2.1. Governing equations

The three-dimensional compressible Navier–Stokes equations for an ideal gas are considered, as follows:

(2.1) \begin{align} \frac {\partial \rho }{\partial t}+\frac {\partial (\rho u_{j})}{\partial x_{j}} &= 0, \end{align}

(2.2) \begin{align} \frac {\partial \rho u_i}{\partial t}+\frac {\partial (\rho u_iu_{\!j}+p\delta _{\textit{ij}})}{\partial x_j} &= \frac {\partial \sigma _{\textit{ij}}}{\partial x_j}, \end{align}

(2.3) \begin{align} \frac {\partial \rho e_{t}}{\partial t}+\frac {\partial (\rho e_{t} u_{j} + p u_{j})}{\partial x_{j}} &= \frac {\partial (u_{i}\sigma _{\textit{ij}}-q_{j})}{\partial x_{j}}. \end{align}

In the equations presented, $\rho$ is the density, and $p$ is the static pressure given by the perfect gas equation of state $p=\rho R T$ , where $R$ is the gas constant. The symbol $u_{i}\ (i=1,2,3 )$ represents the velocity vector components in the streamwise $x$ -direction ( $u$ ), wall-normal $y$ -direction ( $v$ ), and spanwise $z$ -direction ( $w$ ), respectively. The symbol $e_{t} = e + u_{i}u_{i}/2$ is the specific total energy, and $e = c_v T$ is the specific internal energy, where $c_v = R / (\gamma - 1)$ is the specific heat at constant volume, and $\gamma = 1.4$ is the specific heat ratio. The symbols $\sigma _{\textit{ij}}$ in (2.2) and $q_{j}$ in (2.3), which denote the viscous stress tensor and the conductive heat flux, respectively, are given by

(2.4a,b) \begin{equation} \sigma _{\textit{ij}} = \mu \left (\frac {\partial u_{i}}{\partial x_{j}}+\frac {\partial u_{j}}{\partial x_{i}}\right )-\frac 23\mu \frac {\partial u_{k}}{\partial x_{k}}\delta _{\textit{ij}}, \quad q_{j} = -\lambda \frac {\partial T}{\partial x_{j}}, \end{equation}

where $\delta _{\textit{ij}}$ denotes the Kronecker symbol. The dynamic viscosity $\mu$ is calculated using Sutherland’s law:

(2.5) \begin{equation} \frac {\mu }{\mu _{\textit{ref}}} = \frac {T_{\textit{ref}} + S}{T + S}\left ( \frac {T}{T_{\textit{ref}}} \right )^{3/2}, \end{equation}

where $S = 110.4\ \text{K}$ is the Sutherland constant for air, and $T_{\textit{ref}}$ and $\mu _{\textit{ref}}$ are the reference temperature and corresponding dynamic viscosity, respectively. The thermal conductivity $\lambda$ is determined from the constant Prandtl number $\textit{Pr}= c_p \mu / \lambda = 0.71$ , where $c_p = \gamma R/(\gamma - 1)$ is the specific heat at constant pressure.

2.2. The DNS database

The spatially evolving zero-pressure-gradient flat-plate CTBLs are investigated in this research. The DNS of the CTBLs were conducted using the open-source code STREAmS, developed by Bernardini et al. (Reference Bernardini, Modesti, Salvadore and Pirozzoli2021). Detailed numerical methods for the DNS can be retrieved from Yu et al. (Reference Yu, Zhao, Yuan and Xu2024b ), where the validation of the DNS data is conducted in supplementary Appendix A.

The present CTBL simulations comprise three adiabatic cases and two wall-cooling cases, designed to investigate the effects of Mach number and wall cooling on the turbulent transport processes. For each case, the wall temperature $T_w$ is temporally constant and spatially uniform. The adiabatic cases are set at three Mach numbers, $\textit{Ma}_e =2, 4, 6$ , and the adiabatic wall temperature is set as the recovery temperature of the boundary layer, $T_r = T_e [1 + (\gamma - 1) r\, Ma_e^2 /2 ]$ , where $r = Pr^{1/3}$ denotes the recovery factor. Throughout this paper, subscripts $w$ and $e$ denote the wall parameter and the boundary layer edge parameter, respectively. Two cooled walls are set with temperature ratios $T_w/T_r = 0.5$ and $T_w/T_r = 0.25$ at the hypersonic condition $\textit{Ma}_e = 6$ . The wall-cooling configurations represent weak and moderate cooling conditions, respectively, where the classification of wall-cooling conditions follows the definition by Gibis et al. (Reference Gibis, Sciacovelli, Kloker and Wenzel2024).

The entire computational domain is a cuboid aligned with the wall surface. Periodic boundary conditions are employed in the spanwise direction of the computational domain. The streamwise and wall-normal outflow planes are set with unreflected boundary conditions. The inlet flow is composed of the empirical mean flow profile (Musker Reference Musker1979) and the synthetic turbulent fluctuations using digital filtering methods (Klein, Sadiki & Janicka Reference Klein, Sadiki and Janicka2003). The friction Reynolds number ${\textit{Re}}_{\tau } = \rho _{w} u_{\tau } \delta / \mu _{w}$ at the inlet is set as 200, according to which the freestream Reynolds number is calculated. The symbol $u_{\tau } = \sqrt {{{\tau }_{w}}/{\rho _{w}}}$ is the friction velocity, $\delta _{\nu } = \mu _w / (\rho _w u_{\tau })$ is the viscous length scale, and ${\tau }_{w} = \mu _{w}\,({\partial {\overline {u}}/\partial {y}})_{w}$ is the wall shear stress. The no-slip boundary condition for velocity is applied at the wall.

The size of the computational domain is $L_x \times L_y \times L_z = 80\delta _{\textit{in}} \times 9\delta _{\textit{in}} \times 8\delta _{\textit{in}}$ , and the grid number is $N_x \times N_y \times N_z = 2400 \times 280 \times 280$ , where $\delta _{\textit{in}}$ represents the thickness of the boundary layer at the inlet. The grids are distributed uniformly in the streamwise and spanwise directions, and stretched using a hyperbolic sine function in the wall-normal direction, where 240 grids are clustered below $y = 2.5 \delta _{\textit{in}}$ . Hereinafter, the results reported are obtained at the streamwise location $65\delta _{\textit{in}}$ , where the turbulent boundary layer is fully developed. More information about the DNS data at the selected streamwise location, including the Reynolds numbers based on the nominal boundary thickness ( ${\textit{Re}}_\delta$ ), the displacement thickness ( ${\textit{Re}}_{\delta ^{*}}$ ), and the semi-local viscous scaling ( ${\textit{Re}}_{\tau }^{*}$ ), can be found in table 1. The non-dimensional grid intervals in three directions, $\Delta x^{+}$ , $\Delta y_{w}^{+}$ and $\Delta z^{+}$ , are also provided in table 1, where the superscript $+$ indicates the viscous scaling. The semi-local viscous scaling is indicated by superscript $\ast$ , where the semi-local friction velocity and viscous length scale are defined as $u_{\tau }^{*} = \sqrt {\tau _w / \overline {\rho }}$ and $\delta _{\nu }^{*} = \overline {\mu }/(\overline {\rho } u_{\tau }^{*})$ , respectively. The statistical results in the research are averaged over the spanwise range and the time span $240\delta _{\textit{in}}/u_{e}$ . The data are collected after the simulations have been run for $1000\delta _{\textit{in}}/u_{e}$ , by which time the turbulent flows are fully developed.

Table 1.

Dimensionless boundary layer parameters at the selected streamwise location. Here, ${\textit{Re}}_{\delta } = \rho _eU_e\delta /\mu _e$ is the Reynolds number based on the local nominal thickness of the boundary layer $\delta$ , ${\textit{Re}}_{\delta ^{*}} = \rho _eU_e\delta ^{*}/\mu _e$ is the Reynolds number based on the local displacement thickness $\delta ^{*}$ , ${\textit{Re}}_{\tau } = {\rho }_{w}u_{\tau }\delta /{\mu }_{w}$ is the friction Reynolds number, ${\textit{Re}}_{\tau }^{*} = \sqrt {{\rho }_{e} \tau _{w}}\, \delta /{\mu }_{e}$ is the semi-local scaled friction Reynolds number, $H = \delta ^* / \theta$ is the shape factor of the boundary layer, $\theta$ is the local momentum thickness, $\varTheta = ({T}_w - T_e)/({T}_r - T_e)$ is the diabatic parameter, and $\Delta x^{+}$ , $\Delta y_{w}^{+}$ and $\Delta z^{+}$ are dimensionless grid sizes with viscous scaling in the $x$ -direction, $y$ -direction at the wall and $z$ -direction, respectively. The line styles for different cases are also indicated.

4.1. Transport equations

The transport equation for the Reynolds stress tensor $\widetilde {u_i^{\prime \prime} u_{\!j}^{\prime \prime}}$ in compressible flows is written as (Gatski & Bonnet Reference Gatski and Bonnet2013)

(4.2) \begin{equation} \overline {\rho }\frac {\tilde {\textrm{D}} \widetilde {u_{i}^{\prime \prime}u_{j}^{\prime \prime}}}{\tilde {\textrm{D}} t} = \frac {\partial \overline {\rho }\widetilde {u_{i}^{\prime \prime}u_{j}^{\prime \prime}}}{\partial t} + \frac {\partial \overline {\rho }\tilde {u}_{k}\widetilde {u_{i}^{\prime \prime}u_{j}^{\prime \prime}}}{\partial x_{k}} = P_{u_i u_{\!j}} + \varGamma _{u_i u_{\!j}} + D_{u_i u_{\!j}}^{v} - \varepsilon _{u_i u_{\!j}}^{v} + \varPi _{u_i u_{\!j}} + M_{u_i u_{\!j}}, \end{equation}

where

(4.3a \begin{align} P_{u_i u_{\!j}} &= -\left (\overline {\rho }\, \widetilde { u_{i}^{\prime \prime}u_{k}^{\prime \prime}} \frac {\partial \tilde {u}_{j}} {\partial x_{k}}+\overline {\rho }\, \widetilde { u_{j}^{^{\prime \prime}}u_{k}^{^{\prime \prime}}} \frac {\partial \tilde {u}_{i}}{\partial x_{k}}\right )\!, \end{align}

(4.3b \begin{align} \varGamma _{u_i u_{\!j}} &= -\frac {\partial \overline {\rho }\, \widetilde {\big(u_k^{\prime \prime}u_i^{\prime \prime}u_{\!j}^{\prime \prime}\big)}}{\partial x_k}, \end{align}

(4.3c \begin{align} D_{u_i u_{\!j}}^{v} &= \frac {\partial \Big( \overline {{u_{j}^{\prime }\sigma _{ik}^{\prime }}} + \overline {u_{i}^{\prime }\sigma _{jk}^{\prime }} \Big)}{\partial x_{k}}, \end{align}

(4.3d \begin{align} \varepsilon _{u_i u_{\!j}}^{v} &= \left (\overline {{\sigma _{ik}^{\prime }\frac {\partial u_{j}^{\prime }}{\partial x_{k}}}}+\overline {{\sigma _{jk}^{\prime }\frac {\partial u_{i}^{\prime }}{\partial x_{k}}}}\right )\!, \end{align}

(4.3e \begin{align} \varPi _{u_i u_{\!j}} &= -\left (\overline {u_{\!j}^{\prime } \frac {\partial p^\prime }{\partial x_i}} + \overline {u_i^{\prime } \frac {\partial p^\prime }{\partial x_j}}\right )\!, \end{align}

(4.3f \begin{align} M_{u_i u_{\!j}} &= \overline {u_{i}^{\prime \prime}}\frac {\partial \left (\overline {\sigma _{jk}}-\overline {p}\delta _{jk}\right )}{\partial x_{k}}+\overline {u_{j}^{\prime \prime}}\frac {\partial \big(\overline {\sigma _{ik}}-\overline {p}\delta _{ik}\big)}{\partial x_{k}}. \end{align}

The budget terms on the right-hand side of (4.2) include the turbulent production $P_{u_i u_{\!j}}$ , turbulent transport $\varGamma _{u_i u_{\!j}}$ , viscous diffusion $D_{u_i u_{\!j}}^{v}$ , viscous dissipation $\varepsilon _{u_i u_{\!j}}^{v}$ , pressure term $\varPi _{u_i u_{\!j}}$ , and mass flux contribution $M_{u_i u_{\!j}}$ .

Figure 3.

Streamwise Reynolds stress budget terms with semi-local scaling. The budget terms (distinguished by different colours) include the turbulent production $P_{uu}$ (red), turbulent transport $\varGamma _{uu}$ (green), viscous diffusion $D_{uu}^v$ (blue), viscous dissipation $-\varepsilon _{uu}^v$ (orange), pressure term $\varPi _{uu}$ (cyan) and mass flux contribution $M_{uu}$ (purple). (a) Results for different Mach numbers without wall cooling: M2ad ( ), M4ad ( ), M6ad ( ). (b) Results for fixed Mach number with/without wall cooling: M6ad ( ), M6Tw05 ( ), M6Tw025 ( ). The legend for the budget terms of M2ad is presented in (a).

Figure 4.

Reynolds shear stress budget terms with semi-local scaling. The budget terms include the turbulent production $P_{uv}$ (red), turbulent transport $\varGamma _{uv}$ (green), viscous diffusion $D_{uv}^v$ (blue), viscous dissipation $-\varepsilon _{uv}^v$ (orange), pressure term $\varPi _{uv}$ (cyan) and mass flux contribution $M_{uv}$ (purple). (a) Results for different Mach numbers without wall cooling: M2ad ( ), M4ad ( ), M6ad ( ). (b) Results for fixed Mach number with/without wall cooling: M6ad ( ), M6Tw05 ( ), M6Tw025 ( ). The legend for the budget terms of M2ad is presented in (a).

The budget terms of streamwise Reynolds stress $\widetilde {{u^{\prime \prime}}^2}$ are ploted in figure 3 using semi-local scaling ( ${u_{\tau }^{*}}^3 / \delta _{\nu }^*$ ). The profiles of different cases exhibit good collapse. The increase in Mach number slightly raises the turbulent production $P_{uu}$ in the region $10 \lesssim y^* \lesssim 100$ , which Hasan et al. (Reference Hasan, Costa, Larsson, Pirozzoli and Pecnik2025) attribute to the variation of the turbulent shear stress with Mach number. Near $y^{*} = 1$ , the turbulent production $P_{uu}$ vanishes and the viscous diffusion $D_{uu}^{v}$ balances the dissipation $\varepsilon _{uu}^{v}$ . A similar behaviour is observed in the transport of $\widetilde {{T^{\prime \prime}}^2}$ under adiabatic wall conditions, as will be discussed later. In the region $y^{*} \gtrsim 50$ , the destruction mechanism of the pressure term $\varPi _{uu}$ becomes important, exhibiting a magnitude comparable to that of the viscous dissipation $\varepsilon _{uu}^{v}$ . Under wall-cooling conditions, the magnitude of the pressure term $\varPi _{uu}$ decreases, which implies that the ability of the pressure term to redistribute turbulent kinetic energy weakens with increasing wall cooling (Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023).

Figure 4 plots the budget terms of the Reynolds shear stress $\widetilde {\ u^{\prime \prime} v^{\prime \prime}}$ . The pressure term $\varPi _{uv}$ is the main term balancing the turbulent production $P_{uv}$ , which is similar to behaviour observed in the transport of $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ discussed later. A good collapse of the adiabatic cases across different Mach numbers is achieved with the semi-local scaling, but wall cooling disrupts this collapse near the wall. The turbulent production $P_{uv}$ , whose magnitude increases with stronger wall cooling in the region $y^{*} \lesssim 10$ , is directly linked to the increase in $v_{\textit{rms}}^*$ (as shown in supplementary Appendix A).

Figure 5.

Temperature variance budget terms with semi-local scaling. The budget terms include the turbulent production $P_{\textit{TT}}$ (red), turbulent transport $\varGamma _{\textit{TT}}$ (green), thermal diffusion $D_{\textit{TT}}^q$ (blue), thermal dissipation $-\varepsilon _{\textit{TT}}^q$ (orange), work contribution $W_{\textit{TT}}$ (yellow) and mass flux contribution $M_{\textit{TT}}$ (purple). The vertical lines denote the locations $y^{*} = y_{\textit{peak}}^*$ . (a) Results for different Mach numbers without wall cooling: M2ad ( ), M4ad ( ), M6ad ( ). (b) Results for weak wall cooling: M6Tw05 ( ). (c) Results for moderate wall cooling: M6Tw025 ( ). The legend for the budget terms of M2ad is presented in (a).

The transport equation for $\widetilde {{T^{\prime \prime}}^2}$ , derived from the mean internal energy equation, is written as (Gerolymos & Vallet Reference Gerolymos and Vallet2014)

(4.4) \begin{equation} \overline {\rho }\frac {\tilde {\textrm{D}} \widetilde {{T^{\prime \prime}}^2}}{\tilde {\textrm{D}} t} = \frac {\partial \overline {\rho }\widetilde {{T^{\prime \prime}}^2}}{\partial t} + \frac {\partial \overline {\rho }\tilde {u}_{k}\widetilde {{T^{\prime \prime}}^2}}{\partial x_{k}} = P_\textit{TT} + \varGamma _\textit{TT} + D_\textit{TT}^{q} - \varepsilon _\textit{TT}^{q} + W_\textit{TT} + M_\textit{TT}, \end{equation}

where

(4.5a \begin{align} P_\textit{TT} &= -2\overline {\rho }\, \widetilde { u_{k}^{\prime \prime}T^{\prime \prime}} \frac {\partial \tilde {T}} {\partial x_{k}}, \end{align}

(4.5b \begin{align} \varGamma _\textit{TT} &= -\frac {\partial \overline {\rho }\, \widetilde {u_k^{\prime \prime}{T^{\prime \prime}}^2}}{\partial x_k}, \end{align}

(4.5c \begin{align} D_\textit{TT}^{q} &= -\frac {2}{c_v}\frac {\partial \overline {{T^{\prime }q_{k}^{\prime }}}}{\partial x_{k}}, \end{align}

(4.5d \begin{align} \varepsilon _\textit{TT}^{q} &= -\frac {2}{c_v}\,\overline {{q_{k}^{\prime }\frac {\partial T^{\prime }}{\partial x_{k}}}}, \end{align}

(4.5e \begin{align} W_\textit{TT} &= \frac {2}{c_v}\,\overline {T^{\prime }\left [(\sigma _{ik}-p\delta _{ik})\frac {\partial u_{i}}{\partial x_{k}}\right ]^{\prime }}, \end{align}

(4.5f \begin{align} M_\textit{TT} &= \frac {2}{c_v}\overline {T^{\prime \prime}} \left[\overline {{\left (\sigma _{ik}-p\delta _{ik}\right )\frac {\partial u_{i}}{\partial x_{k}}}}-\overline {{\frac {\partial q_{k}}{\partial x_{k}}}}\right ]\!. \end{align}

The budget terms on the right-hand side of (4.4) include the turbulent production $P_{\textit{TT}}$ , turbulent transport $\varGamma _{\textit{TT}}$ , thermal diffusion $D_{\textit{TT}}^{q}$ , thermal dissipation $\varepsilon _{\textit{TT}}^{q}$ , work contribution $W_{\textit{TT}}$ , and mass flux contribution $M_{\textit{TT}}$ . The work contribution $W_{\textit{TT}}$ is the primary term that causes temperature to behave differently from a passive scalar in the transport of $\widetilde {{T^{\prime \prime}}^2}$ . A further decomposition analysis of $W_{\textit{TT}}$ is provided in supplementary Appendix B.

Figure 5 plots the budget terms of the temperature variance $\widetilde {{T^{\prime \prime}}^2}$ . With semi-local scaling ( ${u_{\tau }^{*}}^5 /(R^2 \delta _{\nu }^*)$ ), the adiabatic cases for different Mach numbers show good collapse in the region $y^{*} \lesssim 10$ , where the balance between the thermal dissipation $\varepsilon _{\textit{TT}}^q$ and diffusion $D_{\textit{TT}}^q$ is the dominant physical mechanism. In the region $y^{*} \gtrsim 10$ , the turbulent production $P_{\textit{TT}}$ is the dominant term that balances the thermal dissipation $\varepsilon _{\textit{TT}}^q$ , similar to the transport of $\widetilde {{u^{\prime \prime}}^2}$ (shown in figure 3). A notable difference in the transport of $\widetilde {{T^{\prime \prime}}^2}$ is the non-negligible work contribution term $W_{\textit{TT}}$ , which acts as a secondary generation mechanism. The slight increase in $P_{\textit{TT}}$ can be attributed to the rise in the wall-normal heat flux $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ with increasing Mach number (Gibis et al. Reference Gibis, Sciacovelli, Kloker and Wenzel2024). Under wall-cooling conditions, the magnitudes of the budget terms significantly decrease, leading to a failure of collapse between the diabatic cases and adiabatic cases. In the near-wall region where $y^{*} \lesssim 10$ , the work contribution $W_{\textit{TT}}$ becomes the dominant term that balances the losses from the thermal dissipation $\varepsilon _{\textit{TT}}^q$ and thermal diffusion $D_{\textit{TT}}^q$ . The turbulent production $P_{\textit{TT}}$ vanishes at the location $y^{*} = y_{\textit{peak}}^*$ where the thermal diffusion $D_{\textit{TT}}^{q}$ dominates the generation of $\widetilde {{T^{\prime \prime}}^2}$ . This diffusion-dominated generation mechanism near $y^{*} = y_{\textit{peak}}^*$ may explain the previously observed isotropic behaviour of the temperature fluctuations, which exhibit no organised patterns under wall-cooling conditions (Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023).

The transport equation for the turbulent heat flux $\widetilde {u_i^{\prime \prime} T^{\prime \prime}}$ is written as (Gerolymos & Vallet Reference Gerolymos and Vallet2014)

(4.6) \begin{equation} \overline {\rho }\frac {\tilde {\textrm{D}} \widetilde {u_{i}^{\prime \prime}{T^{\prime \prime}}}}{\tilde {\textrm{D}} t} = \frac {\partial \overline {\rho }\widetilde {u_{i}^{\prime \prime}{T^{\prime \prime}}}}{\partial t} \!+\! \frac {\partial \overline {\rho }\tilde {u}_{k}\widetilde {u_{i}^{\prime \prime}{T^{\prime \prime}}}}{\partial x_{k}} = P_{u_{i} T} \!+\! \varGamma _{u_{i} T} \!+\! D_{u_{i} T}^{v-q} - \varepsilon _{u_{i} T}^{v-q} \!+\! W_{u_{i} T} \!+\! \varPi _{u_{i} T} \!+\! M_{u_{i} T}, \end{equation}

where

(4.7a \begin{align} P_{u_{i} T} &= -\left (\overline {\rho }\,\widetilde {u_{k}^{\prime \prime}T^{\prime \prime}} \frac {\partial \tilde {u_{i}}}{\partial x_{k}}+\overline {\rho }\,\widetilde {u_{i}^{\prime \prime}u_{k}^{\prime \prime}} \frac {\partial \tilde {T}}{\partial x_{k}}\right )\!, \end{align}

(4.7b \begin{align} \varGamma _{u_{i} T} &= -\frac {\partial \overline {\rho } \widetilde {\left(u_{k}^{\prime \prime}u_{i}^{\prime \prime}T^{\prime \prime}\right)}}{\partial x_{k}}, \end{align}

(4.7c \begin{align} D_{u_{i} T}^{v-q} &= \frac {\partial \overline {T^{\prime }\sigma _{ik}^{\prime }}}{\partial x_{k}} - \frac {1}{c_v}\frac {\partial \overline {u_{i}^{\prime }q_{k}^{\prime }}}{\partial x_{k}}, \end{align}

(4.7d \begin{align} \varepsilon _{u_i T}^{v-q} &= \overline {{\sigma _{ik}^{\prime }\frac {\partial T^{\prime }}{\partial x_{k}}}} - \frac {1}{c_v}\, \overline {{q_{k}^{\prime }\frac {\partial u_{i}^{\prime }}{\partial x_{k}}}}, \end{align}

(4.7e \begin{align} \varPi _{u_i T} &= -\overline {{T^{\prime }\frac {\partial p^{\prime }}{\partial x_{i}}}}, \end{align}

(4.7f \begin{align} W_{u_{i} T} &= \frac {1}{c_v}\, \overline {{u_{i}^{\prime }\left [(\sigma _{jk}-p\delta _{jk})\frac {\partial u_{j}}{\partial x_{k}}\right ]^{\prime }}}, \end{align}

(4.7g \begin{align} M_{u_{i} T} &= \overline {{T^{\prime \prime}}}\frac {\partial \overline {\sigma _{ik}-p\delta _{ik}}}{\partial x_{k}} + \frac {1}{c_v} \overline {{u_{i}^{\prime \prime}}}\left [\overline {{(\sigma _{jk}-p\delta _{jk})\frac {\partial u_{j}}{\partial x_{k}}}}-{\frac {\partial \overline {q_{k}}}{\partial x_{k}}}\right ]\!. \end{align}

The budget terms on the right-hand side of (4.6) include the turbulent production $P_{u_i T}$ , turbulent transport $\varGamma _{u_i T}$ , viscous-thermal diffusion $D_{u_i T}^{v-q}$ , viscous-thermal dissipation $\varepsilon _{u_i T}^{v-q}$ , pressure term $\varPi _{u_i T}$ , work contribution $W_{u_i T}$ , and mass flux contribution $M_{u_i T}$ . Similar to $W_{\textit{TT}}$ in (4.5e ), the work contribution $W_{u_iT}$ is the primary term that introduces non-passive scalar behaviour in the transport of $\widetilde {u_i^{\prime \prime} T^{\prime \prime}}$ , and its decomposition is also discussed in supplementary Appendix B.

Figure 6.

Streamwise heat flux budget terms with semi-local scaling. The budget terms include the turbulent production $P_{uT}$ (red), turbulent transport $\varGamma _{uT}$ (green), thermal diffusion $D_{uT}^{v-q}$ (blue), thermal dissipation $-\varepsilon _{uT}^{v-q}$ (orange), pressure term $\varPi _{uT}$ (cyan), work contribution $W_{uT}$ (yellow) and mass flux contribution $M_{uT}$ (purple). The vertical lines denote the locations $y^{*} = y_{\textit{peak}}^*$ . (a) Results for different Mach numbers without wall cooling: M2ad ( ), M4ad ( ), M6ad ( ). (b) Results for weak wall cooling: M6Tw05 ( ). (c) Results for moderate wall cooling: M6Tw025 ( ). The legend for the budget terms of M2ad is presented in (a).

Figure 6 plots the budget terms of the streamwise turbulent heat flux $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ with semi-local scaling ( ${u_{\tau }^{*}}^4 /(R \delta _{\nu }^*)$ ). Under the adiabatic wall condition, a good collapse of the results is achieved across different Mach numbers, as shown in figure 6(a). In the viscous sublayer, the work contribution $W_{uT}$ acts as a destruction term and participates in the balance between the viscous-thermal diffusion $D_{u T}^{v-q}$ and dissipation $\varepsilon _{u T}^{v-q}$ . This phenomenon corresponds to the relatively small correlation coefficient $C_{u^{\prime \prime}T^{\prime \prime}}$ observed near the wall in adiabatic cases, as shown in figure 1(a), and further explained in $\S$ 5.1. In the remaining regions, the turbulent production $P_{uT}$ is almost the sole generation term for $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ . In the region $y^{*} \gtrsim 100$ , the pressure term $\varPi _{uT}$ is the main destruction term balancing the turbulent production $P_{uT}$ , implying that the pressure plays a key role in the velocity–temperature dissimilarity in the outer layer.

Figure 7.

Wall-normal heat flux budget terms with semi-local scaling. The budget terms include the turbulent production $P_{vT}$ (red), turbulent transport $\varGamma _{vT}$ (green), viscous diffusion $D_{vT}^{v-q}$ (blue), viscous dissipation $-\varepsilon _{vT}^{v-q}$ (orange), pressure term $\varPi _{vT}$ (cyan), work contribution $W_{vT}^{v-q}$ (yellow) and mass flux contribution $M_{uv}$ (purple). The vertical lines denote the locations $y^{*} = y_{\textit{peak}}^*$ . (a) Results for different Mach numbers without wall cooling: M2ad ( ), M4ad ( ), M6ad ( ). (b) Results for weak wall cooling: M6Tw05 ( ). (c) Results for moderate wall cooling: M6Tw025 ( ). The legend for the budget terms of M2ad is presented in (a).

As for the influence of the wall cooling, the signs of the turbulent heat fluxes change near $y^{*} = y^*_{\textit{peak}}$ , necessitating additional handling to incorporate this phenomenon in subsequent analysis. Here, $y_{\textit{peak}}^*$ (represented by the black vertical line in figure 6 b,c) is uniformly used to partition the boundary layer into two regions. The budget terms that have the same sign as the turbulent heat flux are labelled as the generation mechanism (gain), while those with the opposite sign are labelled as the destruction mechanism (loss). Under this partition, the turbulent production $P_{uT}$ always represents the generation mechanism. In the region where $y^{*}\lt y_{\textit{peak}}^*$ , the dominant generation mechanism comes from the work contribution $W_{uT}$ , which, together with the turbulent production $P_{uT}$ , balances the losses from the viscous-thermal dissipation $\varepsilon _{u T}^{v-q}$ and diffusion $D_{u T}^{v-q}$ . This phenomenon corresponds to the very high positive correlation between velocity and temperature fluctuations under wall-cooling conditions, as shown in figure 1(a). Additionally, the minimal turbulent production $P_{uT}$ leads to $C_{u^{\prime \prime}T^{\prime \prime}} \approx 0$ near $y^{*} = y_{\textit{peak}}^*$ .

Figure 7 plots the budget terms of the wall-normal turbulent heat flux $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ using semi-local scaling ( ${u_{\tau }^{*}}^4 /(R \delta _{\nu }^*)$ ). Similar to the other turbulent second-order moments, a collapse is achieved across different Mach numbers under the adiabatic wall condition. The profiles of the budget terms for $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ are very similar to those terms for $\widetilde {\ u^{\prime \prime}v^{\prime \prime}}$ shown in figure 4, with the turbulent production $P_{vT}$ and the pressure term $\varPi _{vT}$ nearly balancing each other in the region $y^{*} \gtrsim 10$ . Under the wall-cooling conditions, the boundary layer is likewise divided into two regions using $y^{*}_{\textit{peak}}$ , ensuring that the turbulent production $P_{vT}$ always serves as a generation mechanism. The work contribution $W_{vT}$ serves as the dominant generation term balancing the pressure term $\varPi _{vT}$ in the region where $y^{*} \lt y_{\textit{peak}}^*$ , similar to the behaviour of the work contribution terms for $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ and $\widetilde {{T^{\prime \prime}}^2}$ . Importantly, the work contribution $W_{vT}$ near the wall is a key contributor to the negative values of $C_{{v^{\prime \prime}}{T^{\prime \prime}}}$ under wall-cooling conditions observed in figure 1(b), explaining its different behaviour from the incompressible turbulent boundary layer results in the near-wall region. Near the location $y^{*} = y_{\textit{peak}}^*$ , the minimal turbulent production $P_{vT}$ results in $C_{{v^{\prime \prime}}{T^{\prime \prime}}} \approx 0$ .

4.2. Time scales for transport processes

It is informative to evaluate and compare the different time scales associated with the transport processes of the statistical moments. The time scale for the budget term $\varPsi$ of the statistical moment $\varPhi$ is defined as

(4.8) \begin{equation} t_{\varPsi , \varPhi } = \left |\frac {\overline {\rho } \varPhi }{\varPsi _{\varPhi }}\right |\!. \end{equation}

For example, the turbulent production time scale $t_{P, uu} = |\overline {\rho }\,\widetilde {{u^{\prime \prime}}^2}/P_{uu}|$ measures the time required to produce a given amount of $\widetilde {{u^{\prime \prime}}^2}$ at the constant rate $P_{uu}$ ; similarly, the dissipation time scale $t_{\varepsilon ,uu} = |\overline {\rho }\,\widetilde {{u^{\prime \prime}}^2}/\varepsilon _{uu}^v|$ measures the time to dissipate $\widetilde {{u^{\prime \prime}}^2}$ at the rate $\varepsilon _{uu}^v$ .

Figure 8.

Time scales for transport processes with semi-local scaling. (a, b) The turbulent production time scales for $\widetilde {{u^{\prime \prime}}^2}$ and $\widetilde {{T^{\prime \prime}}^2}$ . (c,d) The dissipation time scales for $\widetilde {{u^{\prime \prime}}^2}$ and $\widetilde {{T^{\prime \prime}}^2}$ . (e, f) The turbulent production time scales for $\widetilde {\ u^{\prime \prime}v^{\prime \prime}}$ and $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ . (g,h) The pressure term time scales for $\widetilde {\ u^{\prime \prime}v^{\prime \prime}}$ and $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ . The black and grey solid lines represent the corresponding power-law fits.

The time scales of the turbulent production for the streamwise Reynolds stress $\widetilde {{u^{\prime \prime}}^2}$ and temperature variance $\widetilde {{T^{\prime \prime}}^2}$ are plotted in figure 8(a,b) with semi-local scaling ( $\delta _{\nu }^*/u_\tau ^*$ ). These turbulence time scales generally increase with the wall-normal distance, and exhibit a distinct piecewise power-law behaviour. In the region $10 \lesssim y^* \lesssim 100$ , indicated by the black solid line, the results for different Mach numbers collapse well under adiabatic wall conditions. This collapse is disrupted by wall cooling for $t_{P, TT}^*$ . For $y^{*} \gtrsim 100$ , indicated by the grey solid line, the trends for different Mach numbers and wall temperatures remain similar under semi-local scaling, although the results do not collapse as closely. The power-law exponents of the time scales for $t_{P,uu}^*$ and $t_{P,TT}^*$ are identical, and their coefficients are also very close. The ratio between the two time scales is nearly constant, $t_{P,TT}/t_{P,uu} \approx 1.18$ , in the region $y^{*} \gtrsim 10$ for the adiabatic cases, and $y^{*} \gt y^*_{\textit{peak}}$ for the wall-cooling cases. This constant ratio between $t_{P,uu}$ and $t_{P,TT}$ is the crux of the RSRA, which will be discussed subsequently in $\S$ 5.

The time scales of the dissipation for $\widetilde {{u^{\prime \prime}}^2}$ and $\widetilde {{T^{\prime \prime}}^2}$ are plotted in figure 8(c,d). A collapse with a power law $t_{\varepsilon ,uu}^* \approx {y^*}^2$ is achieved in the region where $y^{*} \lesssim 10$ . The power law for $t_{\varepsilon ,TT}^*$ in the region $y^{*} \lesssim 10$ is given only for the adiabatic results, due to the direct impact of the wall cooling on the near-wall temperature. The power laws for $t_{\varepsilon ,uu}^*$ and $t_{\varepsilon ,TT}^*$ in the region $y^{*} \gtrsim 10$ have the same exponents, meaning that their ratio approaches a constant in the outer layer at large $y^{*}$ : $(t_{\varepsilon ,TT}/t_{\varepsilon ,uu})_{\textit{outer}} \approx 0.32$ . This ratio is less than unity due to the pressure term $\varPi _{uu}$ for $\widetilde {{u^{\prime \prime}}^2}$ and the work contribution $W_{\textit{TT}}$ for $\widetilde {{T^{\prime \prime}}^2}$ , as shown in figures 3 and 5, respectively. The pressure term $\varPi _{uu}$ primarily serves as an additional destruction mechanism, redistributing energy from the streamwise Reynolds stress $\widetilde {{u^{\prime \prime}}^2}$ to the wall-normal and spanwise components (Cheng & Fu Reference Cheng and Fu2024b ). The work contribution $W_{\textit{TT}}$ , which distinguishes temperature from a passive scalar in (4.4), acts as an additional generation mechanism in the outer layers. As discussed in supplementary Appendix B, the pressure work contribution $W_{\textit{TT}}^p$ dominates the total work contribution $W_{\textit{TT}}$ in the outer layer. Consequently, the opposing roles of the pressure in the transport processes of $\widetilde {{u^{\prime \prime}}^2}$ and $\widetilde {{T^{\prime \prime}}^2}$ lead to their distinct destruction mechanisms. This difference may be the reason for the nonlinear correlation between velocity and temperature fluctuations.

The time scales of the turbulent production for the Reynolds shear stress $\widetilde {\ {u^{\prime \prime}v^{\prime \prime}}}$ and wall-normal heat flux $\widetilde {{{v^{\prime \prime}}{T^{\prime \prime}}}}$ are plotted in figure 8(e, f) with semi-local scaling. The ratio of the turbulent production time scales $t_{P,uv}$ and $t_{P,vT}$ is the turbulent Prandtl number, $t_{P,uv}/t_{P,vT} = \textit{Pr}_T$ . The fact that $\textit{Pr}_T \lt 1$ ( $t_{P,uv} \lt t_{P,vT}$ ) shown in figure 2 indicates that the production process associated with $\widetilde {\ u^{\prime \prime}v^{\prime \prime}}$ occurs faster than the process associated with $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ , assuming that the two production rates $P_{uv}$ and $P_{vT}$ are comparable. This phenomenon explains why $\textit{Pr}_T \lt 1$ in the region $y^{*} \lt y_{\textit{peak}}^*$ under wall-cooling conditions, despite the corresponding $C_{u^{\prime \prime}T^{\prime \prime}}$ being close to unity. The dominant generation mechanism of the wall-normal turbulent heat flux $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ in the region $y^{*} \lt y_{\textit{peak}}^*$ is the work contribution term $W_{vT}$ (primarily the pressure work contribution $W_{vT}^p$ , defined in supplementary Appendix B) rather than the turbulent production term $P_{vT}$ (as shown in figure 7 b,c). The work contribution $W_{vT}$ generates additional turbulent heat flux $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ in the near-wall region, leading to a time scale $t_{P,vT}$ that is significantly larger than $t_{P,uv}$ .

Figure 9.

Assessment of the approximate expression for $\varPi _{vT}/\varPi _{uv}$ (see (4.10)) in (a) outer scaling ( $y/\delta$ ) and (b) semi-local scaling ( $y^{*}$ ).

The trends of the time scales for the pressure terms of $\widetilde {\ u^{\prime \prime}v^{\prime \prime}}$ and $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ , plotted in figure 8(g,h), can each be represented by a single power-law relationship. The ratio between the two time scales is very close to unity near $y^{*} =31$ . Based on the quantitative power-law representation, the relationship between the ratios $t_{P,uv}/t_{P,vT}$ and $t_{\varPi ,uv}/t_{\varPi ,vT}$ can be obtained:

(4.9) \begin{equation} \left (\frac {t_{P,uv}}{t_{P,vT}}\right )^{-{3}/{2}} \frac {t_{\varPi ,uv}}{t_{\varPi ,vT}}\approx \left \{ \begin{array}{ll} 1.41(y^*)^{-0.1}, & 10 \lesssim y^* \lesssim 100 \\[2pt] 0.87(y^*)^{0.02}, & y^* \gtrsim 100 \end{array} \right . \approx 1. \end{equation}

After some algebraic manipulation of (4.9), the approximate expression for the ratio $\varPi _{vT}/\varPi _{uv}$ can be obtained:

(4.10) \begin{equation} \frac {\varPi _{vT}}{\varPi _{uv}} \approx \left (\frac {\widetilde {\ u^{\prime \prime}v^{\prime \prime}}}{\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}}\right )^{{1}/{2}} \left (\frac {P_{vT}}{P_{uv}}\right )^{{3}/{2}} = \sqrt {\textit{Pr}_T}\frac {P_{vT}}{P_{uv}} \approx \sqrt {\textit{Pr}_T}\frac {\partial \tilde {T}/\partial y}{\partial \tilde {u}/ \partial y}. \end{equation}

The deviation of the DNS data from (4.10) is evaluated by dividing the left-hand side by the corresponding right-hand side, and is plotted in figure 9. An accurate prediction can be obtained in the region where $y^{*} \gtrsim 10$ for the adiabatic cases and $y^{*} \gt y_{\textit{peak}}^*$ for the wall-cooling cases. According to (4.10), the magnitude of the pressure term $\varPi _{vT}$ is smaller than the turbulent production $P_{vT}$ in the region $\textit{Pr}_T \lt 1$ , when the ratio $\varPi _{uv}/P_{uv}$ is unity (as confirmed in figure 4). A similar phenomenon can also be observed in Fan et al. (Reference Fan, Kozul, Li and Sandberg2024), and in incompressible turbulent channel flows (Kawamura, Abe & Matsuo Reference Kawamura, Abe and Matsuo1999) for fluids with moderate Prandtl numbers, i.e. $\textit{Pr}=0.71$ . As reported by Kawamura et al. (Reference Kawamura, Ohsaka, Abe and Yamamoto1998), for fluids with low Prandtl numbers, the dissipation term $\varepsilon _{vT}^{v-q}$ is the dominant destruction term, and the pressure term $\varPi _{vT}$ is negligible. In contrast, for fluids with high Prandtl numbers, the magnitude of the pressure term $\varPi _{vT}$ is equal to that of the production term $P_{vT}$ in the log-law region. It should be noted that the relative importance of the pressure and dissipation terms varies with the Prandtl number in the tranport of $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ , even though the present discussion focuses on $\textit{Pr}=0.71$ . Also indicated by (4.10), the dissimilarity of the pressure terms in the transport processes of $\widetilde {\ u^{\prime \prime}v^{\prime \prime}}$ and $\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}$ can be quantitatively characterised by the turbulent Prandtl number $\textit{Pr}_T$ .

5.1. Linear transport equation

In this section, a connection between three transport processes ( $\widetilde {{u^{\prime \prime}}^2}$ , $\widetilde {{T^{\prime \prime}}^2}$ and $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ ) and the SRA is established. The intrinsic assumption of the SRA is the linear relationship between temperature and velocity fluctuations in large-scale motions. Based on this assumption, a variety of SRA relations are established by determining the proportionality factor between the characteristic length scales of velocity and temperature fluctuations (Gaviglio Reference Gaviglio1987; Huang et al. Reference Huang, Coleman and Bradshaw1995; Guarini et al. Reference Guarini, Moser, Shariff and Wray2000; Zhang et al. Reference Zhang, Bi, Hussain and She2013).

Here, we start with an ad hoc assumption that $|C_{u^{\prime \prime}T^{\prime \prime}}|=1$ , which clearly does not hold in actual CTBLs shown in figure 1(a). This assumption, together with (4.1), leads to the equality

(5.1) \begin{align} \notag \overline {\rho }\frac {\tilde {\textrm{D}} \widetilde {u^{\prime \prime}{T^{\prime \prime}}}}{\tilde {\textrm{D}} t} & = \text{sgn}\big(\widetilde {u^{\prime \prime}{T^{\prime \prime}}}\big) \overline {\rho }\frac {\tilde {\textrm{D}} {u_{\textit{rms}}^{\prime \prime}{T_{\textit{rms}}^{\prime \prime}}}}{\tilde {\textrm{D}} t} \nonumber\\ & = \frac {\text{sgn}\big(\widetilde {u^{\prime \prime}{T^{\prime \prime}}}\big)}{2} \left ( \overline {\rho }\frac {\tilde {\textrm{D}} \widetilde {{u^{\prime \prime}}^2}}{\tilde {\textrm{D}} t} \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} + \overline {\rho }\frac {\tilde {\textrm{D}} \widetilde {{T^{\prime \prime}}^2}}{\tilde {\textrm{D}} t} \frac {u_{\textit{rms}}^{\prime \prime}}{T_{\textit{rms}}^{\prime \prime}} \right )\!, \end{align}

where ‘ $\text{sgn}$ ’ denotes the signum function, which returns $1$ for positive values, $-1$ for negative values, and $0$ for zero. It should be noted that (5.1) is valid only when $\text{sgn}(\widetilde {u^{\prime \prime}{T^{\prime \prime}}})$ is constant, i.e. it does not hold near the wall-normal distance corresponding to the mean temperature peak $y_{\textit{peak}}$ , as the sign of $\widetilde {u^{\prime \prime}{T^{\prime \prime}}}$ varies (as shown in figure 1 a).

In light of (5.1), it is reasonable to conjecture that the budget terms for $\widetilde {{u^{\prime \prime}}^2}$ , $\widetilde {{T^{\prime \prime}}^2}$ and $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ have the same relationship under the assumption $|C_{u^{\prime \prime}T^{\prime \prime}}|=1$ . This relationship is referred to as the linear transport equation, as follows:

(5.2) \begin{equation} \varPsi _{uT} = \frac {\text{sgn}\big(\widetilde {u^{\prime \prime}{T^{\prime \prime}}}\big)}{2} \left ( \varPsi _{uu} \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} + \varPsi _{\textit{TT}} \frac {u_{\textit{rms}}^{\prime \prime}}{T_{\textit{rms}}^{\prime \prime}} \right )\!, \end{equation}

where $\varPsi$ denotes the budget terms in the transport equations. The validity of the linear transport equation is explained in supplementary Appendix C. Hereinafter, the budget term computed from the right-hand side of (5.2) is denoted by the superscript ‘linear’, i.e. $\varPsi _{uT}^{\textit{linear}}$ .

Figure 10.

Assessment of the linear transport equations (5.2) for (a) turbulent production $P$ , (b) viscous-thermal diffusion and dissipation $(D-\varepsilon )$ , (c) pressure term $\varPi$ , and (d) work contribution $W$ .

To assess the deviation of the DNS data from the linear transport equation, $\varPsi _{uT}/\varPsi _{uT}^{\textit{linear}}$ for the main budget terms (the turbulent production $P$ , viscous-thermal diffusion and dissipation $(D-\varepsilon )$ , pressure term $\varPi$ and work contribution $W$ ) are presented in figure 10. The turbulent production, plotted in figure 10(a), satisfies the linear transport equation in regions where the mean temperature gradient is non-zero. Under the wall-cooling conditions, the linear transport equation for the turbulent production shows a slight deviation near the wall, and this deviation decreases as the wall cooling intensifies. This perfect satisfaction indicates the linear mechanism in turbulent production of the streamwise velocity and temperature fluctuations. As reported previously, the streamwise velocity and temperature are linearly correlated in large-scale motions in CTBLs (Yu & Xu Reference Yu and Xu2021; Cheng & Fu Reference Cheng and Fu2023), and the turbulent production terms are primarily dominated by these large-scale motions as well. As for the other budget terms plotted in figure 10(b,c,d), deviations from the linear transport equations are evident. The additional gains and losses introduced by these nonlinear budget terms essentially balance each other and hardly affect the turbulent production. This behaviour explains the observation that the ratio of the turbulent production time scales remains close and approximately constant, with $t_{P,TT}/t_{P,uu} \approx 1.18$ shown in figure 8(a,b). The region where the deviations are smallest coincides with the region where the turbulent Prandtl number $\textit{Pr}_T$ approaches unity and the correlation coefficient $C_{u^{\prime \prime}T^{\prime \prime}}$ reaches its negative peak. It can be inferred that the reduction in the magnitude of $C_{u^{\prime \prime}T^{\prime \prime}}$ , as well as $\textit{Pr}_T$ , is due to the nonlinear transport mechanisms of these budget terms.

Figure 11.

Deviation of the linear transport equation (5.2). The budget terms include the turbulent production $P_{uT}$ (red), turbulent transport $\varGamma _{uT}$ (green), thermal diffusion $D_{uT}^{v-q}$ (blue), thermal dissipation $-\varepsilon _{uT}^{v-q}$ (orange), pressure term $\varPi _{uT}$ (cyan), work contribution $W_{uT}$ (yellow) and mass flux contribution $M_{uT}$ (purple). The vertical lines denote the locations $y^{*} = y_{\textit{peak}}^*$ . (a) Results for M2ad ( ), M4ad ( ), M6ad ( ). (b) Results for M6Tw05 ( ). (c) Results for M6Tw025 ( ). The legend for the budget terms of M2ad is presented in (a).

To further investigate the dissimilarity between velocity and temperature in CTBLs, figure 11 presents the deviation of the linear transport equation, $\varPsi _{uT} - \varPsi _{uT}^{\textit{linear}}$ . In the outer layer, the deviations originate from the pressure term $\varPi _{uT}$ , and the viscous-thermal diffusion and dissipation term $(D-\varepsilon )_{uT}$ . It is worth mentioning that $\varPi _{uT}^{\textit{linear}}$ corresponds exclusively to the pressure term $\varPi _{uu}$ of $\widetilde {{u^{\prime \prime}}^2}$ , as shown in supplementary Appendix C. The fact that the transport equation of $\widetilde {{T^{\prime \prime}}^2}$ does not contain a pressure term leads to a greater destruction induced by the pressure term $\varPi _{uT}$ in the transport of $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ compared with $\varPi _{uT}^{\textit{linear}}$ . This greater destruction is identified as the dominant source of the dissimilarity between velocity and temperature fluctuations in the outer layer. The deviation of $(D-\varepsilon )_{uT}$ mainly acts to compensate for the excessive destruction caused by $\varPi _{uT}$ .

Although both $\varPi _{uT}$ and $(D-\varepsilon )_{uT}$ serve as destruction mechanisms in the transport of $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ (as shown in figure 6), they are associated with different turbulent scales. This distinction can be understood qualitatively by referring to the spectral analysis on transport budgets of turbulent heat fluxes in incompressible Couette turbulence conducted by Kawata & Tsukahara (Reference Kawata and Tsukahara2022). Their spectral analysis shows that the pressure-related terms of $\widetilde {{u^{\prime \prime}}^2}$ and $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ primarily act at large scales in the outer layer, within a range of scales similar to those associated with the turbulent productions. In CTBLs, the spectrum of the pressure-related term of $\widetilde {{u^{\prime \prime}}^2}$ behaves similarly to that in incompressible flows (Fan & Li Reference Fan and Li2023). Although the spectrum of the pressure-related term of $\widetilde {u^{\prime \prime}T^{\prime \prime}}$ has not yet been reported, it is also expected to behave similarly to that in incompressible flows. By linear coherence spectrum analysis, Cheng & Fu (Reference Cheng and Fu2023) observed that the scale range exhibiting strong linear coherence between velocity and temperature fluctuations shifts towards larger scales with increasing wall-normal distance, and the amplitude of this large-scale coherence diminishes. These observations can be attributed to the action of the pressure-related terms at large scales, which reduces the similarity between velocity and temperature fluctuations. Consequently, despite the strong similarity in turbulent production, the dissimilarity in the destruction mechanisms contributes to the reduction of $C_{u^{\prime \prime}T^{\prime \prime}}$ and $\textit{Pr}_T$ in the outer layer.

The influence of the work contribution $W_{uT}$ in the outer layer on the nonlinear correlation between velocity and temperature fluctuations is negligible compared to that of the pressure term $\varPi _{uT}$ . The significant deviation from the linear transport behaviour of $W_{uT}$ is primarily observed in the viscous sublayer under the adiabatic wall, which is also compensated by $(D-\varepsilon )_{uT}$ . The linear work contribution $W_{uT}^{\textit{linear}}$ corresponds exclusively to the work contribution $W_{\textit{TT}}$ of $\widetilde {{T^{\prime \prime}}^2}$ , as shown in supplementary Appendix C. Similar to the pressure term $\varPi _{uT}$ , the greater destruction induced by the work contribution $W_{uT}$ compared with $W_{uT}^{\textit{linear}}$ is the primary reason for the reduced similarity between the velocity and temperature fluctuations in the near-wall region $y^{*} \lesssim 10$ under adiabatic wall conditions. Under wall-cooling conditions, however, the relative deviation of $W_{uT}$ is small in the region $y^{*} \lt y^*_{\textit{peak}}$ , as shown in figure 10(d). This small deviation corresponds to the observation that near-wall $C_{u^{\prime \prime}T^{\prime \prime}}$ tends towards unity as the wall-cooling intensity increases.

Figure 12.

Assessment of the (a) GSRA (1.1) and (b) HSRA (1.2). The solid symbols denote the values at the boundary layer edge. For line styles, see table 1.

After manipulating the linear transport equation for the turbulent production, an equation for the fluctuation intensity ratio $T_{\textit{rms}}^{\prime \prime} / u_{\textit{rms}}^{\prime \prime}$ can be obtained as

(5.3) \begin{equation} \left ( \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} \right )^2 - \text{sgn}\big( \widetilde {u^{\prime \prime}{T^{\prime \prime}}} \big) \left ( \frac {\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}}{\widetilde {\ u^{\prime \prime}v^{\prime \prime}}} + \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y} \right ) \left ( \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} \right ) + \frac {\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}}{\widetilde {\ u^{\prime \prime}v^{\prime \prime}}} \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y} = 0, \end{equation}

and the two solutions are

(5.4a,b) \begin{equation} \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} \approx \left | \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y} \right |\!, \quad \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} \approx \left | \frac {\widetilde {{v^{\prime \prime}}{T^{\prime \prime}}}}{\widetilde {\ u^{\prime \prime}v^{\prime \prime}}} \right | = \left | \frac {1}{\textit{Pr}_T} \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y} \right |\!. \end{equation}

These two solutions are the alternative forms of the GSRA (1.1) and HSRA (1.2), respectively. The difference between the two SRAs arises from $\textit{Pr}_T$ not being equal to unity, which is caused by various nonlinear mechanisms in transport processes. Note that there are small differences between the expressions of the GSRA and HSRA in (1.1) and (1.2), and those in (5.4), due to the assumption $\tilde {T}_t \approx \tilde {T} + \tilde {u}^2/(2c_p)$ .

The assessments of the GSRA and HSRA are shown in figure 12 by dividing the left-hand sides of (1.1) and (1.2) by the corresponding right-hand sides. A result greater than unity indicates an underestimation of the DNS results, and conversely, a result less than unity indicates an overestimation. In the region near the wall where $y^{*} \lesssim 10$ , the significant deviations in the adiabatic cases stem from the zero mean temperature gradient. As the wall cooling strengthens, the GSRA performs increasingly well near the wall, while the HSRA, influenced by the Dirichlet boundary conditions, performs poorly. In the region $10 \lesssim y^* \lesssim 100$ , both results show a smooth trend with very small deviations that slightly underestimate the DNS results. This region also coincides with the region where the turbulent Prandtl number $\textit{Pr}_T$ approaches unity, and the correlation coefficient $C_{u^{\prime \prime}T^{\prime \prime}}$ reaches its negative peak, as shown in figure 1(a). This observation demonstrates the equivalence of GSRA and HSRA under strong linear mechanisms. The underestimation is due to the imperfect linear correlation between the streamwise velocity and temperature in the region $10 \lesssim y^* \lesssim 100$ . Further away from the wall, the decrease of $\textit{Pr}_T$ causes the GSRA and HSRA to deviate in different ways within the boundary layer. The GSRA underestimates the DNS results to a greater extent. In contrast, the HSRA provides relatively accurate predictions in the log-law region, but overestimates the DNS results further away from the wall. This is also the reason why the HSRA is more widely applied than the GSRA (Duan & Martin Reference Duan and Martin2011a , Reference Duan and Martinb ; Passiatore et al. Reference Passiatore, Sciacovelli, Cinnella and Pascazio2022; Chen et al. Reference Chen, Cheng, Gan and Fu2023; Yu et al. Reference Yu, Zhou, Dong, Yuan and Xu2023; Gerolymos & Vallet Reference Gerolymos and Vallet2024). However, both solutions are essentially based on the linear mechanism underlying the turbulent production. The opposite deviations that they exhibit result from the influence of nonlinear mechanisms from the other budget terms in transport processes within CTBLs.

5.2. The RSRA

Since the turbulent production terms are governed by linear mechanisms, refining the SRA based on their characteristic scales is expected to produce more consistent and physically grounded results. To this end, the RSRA can be derived according to the similar turbulent production time scales of $\widetilde {{u^{\prime \prime}}^2}$ and $\widetilde {{T^{\prime \prime}}^2}$ mentioned in $\S$ 4.2, with $t_{P,\textit{TT}}/t_{P,uu} \approx 1.18$ :

(5.5a,b) \begin{equation} \frac {T_{\textit{rms}}^{\prime \prime}/\tilde {T}}{\left ( \gamma - 1 \right )\, \textit{Ma}^2\, u_{\textit{rms}}^{\prime \prime}/\tilde {u} } \approx \frac {C}{\sqrt {\textit{Pr}_T} \,( 1 - {\partial \tilde {T}_t}/{\partial \tilde {T}} )}, \quad \frac {T_{\textit{rms}}^{\prime \prime}}{u_{\textit{rms}}^{\prime \prime}} \approx \frac {C_1}{\sqrt {\textit{Pr}_T}} \left | \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y} \right |\!. \end{equation}

The coefficient $C_1 \approx 1.09$ is the square root of the constant ratio 1.18, which reflects the imperfect linear correlation between the streamwise velocity and temperature fluctuations in actual large-scale motions within CTBLs. Under the assumption $\tilde {T}_t \approx \tilde {T} + \tilde {u}^2/(2c_p)$ , (5.5a ) and (5.5b ) are equivalent, but a coefficient $C \approx 1.03$ is required to compensate the deviation introduced by this assumption. Exactly the same RSRA can also be obtained by taking the geometric mean of the GSRA and HSRA, by which the influence of $\textit{Pr}_T \lt 1$ is counteracted, as indicated by the underestimation of the GSRA and overestimation of the HSRA shown in figure 12.

The deviation of the DNS data from the RSRA (5.5a ) is plotted in figure 13, evaluated by dividing the left-hand side by the corresponding right-hand side. The results in outer scaling from Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) are also included in figure 13(a). The RSRA accurately captures the quantitative relationship between the streamwise velocity and temperature fluctuation intensities, especially in the outer layer of the boundary layer, as shown in figure 13(a). In semi-local scaling, shown in figure 13(b), the RSRA demonstrates better validity compared with the GSRA and HSRA shown in figure 12. Near the location $y^{*} = y_{\textit{peak}}^*$ , where the correlation coefficient $C_{u^{\prime \prime}T^{\prime \prime}}$ is nearly zero, all the SRAs fail to capture the temperature fluctuation intensity.

Figure 13.

Assessment of the RSRA (5.5a ) in (a) outer scaling ( $y/\delta$ ) and (b) semi-local scaling ( $y^{*}$ ). The scatter symbols in (a) are adiabatic results from Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011), with $\textit{Ma}_e=2$ at ${\textit{Re}}_\tau=251$ , 497 and 1116. The solid symbols in (b) denote the values at the boundary layer edge. For line styles, see table 1.

Another factor limiting the application of the SRA is the turbulent Prandtl number $\textit{Pr}_T$ . In previous studies, $\textit{Pr}_T$ was often assumed to be a constant (Martin Reference Martin2007; Duan & Martin Reference Duan and Martin2011b ; Sciacovelli, Cinnella & Gloerfelt Reference Sciacovelli, Cinnella and Gloerfelt2018; Cheng & Fu Reference Cheng and Fu2024a ) or replaced by an empirical relation (Cebeci Reference Cebeci1973; Pirozzoli et al. Reference Pirozzoli, Grasso and Gatski2004; Huang et al. Reference Huang, Coleman, Spalart and Yang2023; Cheng & Fu Reference Cheng and Fu2024b ; Gerolymos & Vallet Reference Gerolymos and Vallet2024). As mentioned above, using only outer scaling or semi-local scaling cannot correctly scale $\textit{Pr}_T$ under different Mach numbers and wall temperatures. Here, an approximate expression for $\textit{Pr}_T$ in terms of the mean quantities can be derived using the RSRA. The derivation begins with the original SRA (Morkovin Reference Morkovin1962):

(5.6) \begin{equation} \frac {T_{\textit{rms}}^{\prime \prime}/\tilde {T}}{( \gamma - 1 )\, \textit{Ma}^2\, u_{\textit{rms}}^{\prime \prime}/\tilde {u} } \approx 1. \end{equation}

The validity of the original SRA suggested by Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000) relies on the assumption $\widetilde {{T^{\prime \prime}}^2} \gg \widetilde {{T_t^{\prime \prime}}^2} - 2\,\widetilde {T^{\prime \prime}T_t^{\prime \prime}}$ . Although this assumption holds only under adiabatic boundary conditions, and has been shown by DNS data to be only approximately satisfied (Guarini et al. Reference Guarini, Moser, Shariff and Wray2000; Zhang et al. Reference Zhang, Bi, Hussain and She2013), it will still be adopted here as it provides a useful framework. Since the underlying assumptions of the RSRA and the original SRA are non-contradictory, their equivalence under adiabatic wall conditions can be expected, yielding the following expression for $\textit{Pr}_T$ :

(5.7) \begin{equation} {\sqrt {\textit{Pr}_T}} \approx C\left ( 1 - \frac {\partial \tilde {T}_t}{\partial \tilde {T}} \right )^{-1}. \end{equation}

The deviation of the DNS data from (5.7) is plotted in figure 14(a). The approximate expression for $\textit{Pr}_T$ demonstrates good predictive accuracy across most of the adiabatic boundary layer ( $y/\delta \lesssim 0.8$ ), with a slight underestimation of $\textit{Pr}_T$ observed. When entering the entrainment/intermittent region (Duan et al. Reference Duan, Beekman and Martin2011), the expression overestimates the DNS results. The discrepancy in this approximate expression primarily arises from the assumption of original SRA proposed by Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000). An empirical expression, similar in form to the expression for $\textit{Pr}_T$ proposed by Huang et al. (Reference Huang, Coleman, Spalart and Yang2023), $ \epsilon = 0.7 - 0.35 \tanh[5(y/\delta - 1)]$ , is suggested here to correct for this discrepancy.

Figure 14.

Assessment of the approximate expressions for $\textit{Pr}_T$ : (a) expression (5.7) and (b) expression (5.8). The coefficient $g$ is calculated using (5.13). The grey solid line represents the empirical expression $\epsilon = 0.7 - 0.35\tanh[5(y/\delta -1)]$ . The scatter symbols in (b) are adiabatic results from Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011), with $\textit{Ma}_e=2$ at ${\textit{Re}}_\tau=251$ , 497 and 1116. For line styles, see table 1.

The remaining issue that requires further consideration is the wall-cooling effect, where the assumption proposed by Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000) does not hold. As shown in figure 14(a), the approximate expression increasingly deviates as the wall cooling intensifies. However, except for the region near the wall where the mean temperature reaches its peak, the overall trend remains largely unchanged. Therefore, we introduce a coefficient $g = g(\varTheta )$ , which is a function of diabatic parameter $\varTheta = (T_w - T_e)/(T_r - T_e)$ , to incorporate the effects of wall cooling. The approximate expression for $\textit{Pr}_T$ can be rewritten as

(5.8) \begin{equation} {\sqrt {\textit{Pr}_T}} \approx C\, g ( \varTheta )\, \epsilon \left ( \frac {y}{\delta } \right ) \left ( 1 - \frac {\partial \tilde {T}_t}{\partial \tilde {T}} \right )^{-1}, \end{equation}

where $g(1)=1$ . To obtain the coefficient $g$ , the generalised Reynolds analogy (GRA) (Zhang et al. Reference Zhang, Bi, Hussain and She2013) is applied:

(5.9) \begin{equation} \frac {\tilde {T}}{T_e} \approx \frac {T_w}{T_e} + \frac {T_r - T_w}{T_e} f \left ( \frac {\tilde {u}}{u_e} \right ) + \frac {T_e - T_r}{T_e} \left ( \frac {\tilde {u}}{u_e} \right )^2, \end{equation}

where

(5.10) \begin{equation} f \left ( \frac {\tilde {u}}{u_e} \right ) = (1 - s\,Pr)\left ( \frac {\tilde {u}}{u_e} \right )^2 + s\,Pr \left ( \frac {\tilde {u}}{u_e} \right )\!. \end{equation}

In (5.10), $s \equiv (q_{y,w} u_e)/[\tau _w (h_w-h_r)]$ denotes the Reynolds analogy factor, and $s\,Pr$ is approximately 0.8 for CTBLs (Zhang et al. Reference Zhang, Bi, Hussain and She2013; Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023). The GRA in (5.9) can be reformulated in terms of the diabatic parameter $\varTheta$ as

(5.11) \begin{equation} \frac {\tilde {T} - T_e}{T_r - T_e} \approx \left [ 1 - f \left ( \frac {\tilde {u}}{u_e} \right ) \right ] \varTheta + s\,Pr \left ( \frac {\tilde {u}}{u_e} \right ) \left ( 1 - \frac {\tilde {u}}{u_e} \right )\!. \end{equation}

Then the ratio of the mean temperature gradient to the velocity gradient at the boundary layer edge can be obtained:

(5.12) \begin{equation} \left ( \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y} \right )_e \approx \frac {T_r - T_e}{u_e} \left [ \left ( s\,Pr - 2 \right ) \varTheta - s\,Pr \right ]\!. \end{equation}

Assuming that the turbulent Prandtl number at the boundary layer edge, $\textit{Pr}_{T,e}$ , is constant, as shown in figure 2, the coefficient $g$ in (5.8) can be calculated by the ratio of the derivative under diabatic and adiabatic wall conditions:

(5.13) \begin{align} \notag g &\approx \frac {\big(1 - {\partial \tilde {T}_t}/{\partial \tilde {T}}\big)_{e,{\textit{dia}}}}{\big(1 - {\partial \tilde {T}_t}/{\partial \tilde {T}}\big)_{e,{\textit{adi}}}} \approx {\left ( \frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y}\right )_{\!e,{\textit{adi}}}}{\left (\frac {\partial \tilde {T} / \partial y}{\partial \tilde {u} / \partial y}\right )_{\!e,{\textit{dia}}}^{-1}} \nonumber\\ &\approx \frac {2}{\left (2 - s\,Pr\right )\varTheta + s\,Pr}. \end{align}

By substituting the expression for $g$ given by (5.13), the empirical formula for $\epsilon$ , and $C \approx 1.03$ into (5.8), we can obtain the explicit expression for $\textit{Pr}_T$ .

Figure 14(b) plots the deviation of the DNS data from the approximate expression for $\textit{Pr}_T$ in (5.8). The results from Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) are also included. Good predictive performance is achieved in regions away from $y=y_{\textit{peak}}$ under different Mach numbers and wall temperatures. It is clear that the difference in $\textit{Pr}_T$ between a cooled wall and an adiabatic wall can be reasonably measured by the diabatic parameter $\varTheta$ , which echoes the finding in Cogo et al. (Reference Cogo, Baù, Chinappi, Bernardini and Picano2023).

Figure 15.

Wall-normal profiles of percentage errors $\varepsilon _{\scriptscriptstyle {\textit{Pr}_T}}$ of the approximate relations (5.8), $\textit{Pr}_T = 0.9-0.3(y/\delta )^2$ (Abe & Antonia Reference Abe and Antonia2017) and $\textit{Pr}_T=1.05-0.2\tanh^3(y^*/A_\textit{Pr}^*)$ (Huang et al. Reference Huang, Coleman, Spalart and Yang2023), for cases (a) M2ad, (b) M4ad, (c) M6ad, (d) M6Tw05, and (e) M6Tw025. The inset in each panel plots the wall normal profile of $\textit{Pr}_T$ and the approximate relations.

Figure 15 presents a quantitative comparison between the newly proposed relation for $\textit{Pr}_T$ , as given by (5.8), $\textit{Pr}_T = 0.9-0.3(y/\delta )^2$ proposed by Abe & Antonia (Reference Abe and Antonia2017), and $\textit{Pr}_T=1.05-0.2\tanh^3(y^*/A_\textit{Pr}^*)$ proposed by Huang et al. (Reference Huang, Coleman, Spalart and Yang2023). The percentage error $\varepsilon _{\textit{Pr}_T}$ is given by

(5.14) \begin{equation} \varepsilon _{\textit{Pr}_T} = \frac {\left | \textit{Pr}_{T, {\textit{DNS}}} - \textit{Pr}_{T, {\textit{approx}}} \right |}{\left | \textit{Pr}_{T, {\textit{DNS}}} \right |} \times 100\,\%. \end{equation}

Under adiabatic wall conditions, the relative error of (5.8) in the near-wall region ( $y/\delta \lt 0.2$ ) is comparable to that of the relation proposed by Huang et al. (Reference Huang, Coleman, Spalart and Yang2023), remaining within 10 %. When $y/\delta \gt 0.4$ , the relation proposed by Huang et al. (Reference Huang, Coleman, Spalart and Yang2023) gradually deviates from the DNS results, whereas (5.8) maintains an error level comparable to that proposed by the Abe & Antonia (Reference Abe and Antonia2017) relation under adiabatic wall conditions. Under wall-cooling conditions, (5.8) fails near the wall, because (5.8) is based on the RSRA, which is unable to predict the temperature fluctuation intensity $T_{\textit{rms}}^{\prime \prime}$ near $y = y_{\textit{peak}}$ . However, (5.8) yields good performance in the region away from $y = y_{\textit{peak}}$ (especially in the region $y/\delta \gt 0.4$ ).

The newly proposed relation for $\textit{Pr}_T$ in (5.8) exhibits clear superiority in practical applications, as it is derived based on physical reasoning rather than mere empirical fitting. It maintains validity across different Reynolds numbers, Mach numbers and wall-cooling conditions, providing a unified approximation for $\textit{Pr}_T$ in CTBLs. This broad applicability significantly enhances its potential for engineering modelling and further theoretical analysis, even though further investigation is required to improve (5.8) in the region near the mean temperature peak.