2.1. Direct numerical simulation of turbulent C–P flow

The compressible C–P flow is configured as described in figure 1( $a$ ), driven by both moving-wall frictions and an external pressure gradient between two isothermal walls with fixed wall temperature $T_w=293.15\,\text{K}$ . It is statistically stationary with $\bar {\rho }$ , $\overline {U}$ and $\overline {T}$ representing the Reynolds-averaged density, velocity and temperature, respectively. We define the bulk density $\rho _{m}=\int _{-h}^{h}\bar {\rho }(y)\textrm {d}y/(2h)$ and the bulk velocity $U_{m}=\int _{-h}^{h}\widetilde {U}(y)\textrm {d}y/(2h)$ with $\widetilde {U}$ the Favre-averaged streamwise velocity. Then two independent Reynolds numbers can be defined, ${\textit{Re}}_w = \rho _{m}U_w h / \bar {\mu }_w$ and ${\textit{Re}}_{m} = \rho _{m}U_{m}h / \bar {\mu }_w$ , with $\bar {\mu }_w$ the dynamic viscosity at the wall. It is convenient to employ polar coordinates $({\textit{Re}}_0,\theta )$ defined by

(2.1) \begin{equation} {\textit{Re}}_0 = \frac {\rho _{m}U_{0}h }{ \bar {\mu }_w }, \quad \tan \theta = \frac {{\textit{Re}}_m}{{\textit{Re}}_w}, \quad U_{0} = \sqrt { U_w^2+U_{m}^2} , \end{equation}

with ${\textit{Re}}_0$ the defining Reynolds number, $\theta$ the ratio between the Couette part and the Poiseuille part and a resultant velocity scale $U_{0}$ . The PC and PP limits correspond to $\theta =0^{\circ }$ and $90^{\circ }$ , respectively. The Mach number is defined as $\textit{Ma}_0=U_{0}/({\gamma R T_w})^{1/2}$ . Here $R$ is the gas constant and $\gamma =C_p/(C_p-R)$ is the specific heat ratio with $C_p$ the specific heat at constant pressure. The set ( ${\textit{Re}}_0,\theta ,\textit{Ma}_0$ ) suffices to describe compressible C–P flow for specified fluid properties.

Skin-friction velocities for the two walls are defined as $ u_{\tau ,t} = \textrm {sgn}(-\eta _{t}) \sqrt {\bar {\mu }_w\lvert \eta _{t}\rvert / \bar {\rho }_w}$ and $ u_{\tau ,b} = \textrm {sgn}(\eta _{b}) \sqrt {\bar {\mu }_w\lvert \eta _{b}\rvert / \bar {\rho }_w}$ with $\eta = \textrm {d}\overline {U}/\textrm {d}y$ and $\bar {\rho }_w$ the wall density. Here we denote top/bottom walls by subscripts ‘ $t$ ’/‘ $b$ ’. Wall heat fluxes are defined as $\overline {q}_{w,t}=\lambda \textrm {d}\overline {T}/\textrm {d}y|_{y=h}$ and $\overline {q}_{w,b}=-\lambda \textrm {d}\overline {T}/\textrm {d}y|_{y=-h}$ with $\lambda =C_p \bar {\mu }_w / Pr$ the thermal conductivity. Skin Mach number, skin-friction Reynolds number and dimensionless wall heat flux for wall ‘ $i$ ’ ( $i=t$ , $b$ ) are

(2.2) \begin{equation} \textit{Ma}_{\tau ,i} = {u_{\tau ,i}}/({\gamma R T_w})^\frac{1}{2}, \quad {\textit{Re}}_{\tau ,i} = \frac {\bar {\rho }_{w}u_{\tau ,i}h}{\bar {\mu }_w} , \quad B_{q,i} = \frac {\bar {q}_{w,i}}{C_{p}\bar {\rho }_wu_{\tau ,i}T_w}. \end{equation}

A DNS for turbulent C–P flows at ${\textit{Re}}_0=4000$ with $\textit{Ma}_0=0.8$ and $1.5$ , which are determined with reference to Yao & Hussain (Reference Yao and Hussain2023), is performed. For specified ${\textit{Re}}_0$ and $\textit{Ma}_0$ , we implement $\theta = 15^{\circ }$ , $30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ . The in-house code employs a central, nominally zero-dissipative energy-preserving method (Pirozzoli Reference Pirozzoli2011) combining with a seventh-order WENO scheme. The eighth-order centred scheme is used for viscous terms and the low-storage third-order Runge–Kutta scheme for time integration. The domain is $20\pi h \times 2h \times 6\pi h$ for all cases, which, based on Cheng et al. (Reference Cheng, Pullin, Samtaney and Luo2023), is sufficient to provide promising statistics. For DNS at $\textit{Ma}_0=0.8$ , the grid is $1280\times 180\times 384$ , while for $\textit{Ma}_0=1.5$ , the grid is $1920\times 200\times 576$ . Verification is implemented at $\theta =0^\circ$ and $\textit{Ma}_0=0.8$ . Both the scaled mean velocity $\overline {U}^\ast$ based on the transformation by Trettel & Larsson (Reference Trettel and Larsson2016) and $\overline {T}/T_w$ are shown to reasonably match with DNS of Yao & Hussain (Reference Yao and Hussain2023), as shown in figure 1(b).

Table 1 lists some statistical quantities, including $\textit{Ma}_{\tau ,t}$ , $\textit{Ma}_{\tau ,b}$ , ${\textit{Re}}_{\tau ,t}$ , ${\textit{Re}}_{\tau ,b}$ , $B_{q,t}$ and $B_{q,b}$ . Similar to incompressible C–P flow physics (Cheng et al. Reference Cheng, Pullin, Samtaney and Luo2023), we note ${\textit{Re}}_{\tau ,b}\gt 0$ for all C–P flows, while ${\textit{Re}}_{\tau ,t}\lt 0$ for PC flow ( $\theta =0^{\circ }$ ) and ${\textit{Re}}_{\tau ,t}\gt 0$ for PP flow (not shown in the table). When $\theta$ increases in ( $0^{\circ }$ , $90^{\circ }$ ) with $Re$ specified, ${\textit{Re}}_{\tau ,t}$ must change sign at some critical $\theta _c$ . The skin Mach number and the wall heat flux have the same sign as the corresponding skin-friction Reynolds number. Two distinct flow regimes are then identified: one is the Couette regime (C regime), defined as ${\textit{Re}}_{\tau ,t}\lt 0$ , and the other is the Poiseuille regime (P regime), defined as ${\textit{Re}}_{\tau ,t}\gt 0$ . A key distinction lies in the existence of a velocity maximum between the walls in the P-regime flow, which is absent in the C-regime flow.

Table 1.

The DNS cases implemented. Name convention: M followed by the value of $\textit{Ma}_0$ ( $0.8$ , $1.5$ ), then T and finally the value of $\theta$ ( $15^{\circ }$ , $30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ ). Results of skin Mach numbers ( $\textit{Ma}_{\tau ,t}$ , $\textit{Ma}_{\tau ,b}$ ), skin-friction Reynolds numbers ( ${\textit{Re}}_{\tau ,t}$ , ${\textit{Re}}_{\tau ,b}$ ) and wall heat fluxes ( $B_{q,t}$ , $B_{q,b}$ ). Symbols for $\overline {T}$ in figures thereafter. Regime definitions: ${\textit{Re}}_{\tau ,t}\lt 0$ , C regime; ${\textit{Re}}_{\tau ,t}\gt 0$ , P regime.

Figure 2.

Mean flow features of turbulent C–P flows. (a,c) For C-regime case M0.8T15. (b,d) For P-regime case M0.8T60. Profiles of ( $a$ , $b$ ) $\overline {U}$ and $\overline {T}$ ; ( $c$ , $d$ ) $\text{d}\overline {T}/\text{d}\overline {U}$ . Two-zone modelling of $\text{d}\overline {T}/\text{d}\overline {U}$ with Zhang et al. (Reference Zhang, Bi, Hussain and She2014) based on dividing point at $y_1$ () or $y_2$ ().

2.2. Physical features and induced challenges

A key feature of compressible C–P flow is that the profiles are non-symmetric and complex, as shown in figure 2, which includes $\overline {U}$ and $\overline {T}$ distributions for a C-regime case ( $\theta =15^{\circ }$ ) of figure 2( $a{,}c$ ) and a P-regime case ( $\theta =60^{\circ }$ ) of figure 2( $b{,}d$ ).

For the C-regime case, the locations of temperature maximum $\overline {T}_{\textit{max}}$ and the minimum of velocity gradient $|\text{d}\overline {U}/\text{d}y|_{\textit{min}}$ , denoted as $y_1$ and $y_2$ , respectively, are positioned on opposite sides of centreline $y=0$ . This clearly shows a mismatch phenomenon between $\overline {T}$ and $\overline {U}$ . For the P-regime case of figure 2(b), $\overline {T}_{\textit{max}}$ and $\overline {U}_{\textit{max}}$ are also located on different sides, as shown. We note that due to the existence of $\overline {U}_{\textit{max}}$ , evaluation of $\text{d}\overline {T}/\text{d}\overline {U}$ , which is essential in modelling the T–V relation, shows a sharp discontinuity at the location of velocity maximum, as shown in figure 2(d) , which brings an essential challenge for temperature modelling. Such discontinuity disappears in C-regime flow due to the absence of maximum velocity between the plates, and the profile of $\text{d}\overline {T}/\text{d}\overline {U}$ is smooth as shown in figure 2(c).

A straightforward way of mean profile modelling is to directly apply the state-of-the-art theory, such as GRA by Zhang et al. (Reference Zhang, Bi, Hussain and She2014), for both sides of the channel, which results in two zones. The dividing point of this two-zone decomposition can be either $\overline {T}$ or $\overline {U}$ . Both dividing approaches are tested for modelling $\text{d}\overline {T}/\text{d}\overline {U}$ according to Zhang et al. (Reference Zhang, Bi, Hussain and She2014) for both P-regime and C-regime flows. Significant discrepancies from DNS data are observed, especially around the dividing point. In P-regime cases, the discontinuity at $y=y_2$ is not captured by the models. These deviations, which are independent of the selected dividing point, clearly indicate the defect of directly applying GRA theory.

3.1. Basic formula of the GRA theory

Assume a nominal zone enclosed by two boundaries: the near-wall turbulence boundary (denoted as ‘ $n$ ’) and the far-wall turbulence boundary (denoted as ‘ $f$ ’). In the context of a channel flow, ‘ $n$ ’ and ‘ $f$ ’ refer to the wall and the centre, respectively; while for a turbulent boundary layer, they correspond to the wall and the outer edge. Accordingly, the mean velocity and temperature at these boundaries are denoted by $\overline {U}_n$ , $\overline {U}_f$ and $\overline {T}_n$ , $\overline {T}_f$ , respectively. For clarity, we use non-dimensional velocity $\varXi$ and temperature $\varTheta$ :

(3.1) \begin{equation} \varXi = \frac {\overline {U}- \overline {U}_n}{\overline {U}_f-\overline {U}_n}, \quad \varTheta = \frac {\overline {T}-\overline {T}_n}{\overline {T}_f-\overline {T}_n}, \quad F_n = \frac { \overline {U}_f-\overline {U}_n }{\overline {T}_f-\overline {T}_n } \left . \frac {\partial \overline {T}}{\partial \overline {U}} \right |_n. \end{equation}

In the original GRA, by defining a recovery factor and further introducing an effective Prandtl number ${\textit{Pr}}_e$ , Zhang et al. (Reference Zhang, Bi, Hussain and She2014) developed an ordinary differential equation for the wall-bounded turbulence, which can be rewritten as

(3.2) \begin{equation} \varTheta = \frac 12 \varXi \left ( F_n + {\textit{Pr}}_e^{-1} \frac {\partial \varTheta }{\partial \varXi } \right )\!. \end{equation}

They further assume ${\textit{Pr}}_e=1$ based on existing DNS data and derive a quadratic T–V relation:

(3.3) \begin{equation} \varTheta - F_n( \varXi - \varXi ^2 ) - \varXi ^2 = 0, \end{equation}

where $F_n$ represents a non-dimensional ratio between the heat flux and nominal friction at bound $n$ . For a canonical channel, it recovers to the wall parameters. For an adiabatic wall, where the wall heat flux is zero, the relation further reduces to $\varTheta - \varXi ^2 = 0$ .

Figure 3.

Profile of ${\textit{Pr}}_e$ for C-regime case M0.8T15: (a) whole zone, (b) zone I, (c) zone II and (d) zone III. The solid red line represents ${\textit{Pr}}_e = 1$ . Profile of $\mathcal{E}$ for C-regime cases: (e) zone I, (f) zone II and (g) zone III.

3.2. Three-zone modelling of C-regime flow

According to figure 2(c), two-zone modelling of $\text{d}\overline {T}/\text{d}\overline {U}$ is clearly insufficient. A natural way is to divide the whole channel into three zones, denoted I, II and III. The dimensions are:

1. (i) $-h\lt y\lt y_1$ , from bottom wall $y_n=-h$ to location of $\overline {T}_{\textit{max}}$ at $y_f=y_1$ ,

2. (ii) $y_1\lt y\lt y_2$ , from $y_n=y_1$ to the location of $|\text{d}\overline {U}/\text{d}y|_{\textit{min}}$ at $y_f=y_2$ ,

3. (iii) $y_2\lt y\lt h$ , from top wall at $y_n=h$ to the location of $|\text{d}\overline {U}/\text{d}y|_{\textit{min}}$ at $y_f=y_2$ .

Physically, zone I covers the temperature boundary layer and zone III is the velocity boundary layer. While zone II is a virtual boundary layer with ‘wall’ at $y_1$ and outer edge at $y_2$ . As described above, ${\textit{Pr}}_e$ is a convenient parameter in understanding the profile of temperature and can be computed inversely using the ordinary differential equation (3.2). The distribution ${\textit{Pr}}_e$ for C-regime case M0.8T15 is shown in figure 3(a). Note that for the three zones, $\varXi$ , $\varTheta$ and $F_n$ are defined by local reference physical quantities $\overline {U}_n$ , $\overline {U}_f$ , $\overline {T}_n$ and $\overline {T}_f$ . It is observed that generally ${\textit{Pr}}_e$ is around $1.0$ in the three zones with some deviation in zone I. Fortunate information as described by Zhang et al. (Reference Zhang, Bi, Hussain and She2014) is that the deviation of ${\textit{Pr}}_e$ from unity has a negligible effect on the final T–V relation. Figures 3(b), 3(c) and 3(d) show ${\textit{Pr}}_e$ for zones I, II and III, respectively. Each figure includes all four C-regime cases with varying $\textit{Ma}_0$ and $\theta$ . To better quantify the effects of ${\textit{Pr}}_e$ variations, a sensitivity study of (3.2) is conducted for each zone using an approach similar to that of Chen et al. (Reference Chen, Gan and Fu2025). Specifically, we compute the global variation of $\varTheta$ in each zone when ${\textit{Pr}}_e$ at a specific $y_0$ is perturbed:

(3.4) \begin{equation} \mathcal{E} (y_0) = \frac {1}{\int ^{1}_{0}\varTheta \text{d}\varXi } \int ^{1}_{0}\left | \frac {\partial \varTheta \big({\textit{Pr}}_e^{-1};\varXi \big)}{\partial {\textit{Pr}}_e^{-1}(y_0)} \right | \text{d} \varXi , \end{equation}

where $\varTheta ({\textit{Pr}}_e^{-1};\varXi )$ represents the solution of (3.2) on the $\varXi$ grid, under the far-wall boundary condition $\varTheta |_{\varXi =1}=1$ . The values of $\mathcal{E}$ of C-regime flow are plotted in figure 3(e–g). It shows that $\mathcal{E}$ is primarily amplified at $\varXi \gt 0.6$ , where ${\textit{Pr}}_e$ is close to unity. This generally implies that the unity- ${\textit{Pr}}_e$ assumption can be used for C-regime flow.

The T–V relation for C-regime flow can be constructed by directly using (3.3):

(3.5) \begin{align} \text{I:} \quad \frac {\overline {T}}{\overline {T}_1} &= \frac {T_w}{\overline {T}_1} + \frac {\overline {U}_1 + U_w}{\overline {T}_1} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{b} \frac {\overline {U} + U_w}{\overline {U}_1 + U_w} + \left ( \frac {\overline {T}_1 - T_w}{\overline {T}_1} - \frac {\overline {U}_1 + U_w}{\overline {T}_1} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{b} \right )\nonumber\\&\quad \times \left ( \frac {\overline {U} + U_w}{\overline {U}_1 + U_w} \right )^2 , \nonumber \\ \text{II:} \quad \frac {\overline {T}}{\overline {T}_2} &= \frac {\overline {T}_1}{\overline {T}_2} + \frac {\overline {T}_2 - \overline {T}_1}{\overline {T}_2} \left ( \frac {\overline {U} - \overline {U}_1}{\overline {U}_2 - \overline {U}_1} \right )^2, \nonumber \\ \text{III:} \quad \frac {\overline {T}}{\overline {T}_2} &= \frac {T_w}{\overline {T}_2} + \frac {\overline {U}_2 - U_w}{\overline {T}_2} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{t} \frac {\overline {U} - U_w}{\overline {U}_2 - U_w} + \left ( \frac {\overline {T}_2 - T_w}{\overline {T}_2} - \frac {\overline {U}_2 - U_w}{\overline {T}_2} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{t} \right )\nonumber \\&\quad \times \left ( \frac {\overline {U} - U_w}{\overline {U}_2 - U_w} \right )^2 . \end{align}

Here, $\overline {T}_i$ and $\overline {U}_i$ denote the mean temperature and velocity at $y_i$ ( $i = 1,2$ ).

Figure 4.

Profile of ${\textit{Pr}}_e$ for P-regime case M0.8T60. (a) Whole zone, (b) zone I, (c) zone II and (d) zone III. : (3.7); : (3.8).

3.3. Three-zone modelling of P-regime flow

Considering the singularity of $\text{d}\overline {T}/\text{d}\overline {U}$ near the maximum velocity, modelling of P-regime flow cannot be completed based on the existing GRA theory, and more considerations regarding the specific P-regime flow are needed. We first still divide the channel into three zones based on the locations of temperature maximum and velocity maximum:

1. (i) $-h\lt y\lt y_1$ , from the bottom wall at $y_n=-h$ to location of $\overline {T}_{\textit{max}}$ at $y_f=y_1$ ,

2. (ii) $y_1\lt y\lt y_2$ , from $y_n=y_1$ to location of $\overline {U}_{\textit{max}}$ at $y_f=y_2$ ,

3. (iii) $y_2\lt y\lt h$ , from the top wall at $y_n=h$ to the location of $\overline {U}_{\textit{max}}$ at $y_f=y_2$ .

The distribution ${\textit{Pr}}_e$ for P-regime case M0.8T60 is shown in figure 4(a). It shows that zone I still accords with the GRA theory. While ${\textit{Pr}}_e$ in both zones II and III exhibits infinity at $y_2$ , indicating the failure of the constant-Prandtl-number assumption for the present case. We note that recently the constant- ${\textit{Pr}}_e$ assumption has been argued for other flows (Chen et al. Reference Chen, Gan and Fu2025).

For zones II and III, we note that zone III has a solid wall at $y=h$ with wall heat flux while zone II has a nominal ‘wall’ at $y=y_1$ with zero heat flux. Considering the common $\text{d}\overline {T}/\text{d}\overline {U}$ at $y_2$ , zone II can be considered as a special case of zone III. Note that for velocity, unlike the near-wall scaling of $\overline {U}/u_\tau \propto yu_\tau$ , the profile near $\text{d}\overline {U}/\text{d}y=0$ generally shows $\overline {U}/u_p \propto (yu_p)^2$ according to Cheng et al. (Reference Cheng, Pullin, Samtaney and Luo2023) with $u_p$ some measurement of characteristic velocity. For temperature, as described by Zhang, Song & Xia (Reference Zhang, Song and Xia2024), a linear form of $\overline {T} \propto y$ is found around the zero-velocity-gradient region. Thus, the classic T–V relation is no longer valid for the region around the far-from-wall edge. This generally suggests a quadratic relation between $\varXi$ and $\varTheta$ and gives the following form:

(3.6) \begin{equation} \varTheta + m \varTheta ^2 - F_n( \varXi - \varXi ^2 ) - (1+m) \varXi ^2 = 0, \end{equation}

due to the conditions $\varTheta =1$ and $\varXi =1$ at $y=y_2$ . Here $m$ is a parameter to quantify the relationship between $\varXi$ and $\varTheta$ around the far-from-wall edge. We note it also requires the maximum $\overline {T}$ to occur at the boundary of the zone. Applying (3.6) in the ordinary differential equation gives an analytical ${\textit{Pr}}_e$ :

(3.7) \begin{equation} {\textit{Pr}}_e = \frac {\varXi \left [F_n\left (1-2\varXi \right )+2(1+m)\varXi \right ] }{\left (2\varTheta -F_n\varXi \right ) \left (1+2m\varTheta \right ) }. \end{equation}

With $m = -1/2$ , ${\textit{Pr}}_e$ approaches $+\infty$ at $y_2$ . For zone II with $F_n=0$ , the T–V relation reduces to

(3.8) \begin{equation} \varTheta - \frac {1}{2} \varTheta ^2 - \frac {1}{2} \varXi ^2 = 0 , \quad \quad {\textit{Pr}}_e = \frac { \varXi ^2}{ 2 \varTheta (1-\varTheta ) }. \end{equation}

The above expressions for ${\textit{Pr}}_e$ are also shown in figure 4(a) with $m=-1/2$ . It can be found that this produced tendency ${\textit{Pr}}_e(y)$ reasonably follows the DNS data profile in both zones. Further, figures 4(b), 4(c) and 4(d) show ${\textit{Pr}}_e$ for zones I, II and III, respectively, for all four P-regime cases with varying $\textit{Ma}_0$ and $\theta$ . The predictions of ${\textit{Pr}}_e$ given by the model are basically consistent with the DNS data.

The T–V relations for P-regime flow can be formulated considering wall velocity $U_w$ :

(3.9) \begin{align} \text{I:} \quad &\frac {\overline {T}}{\overline {T}_1} = \frac {T_w}{\overline {T}_1} + \frac {\overline {U}_1 + U_w}{\overline {T}_1} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{b} \frac {\overline {U} + U_w}{\overline {U}_1 + U_w} + \left ( \frac {\overline {T}_1 - T_w}{\overline {T}_1} - \frac {\overline {U}_1 + U_w}{\overline {T}_1} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{b} \right ) \nonumber \\&\qquad\,\, \times \left ( \frac {\overline {U} + U_w}{\overline {U}_1 + U_w} \right )^2, \nonumber \\ \text{II:} \quad & \left ( \frac {\overline {T} - \overline {T}_2}{\overline {T}_1 - \overline {T}_2} \right )^2 + \left ( \frac {\overline {U} - \overline {U}_1}{\overline {U}_2 - \overline {U}_1} \right )^2 = 1, \nonumber \\ \text{III:} \quad & \left ( \frac {\overline {T} - \overline {T}_2}{T_w - \overline {T}_2} \right )^2 + \left ( \frac {\overline {U} - U_w}{\overline {U}_2 - U_w} \right )^2 - \frac {2}{\overline {T}_2 - T_w} \frac {\partial \overline {T}}{\partial \overline {U}}\bigg |_{t} \frac {(\overline {U} - U_w)(\overline {U} - \overline {U}_2)}{\overline {U}_2 - U_w} = 1 . \end{align}

4.1. The $\overline {T}$ comparison for $\textit{Ma}_0=0.8$ and $1.5$

Figure 5 shows the comparison for the C-regime flows ( $\theta =15^{\circ },30^{\circ }$ ). We observe that the present model for C-regime flow, which employs (3.5), exhibits a notable improvement as expected, which attains a close agreement with the DNS results. Figure 6 shows the comparison between the DNS data of P-regime flow ( $\theta =45^{\circ },60^{\circ }$ ) and the models. It can be found that all existing theories failed in zones II and III, especially near $y_2$ where $\text{d}\overline {T}/\text{d}\overline {U}$ approaches $\pm \infty$ . Figure 7 shows the comparison of $\overline {T}/T_w$ versus $\overline {U}/U_{0}$ for C-regime flow (M $0.8$ T $15$ ) and P-regime flow (M $0.8$ T $60$ ). In contrast, the new T–V relation (3.9) overcomes the challenge and smoothly captures the $\overline {T}$ gradient at $y_2$ , providing a better prediction of $\overline {T}$ .

Figure 7.

Comparison of $\overline {T}/T_w$ versus $\overline {U}/U_{0}$ for turbulent C-P flows: (a) M $0.8$ T $15$ ; (b) M $0.8$ T $60$ . The solid red line represents the present model. See figure 5 for the line keys.

4.2. The P-regime cases at $\textit{Ma}_0=2.0$ and $2.5$

In order to test the generality of the new T–V relation (3.9), two DNS cases with higher Mach number for turbulent P-regime flows at $\theta =60^{\circ }$ with $Re=4000$ and $\textit{Ma}_0=2.0$ , $2.5$ are performed here. The domain is $6\pi h \times 2h \times 2\pi h$ for the two cases, which is based on Yao & Hussain (Reference Yao and Hussain2020), and the grid is $384\times 210\times 256$ . Table 2 lists some statistical quantities, including $\textit{Ma}_{\tau ,t}$ , $\textit{Ma}_{\tau ,b}$ , ${\textit{Re}}_{\tau ,t}$ , ${\textit{Re}}_{\tau ,b}$ , $B_{q,t}$ and $B_{q,b}$ . Figure 8 shows the comparison between the DNS data of P-regime flow ( $\textit{Ma}_0=2.0$ and $2.5$ ) and the models. We note that for the case with $\textit{Ma}_0 = 2.5$ , the maximum mean temperature is generally twice the wall temperature. This highlights the significant influence of compressibility on the flow field. Given this observation, the newly proposed T–V relation still achieves accurate prediction of the mean temperature profiles across the current Mach number range, partially demonstrating the present model’s applicability.

Table 2.

The DNS cases implemented. Results of skin Mach numbers ( $\textit{Ma}_{\tau ,t}$ , $\textit{Ma}_{\tau ,b}$ ), skin-friction Reynolds numbers ( ${\textit{Re}}_{\tau ,t}$ , ${\textit{Re}}_{\tau ,b}$ ) and wall heat fluxes ( $B_{q,t}$ , $B_{q,b}$ ). Symbols for $\overline {T}$ in figures thereafter.

Figure 8.

Comparison of $\overline {T}/T_w$ for P-regime flows: (a) M $2.0$ T $60$ ; (b) M $2.5$ T $60$ . The solid red line represents the present model (3.9). See figure 5 for the line keys.

4.3. Error distribution

For more quantitative assessments, the relative error of temperature prediction $\varepsilon _{T}$ and integrated relative error $\varepsilon$ are defined as

(4.1) \begin{equation} \varepsilon _{T}(y)=\left \lvert {\overline {T}_{\textit{DNS}}-\overline {T}_{\textit{Pre}}}\right \rvert / {\overline {T}_{\textit{DNS}}} \times 100 \,\%, \quad \varepsilon = \int _{-h}^{h} \varepsilon _{T}(y) \,\text{d}y, \end{equation}

with $\overline {T}_{\textit{Pre}}$ being the profile predicted by models and $\overline {T}$ the results from DNS. The distribution of $\varepsilon _{T}$ for P-regime case M0.8T60 is shown in figure 9(a). The maximum value of $\varepsilon _{T}$ for the present model remains below $0.5\,\%$ overall, and mainly occurs in zone III. While others are generally above $1\,\%$ . The integrated relative error is shown in figure 9(b). Compared with existing theories, the present model predicts the temperature profile with smaller relative error, with $\varepsilon$ less than 1 %, indicating the present modelling provides a better collapse in temperature scaling in compressible C–P turbulence, especially in P-regime flow.

Figure 9.

(a) Relative error $\varepsilon _{T}$ for P-regime case M0.8T60 and (b) integrated relative error $\varepsilon$ in temperature prediction for C–P flows: , present model. Two-zone modelling with other T–V relations: , Walz (Reference Walz1962); , Duan & Martin (Reference Duan and Martin2011); , Zhang et al. (Reference Zhang, Bi, Hussain and She2014).