{equation} \epsilon _T = \dfrac {\int _{0}^{\bar {u}_e}|\bar {T}_{NS}-\bar {T}_{TV}|\mathrm {d}\bar {u}} {\int _{0}^{\bar {u}_e}\bar {T}_{NS}\,\mathrm {d}\bar {u}}, \end{equation}

is computed. The errors from the two models are compared in figure 3(c) as a function of $T_w/\bar {T}_r$ for $Ma_\infty =3,6,10$ . Although $r_g$ tends to be wall-normally invariant at low $T_w/\bar {T}_r$ , the error of (1.1) for colder-wall cases is not lower but higher. At $Ma_\infty \gt 5$ , $\epsilon _T$ is higher than 3 % for nearly all $T_w/\bar {T}_r$ conditions and its maximum reaches 7 %. In comparison, a notable reduction in $\epsilon _T$ is realised by using (2.13) for all the $Ma_\infty$ and $T_w/\bar {T}_r$ shown. The improvement is most evident for cold-wall cases where $\epsilon _T$ are all less than 2 %, which has practical significance for high-speed realistic applications.

Finally, it is important to explain the different behaviours of $r_g$ and $Pr_e$ between channel and boundary layer cases. The channel flow is driven by the streamwise pressure gradient, which is a wall-normal constant, whereas for flat-plate boundary layers, the convection term acts as the driving force. In the latter case, the convection term is minor near the wall, so the rough balance between the viscous terms and molecular heat flux leads to $r_g\approx Pr$ in the near-wall region, as noted above. In contrast, the constant pressure gradient for channels disrupts the condition for constancy $r_g$ ( $\approx Pr$ ) near the wall, so $r_g$ exhibits a converse trend that it varies more rapidly close to the wall than in the outer region. Another region worth discussing is that close to the boundary layer edge or the channel centreline, where the sensitivity of $Pr_e$ largely differs between the two cases. For explanation, the numerator and denominator of $Pr_e^{-1}$ in (2.8) are plotted in figure 4 for channel, and adiabatic and cold-wall boundary layers. The two curves in the channel case are nearly linear functions of $\bar {u}$ throughout the field. For the two boundary layer cases (figures 4 b and 4 c), however, $\partial \bar {T}/\partial \bar {u}$ declines dramatically near the edge ( $\bar {u}\gtrsim 0.8\bar {u}_e$ ), leading to the uncertainty and sensitivity of $Pr_e$ . The sudden mismatch of the two curves near the edge is due to the thickness difference between the momentum and thermal boundary layers, i.e. $\delta _u\ne \delta _T$ , since $Pr\ne 1$ . Therefore, the TV relation is difficult to follow in the transition region between $\delta _u$ and $\delta _T$ . This transition region does not exist in the channel case because the thicknesses of the momentum and thermal layers are forced to be the same, equal to $h$ . In short, two primary factors responsible for the differences between channel and boundary layers are the spatial distribution of the driving force, and the thicknesses of the momentum and thermal boundary layers.

Figure 4. Numerator and denominator of $Pr_e^{-1}$ in (2.8) for (a) laminar channel case $Ma_b=5$ , and laminar boundary layer cases (b) $Ma_\infty =5$ , $T_w=\bar {T}_r$ and (c) $Ma_\infty =5$ , $T_w=0.1\bar {T}_r$ . All the curves are normalised by $\bar {T}_c/\bar {u}_c$ or $\bar {T}_e/\bar {u}_e$ , accordingly.

2.4. Turbulent boundary layers

In the following, turbulent cases are dealt with using wide elaborated DNS data. For turbulent boundary layers, 17 DNS cases from five different sources are employed with $Ma_\infty$ from 2 to 14 under cold, adiabatic and heated walls ( $T_w/\bar {T}_r=0.18{-}1.9$ ). The specific parameters and data sources of these cases are listed in table 2.

Table 2. Parameters of the turbulent boundary layer DNS datasets, where $Re_{\tau }$ and $Re_\theta$ are the friction and momentum-thickness-based Reynolds numbers. The abbreviations for data sources are: PB for Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011, Reference Pirozzoli and Bernardini2013); ZDC for Zhang et al. (Reference Zhang, Duan and Choudhari2018); VBL for Volpiani et al. (Reference Volpiani, Bernardini and Larsson2018, Reference Volpiani, Bernardini and Larsson2020); ZWLSL for Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022); and CSPB for Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). The notation expresses $Ma_\infty$ , $T_w/\bar {T}_r$ and $Re_{\delta _2}=\rho _\infty u_\infty \delta _{99}/\bar {\mu }_w$ (divided by 100) with $\rho$ the density and $\delta _{99}$ the nominal thickness.

The mean flow $\bar {u}$ and $\bar {T}$ are employed to compute $Pr_e$ and $r_g$ . The adiabatic and heated wall cases are considered first, whose $Pr_e^{-1}$ distributions are plotted in figure 5(a). Here, $Pr_e^{-1}$ is indeed around unity in most regions, which explains the wide success of (1.1) for these cases. Meanwhile, $Pr_e^{-1}$ declines markedly near the boundary layer edge ( $\bar {u}\gtrsim 0.9\bar {u}_e$ ) for all cases, in line with the laminar case in figure 2(c). This is not unexpected because the flow has become episodic and intermittent near the edge (Zhang et al. Reference Zhang, Bi, Hussain and She2014), and the momentum and thermal boundary layers differ in thicknesses as $Pr\ne 1$ (see § 2.3). The cold wall cases are considered separately in figure 5(b) because there are temperature peaks inside the boundary layer, around which $Pr_e^{-1}$ becomes singular due to the diminishing $\partial \bar {T}/\partial \bar {u}$ . The $Pr_e^{-1}$ varies dramatically around the temperature peak, but it becomes close to unity again further away from the wall, the same as in figure 5(a). The sensitivity examination is similarly conducted as for laminar flows to study the influence of $Pr_e^{-1}$ variations on $\bar {T}$ . As shown in figure 5(d), $\mathcal {E}$ is mostly amplified at $\bar {u}\gtrsim 0.7\bar {u}_e$ , where $Pr_e^{-1}$ is close to unity. Also, the dramatic variation of $Pr_e^{-1}$ near the temperature peak for cold-wall cases limitedly influences the accuracy of the TV quadratic relation owing to the relatively low $\mathcal {E}$ . The above two facts support the wide accuracy of (1.1) for turbulent boundary layers.

Figure 5. Turbulent boundary layer results: effective Prandtl number for (a) adiabatic and heated-wall cases and (b) cold-wall cases, and (c) general recovery factor and (d) the sensitivity factor for all cases. The legends (see notation in table 2) are the same for all panels, separately shown in the two boxes.

In addition to $Pr_e^{-1}$ , the $r_g$ profiles of all cases are collectively displayed in figure 5(c). We note that the near-wall $r_g$ (at $\bar {u}\lesssim 0.1\bar {u}_e$ ) can be highly sensitive to $\partial \bar {T}/\partial \bar {u}|_w$ and thus the post-processing discretisation schemes, because both the numerator and denominator in (2.2 b) and their first-order derivatives are zero at the wall. We use the Chebyshev collocation point method to compute these derivatives and ensure that the results are grid independent by increasing $N_y$ up to 401. Using other schemes may lead to different near-wall $r_g$ , but the primary conclusions remain unaltered. For all the cases, $r_g$ tends to be invariant at $\bar {u}\gtrsim 0.5\bar {u}_e$ , but the near-wall $r_g$ rapidly changes, not following constancy.

The velocity and temperature transformations can provide insights into the near-wall behaviour of $r_g$ . The major difference between the temperature transformation and the TV relation is that, the former is derived from the energy equation (Patel, Boersma & Pecnik Reference Patel, Boersma and Pecnik2017; Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022a ), while the latter is built upon the analogy between momentum and energy equations. Therefore, the three relations (two transformations and the TV relation) firmly connect with each other. We first consider the TL velocity transformation of Trettel & Larsson (Reference Trettel and Larsson2016) and the semi-local temperature transformation of Chen et al. (Reference Chen, Huang, Shi, Yang and Lv2022a ). In the viscous sublayer, these two transformations can reproduce the incompressible scalings of velocity and temperature, under the semi-local units. The transformed velocity and temperature ( $\bar {\theta }=\bar {T}-T_w$ ) follow

(2.15a) ${\bar {u}_{TL}^+(y^*) = \int _{0}^{y^*} {\bar {\mu }^+ \dfrac {\partial \bar {u}^+}{\partial y^+} \mathrm {d} y^*} \approx y^*, }$

(2.15b) ${\bar {\theta }_{SL}^+(y^*) = \int _0^{y^*} {\dfrac {\frac {y^+}{y^*}\sqrt {\bar {\rho }^+}\frac {\partial \bar {\theta }^+}{\partial y^+}}{1-\frac {\bar {u}\bar {\tau }_w}{\bar {q}_w}} \mathrm {d} y^*} \approx Pr\,y^*, }$

where $y^*=y^+\bar {\rho }^{+1/2}/\bar {\mu }^+$ is the semi-local coordinate; $\bar {u}^+$ , $\bar {\rho }^+$ , $y^+$ , $\bar {\mu }^+$ and $\bar {\theta }^+$ are normalised by the wall viscous units $u_\tau =(\bar {\tau }_w/\bar {\rho }_w)^{1/2}$ , $\bar {\rho }_w$ , $\bar {\mu }_w/(\bar {\rho }_w u_\tau )$ , $\bar {\mu }_w$ and $T_\tau =\bar {q}_w/(\bar {\rho }_w c_p u_\tau )$ , respectively. Note that (2.15b ) is the re-derived version by Cheng & Fu (Reference Cheng and Fu2024a ) for channels and boundary layers, essentially the same as the original version for Couette flows. Also, for adiabatic walls with $\bar {q}_w\approx 0$ and $T_\tau \approx 0$ , (2.15b ) can still be defined by multiplying the numerator and denominator by a factor $\bar {q}_w$ (Chen et al. Reference Chen, Huang, Shi, Yang and Lv2022a ). The combination of (2.15a ) and (2.15b ) yields a simple TV relation (derivation in Appendix A),

(2.16) \begin{equation} \frac {\partial \bar {\theta }^+}{\partial \bar {u}^+} = Pr \left (1 - \frac {u_\tau ^2}{c_p T_\tau } \bar {u}^+\right ) \quad \text {or}\quad \ \frac {\partial \bar {T}}{\partial \bar {u}} = \frac {Pr}{c_p} \left (\frac {\bar {q}_w}{\bar {\tau }_w} - \bar {u}\right ) = \frac {\partial \bar {T}}{\partial \bar {u}}|_w - \frac {Pr}{c_p} \bar {u}. \end{equation}

Therefore, (2.16) by itself suggests a quadratic relation between $\bar {T}$ and $\bar {u}$ near the wall. An analytical relation is further obtained by substituting (2.16) into (2.2 a) that $r_g=Pr$ in the viscous sublayer, irrespective of wall thermal boundary conditions (see Appendix A). This relation results directly from the balance between viscous terms and molecular heat flux, the same as the laminar case in § 2.3. Nevertheless, the near-wall $r_g\approx Pr$ is not in line with figure 5(c), so we conclude that the accuracy of these transformations (2.15) is only guaranteed integrally. Their point-by-point derivatives may not be accurate enough. Similar analyses for the buffer and logarithmic layers can be performed using the GFM velocity transformation of Griffin et al. (Reference Griffin, Fu and Moin2021b ), and the temperature transformation of Cheng & Fu (Reference Cheng and Fu2024a ), which will be used in § 3 to implement a new temperature wall model.

2.5. Turbulent channel flows

The turbulent channel flows are studied using 12 DNS cases with $Ma_b\leqslant 4$ , whose parameters and data sources are detailed in table 3. As shown in figure 6(a), the $r_g$ of these cases generally experiences a decreasing trend away from the wall, and the Reynolds number effect on $r_g$ is weak. The near-wall $r_g$ also does not follow the constancy deduced from (2.16), i.e. $r_g\approx Pr$ , like the boundary layer cases, though the two transformations in (2.15) are generally applicable to the channel cases in the viscous sublayer. Moreover, the centrelines $r_g$ ( $r_{g,c}$ ) all fall within $0.62\pm 0.01$ for these cases, which is also reflected from the weakly varied $\Theta$ in table 3 (see (2.5a )). Nonetheless, substituting the $\bar {T}_c$ scaling derived by Song et al. (Reference Song, Zhang, Liu and Xia2022) into (2.5a ) suggests a weak dependence of $r_{g,c}$ on $\bar {u}_c/\bar {u}_b$ , as

(2.17) \begin{equation} r_{g,c} = Pr \left [ sr - 1.034 (1 - s Pr) {\bar {u}_b}/{\bar {u}_c} \right ]. \end{equation}

Table 3. Parameters of the turbulent channel DNS datasets. The abbreviations for data sources are: TL for Trettel & Larsson (Reference Trettel and Larsson2016); YH for Yao & Hussain (Reference Yao and Hussain2020); and CF for Cheng & Fu (Reference Cheng and Fu2022). The notation expresses $Ma_b$ and $Re_\tau$ (divided by 100). Note that TL uses the power viscous law instead of sutherland’s law.

Figure 6. Turbulent channel results: (a) general recovery factor; (b) effective Prandtl number; (c) mean temperature from DNS and (1.1); and (d) the sensitivity factor for different cases. The legends (see notation in table 3) are the same for panel (a,b,d). The black dashed line in panel (b) is a fitted curve detailed in Appendix B.

As shown in figure 6(b), $Pr_e^{-1}$ systematically deviates from unity in the middle regime $0.2\!\lesssim \!\bar {u}/\bar {u}_c\!\lesssim \!0.7$ , different from the boundary layer cases in § 2.4. Meanwhile, there is a pronounced sensitivity of $\bar {T}$ on $Pr_e^{-1}$ in that region, as seen from figure 6(d), and the sensitivity ( $\mathcal {E}$ ) rapidly grows with the rise of $Ma_b$ . Consequently, the quadratic relation (1.1) has a modest deviation (overprediction) in the middle regime, especially for high- $Ma_b$ cases, which is clearly demonstrated in figure 6(c). The fact that (1.1) performs better for turbulent boundary layers than turbulent channel flows is reasonable, because the constant pressure gradient term in the channel momentum equation hinders the GRA between momentum and energy equations. For modelling usage, a curve fitting of $Pr_e^{-1}$ in figure 6(b) can be made, similar to the implementation for (2.13) and figure 3, to improve the accuracy of the TV relation for turbulent channel flows. A sample is provided in Appendix B.

Figure 7. Numerator and denominator of $Pr_e^{-1}$ in (2.8) for turbulent (a) channel case TL-M30R19, and boundary layer cases (b) adiabatic PB-M3Tw10R14 and (c) cold-wall ZDC-M6Tw025R11. All the curves are normalised by $\bar {T}_c/\bar {u}_c$ or $\bar {T}_e/\bar {u}_e$ , accordingly.

Considering the different behaviours of $Pr_e$ between laminar and turbulent cases, the numerator and denominator of $Pr_e^{-1}$ in (2.8) for three turbulent cases are plotted in figure 7, in a similar style to figure 4. For turbulent boundary layers, the major difference from the laminar counterpart is that ${\partial \bar {T}}/{\partial \bar {u}}$ in figures 4(b) and 4(c) follow more closely the denominator, until $\bar {u}\approx 0.97\bar {u}_e$ , so the TV relation is more accurate than the laminar boundary layers. This enhanced TV connection in the outer region in turbulent cases can be attributed to the intense wall-normal eddy transport of momentum and energy, and the strong TV fluctuation coupling of very large scale motions (Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Cheng & Fu Reference Cheng and Fu2023, Reference Cheng and Fu2024b ).