3.1. Transformed mean velocity profiles

To investigate the effect of compressibility on the mean velocity profiles of compressible turbulent channels, pipes and boundary layers at high Mach numbers and/or low wall temperatures, various transformed mean velocity profiles have been proposed (Cheng et al. Reference Cheng, Chen, Zhu, Shyy and Fu2024). Through these approaches, the transformed mean velocity profiles are designed to collapse onto the incompressible laws. The incompressible laws are defined as

(3.1a)$$\begin{gather} u^{+}=y^{+} , \end{gather}$$

(3.1b)$$\begin{gather}u^{+}=( 1/k )\log(\kern1.5pt y^{+} ) +C, \end{gather}$$

where $k=0.41$ and $C=5.2$.

One of the most classical and widely used mean velocity transformations was proposed by van Driest based on the Morkovin assumption (van Dreist Reference van Dreist1956; Morkovin Reference Morkovin1962). Its definition is

(3.2)\begin{equation} u_{VD}^{+}=\int_{0}^{{u} ^{+} }(\bar{\rho}/\bar{\rho}_{w})^{1/2} \,\mathrm{d}\bar{u}^{+}. \end{equation}

To improve the performance of the van Driest transformation under the wall-cooling condition, Trettel & Larsson (Reference Trettel and Larsson2016) proposed a new form of mean velocity transformation based on the log law and the conservation of near-wall momentum, which is defined as

(3.3a)$$\begin{gather} u_{TL}^{+}=\int_{0}^{{u} ^{+} }\sqrt{\frac{\bar{\rho}}{\rho_{w}}}\left [ 1+\frac{1}{2}\,\frac{1}{\bar{\rho}}\,\frac{{\rm d}\bar{\rho}}{{{\rm d}}y}\,y-\frac{1}{\bar{\mu}}\,\frac{\mathrm{d} \bar{\mu}}{\mathrm{d} y}\,y \right ] {{\rm d}}\bar{u}^{+} , \end{gather}$$

(3.3b)$$\begin{gather}y^{*}=\frac{\bar{\rho}( \tau_{w}/\bar{\rho} )^{1/2}y }{\bar{\mu}} . \end{gather}$$

This transformation can make the transformed mean velocity profiles collapse well with the incompressible linear law in the viscous sublayer and the buffer layer, while in the log region, it deviates significantly from the incompressible log law. Therefore, Volpiani et al. (Reference Volpiani, Iyer, Pirozzoli and Larsson2020) proposed a new mean velocity transformation suitable for non-adiabatic wall turbulence by further analysing the universality in the viscous sublayer and Morkovin-scaled shear stress, and combining data-driven methods to determine the non-dimensionalizations, which are defined as

(3.4a)$$\begin{gather} u_{V}^{+}=\int_{0}^{{u} ^{+} }\frac{( \bar{\rho}/{\rho_{w}} )^{1/2} }{ (\bar{\mu}/{\mu_{w}} )^{1/2}}\, {{\rm d}}\bar{u}^{+} , \end{gather}$$

(3.4b)$$\begin{gather}y_{V}^{+}=\int_{0}^{{y} ^{+} }\frac{( \bar{\rho}/{\rho_{w}})^{1/2} }{ (\bar{\mu}/{\mu_{w}} )^{3/2}}\, {{\rm d}}\bar{y}^{+} . \end{gather}$$

Recently, Griffin, Fu & Moin (Reference Griffin, Fu and Moin2021) scaled the viscous stress using the semi-local scaling in the viscous sublayer, and analysed the Reynolds stress based on the principle of approximate equilibrium between the production term and dissipation term of turbulent kinetic energy in the log layer. They further proposed a velocity-transformed form based on the total stress equation

(3.5)\begin{equation} u_{TS}^{+}=\int_{0}^{{y} ^{*} }\frac{\tau^{+}S_{eq}^{+}}{\tau^{+}+S_{eq}^{+}-S_{TL}^{+}} \,{{\rm d}}y^{*} , \end{equation}

where $\tau ^{+} = \tau _{v} ^{+} + \tau _{R} ^{+}$ is total stress, while $\tau _{v} ^{+}= ( \mu {\partial u}/{\partial y} ) /\tau _{w}$ and $\tau _{R} ^{+}= ( -\bar {\rho } \widetilde {u^{\prime \prime }v ^{\prime \prime }} ) /\tau _{w}$ are the scaled viscous and Reynolds shear stresses, respectively. Here, $S_{TL}^{+}= ( \bar {\mu }/\mu _{w} ) ( {\partial u^{+}}/{\partial y^{+}} )$ and $S_{eq}^{+}= ( \mu _{w}/\bar {\mu } ) ( {\partial u^{+}}/{\partial y^{*}} )$ are the generalized non-dimensional mean shear stresses derived for the viscous sublayer and the log layer, respectively, and $y^{*}$ is the wall-normal coordinate in the semi-local scaling (Huang, Coleman & Bradshaw Reference Huang, Coleman and Bradshaw1995). The semi-local scaling is defined as $y^{*}=y/y^{*}_{\tau }$, where $y^{*}_{\tau }=\bar {\mu }/(\bar {\rho } \tau _{w})^{1/2}$.

Figure 2 presents the transformed mean velocity profiles $u_{VD}^{+}$, $u_{TL}^{+}$, $u_{V}^{+}$ and $u_{TS}^{+}$ at two different bulk-to-shear viscosity ratios for each case of the DNS. The van Driest transformed mean velocity profile $u_{VD}^{+}$ adheres to an incompressible linear law only in the region very close to the wall in the wall-cooling cases. This behaviour is attributed to the increased gradient of relevant parameters in the near-wall region and the rapid changes in mean density and viscosity coefficient in the viscous sublayer. The Trettel and Larsson transformation $u_{TL}^{+}$ can make the mean velocity profiles of the compressible turbulent boundary layer with low wall temperature similar to the incompressible linear law in the viscous sublayer, but both the van Driest transformation $u_{VD}^{+}$ and Trettel and Larsson transformation $u_{TL}^{+}$ are significantly different from the incompressible log law in the log region. Xu et al. (Reference Xu, Wang, Wan, Yu, Li and Chen2021b), Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022) and Huang et al. (Reference Huang, Duan and Choudhari2022) have respectively reached similar conclusions. In contrast, the Volpiani velocity transformation $u_{V}^{+}$ based on physical analysis and data driven by Volpiani et al. (Reference Volpiani, Iyer, Pirozzoli and Larsson2020) and the total-stress-based mean velocity transformation $u_{TS}^{+}$ by Griffin et al. (Reference Griffin, Fu and Moin2021) can collapse the mean velocity profiles of the compressible turbulent boundary layer with high Mach number and low wall temperature to the incompressible laws of the wall in the viscous sublayer, buffer layer and log layer, which is consistent with the results of Huang et al. (Reference Huang, Duan and Choudhari2022) and Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). Additionally, as the Mach number increases and the wall temperature decreases, the Volpiani velocity transformation $u_{V}^{+}$ and the total-stress-based mean velocity transformation $u_{TS}^{+}$ collapse better with the incompressible scaling laws in the log region by considering the bulk viscosity, indicating that bulk viscosity makes the production term and dissipation term of turbulent kinetic energy more balanced in the log region. However, the transformed mean velocity profiles considering bulk viscosity show no significant differences compared to those without bulk viscosity, closely resembling the incompressible results. This observation indicates that bulk viscosity has minimal effect on the compressible transformed mean velocity profiles.

Figure 2.

Transformed mean velocity profiles: (a) the van Driest transformation; (b) the Trettel and Larsson transformation; (c) the data-driven-based transformation of Volpiani et al.; and (d) the total-stress-based transformation of Griffin et al. The results are compared with linear law $u^{+}=y^{+}$ and log law $u^{+}=(1/k)\log (\kern0.7pt y^{+})+C$, where $k=0.41$, $C=5.2$ in (a), and $k=0.41$, $C=5.5$ in (b), (c) and (d).

3.2. Fluctuations of velocity

Figure 3 shows the variation of the turbulence intensities in the streamwise, wall-normal and spanwise directions using the Morkovin transformation (Morkovin Reference Morkovin1962) with wall-normal distance in inner scaling ($\kern1.5pt y^{+}$) and semi-local scaling ($\kern1.5pt y^{*}$), respectively. Figure 4 displays the variation of Reynolds shear stress with the wall-normal distance in inner scaling and semi-local scaling, respectively. It is worth noting that the streamwise ($u_{rms}^{\prime \prime }$), wall-normal ($v_{rms}^{\prime \prime }$) and spanwise ($w_{rms}^{\prime \prime }$) r.m.s. velocity fluctuations exhibit better similarity in the outer boundary layer than those in the inner layer for cases with different Mach numbers and wall temperature conditions in the inner scaling. Conversely, in the semi-local scaling, the turbulence intensities in the three directions for cases with different inflow conditions collapse well in the viscous sublayer and log layer. These findings are consistent with previous studies by Zhang et al. (Reference Zhang, Duan and Choudhari2018) and Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). In addition, the peak of turbulence intensities in the streamwise, wall-normal and spanwise directions of different cases can have a better collapse by considering the variation of the mean density, which proves the validity of the Morkovin transformation (Morkovin Reference Morkovin1962). A similar trend can be found in the Reynolds shear stress in figure 4. As shown in the figures, after considering the bulk viscosity, the r.m.s. wall-normal velocity fluctuations decrease in the viscous sublayer because of the increase of the dilatational dissipation. And as shown in figure 4, the bulk viscosity causes the Reynolds shear stress to decrease slightly in the log layer.

Figure 3.

Streamwise, wall-normal and spanwise turbulence intensities scaled according to the Morkovin transformation in (a,c,e) the inner scaling, and (b,d,f) the semi-local scaling.

Figure 4.

Reynolds shear stress scaled by the Morkovin transformation in the (a) inner scaling and (b) semi-local scaling.

Figure 5 shows the turbulent Mach number for different cases in the inner scaling and semi-local scaling, respectively. It can be seen that the bulk viscosity slightly reduces the peak value of turbulent Mach number $Ma_t$, indicating that the bulk viscosity weakens the compressibility effects of the hypersonic turbulent boundary layer. The maximum value of the turbulent Mach number increases as the Mach number increases and the wall temperature decreases, in agreement with the results of Duan et al. (Reference Duan, Beekman and Martín2010), Lagha et al. (Reference Lagha, Kim, Eldredge and Zhong2011) and Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022).

Figure 5.

Turbulent Mach number for different cases in (a) the inner scaling, and (b) the semi-local scaling.

3.3. Turbulent kinetic energy budget

Turbulent kinetic energy is analysed to quantify the influence of the bulk viscous term on the hypersonic boundary layer turbulence flow, which is commonly used to characterize the fluctuating motion of a fluid per unit mass and is defined as

(3.6)\begin{equation} \tilde{k}=\frac{1}{2}\,\frac{\overline{\rho u_{k}^{\prime \prime } u_{k}^{\prime \prime } } }{\bar{\rho}} . \end{equation}

The budget equation of turbulent kinetic energy is

(3.7)\begin{equation} \frac{\partial ( \bar{\rho} \tilde{k} ) }{\partial t} +\frac{\partial ( \bar{\rho} \tilde{k} \widetilde{u_{j} } ) }{\partial x_{j} }=P+T+\varPi +\varPhi_{dif}+ \varPhi_{dis}+ST , \end{equation}

where

(3.8)$$\begin{gather} P={-}\overline{\rho u_{i}^{\prime \prime } v^{\prime \prime }}\,\frac{\partial \widetilde{u_{i}} }{\partial y } , \end{gather}$$

(3.9)$$\begin{gather}T={-}\frac{1}{2}\,\frac{\partial}{\partial y } \overline{\rho u_{i}^{\prime \prime } u_{i}^{\prime \prime } v^{\prime \prime }} , \end{gather}$$

(3.10)$$\begin{gather}\varPi =\varPi_{t} +\varPi_{d}={-} \frac{\partial}{\partial y } \overline{ v^{\prime \prime } p^{\prime}}+\overline{ p^{\prime}\,\frac{\partial u_{i}^{\prime \prime } }{\partial x_{i}}}, \end{gather}$$

(3.11)$$\begin{gather}\varPhi_{dif} = \frac{\partial}{\partial y } \overline{u_{i}^{\prime \prime } \sigma_{iy}^{\prime}}, \end{gather}$$

(3.12)$$\begin{gather}\varPhi_{dis} ={-} \overline{ \sigma_{ij}^{\prime}\,\frac{\partial u_{i}^{\prime \prime } }{\partial x_{j} }} , \end{gather}$$

(3.13)$$\begin{gather}ST ={-} \overline{v^{\prime \prime }}\,\frac{\partial \bar{p} }{\partial y } +\overline{u_{i} ^{\prime \prime }}\,\frac{\partial \overline{\sigma_{ij}} }{\partial x_{j} }-\bar{\rho}\tilde{k}\,\frac{\partial \tilde{v} }{\partial y} . \end{gather}$$

The physical significance of each term in the turbulent kinetic energy budget equation is: $P$ is the production term of turbulent kinetic energy, which characterizes the energy input from Reynolds stress to the turbulent fluctuating motion through the deformation rate of the mean motion; $T$ denotes the turbulent diffusion term, which characterizes the diffusion of turbulent kinetic energy generated by the fluctuating motion; $\varPi$ includes the pressure diffusion term and pressure dilatation term, where $\varPi _{t}$ is the pressure diffusion term, and $\varPi _{d}$ is the pressure dilatation term; $\varPhi _{dif}$ represents the viscous diffusion term, which characterizes the spatial transport of turbulent kinetic energy due to the viscous stress; and $\varPhi _{dis}$ denotes the viscous dissipation term, which represents the viscous dissipation due to the turbulent fluctuating motion.

Figures 6(a–c) present the budget terms of turbulent kinetic energy normalized by the conventional inner scaling, and figure 6(d) shows the turbulent kinetic energy transport in the semi-local scaling. Compared to the inner scaling, the turbulent kinetic energy transport in semi-local scaling for different Mach numbers and wall temperature conditions collapses better, which is consistent with the results of Duan et al. (Reference Duan, Beekman and Martín2011) and Zhang et al. (Reference Zhang, Duan and Choudhari2018). In addition, the ratio of the turbulent kinetic energy production term $P$ to the viscous dissipation term $\varPhi _{dis}$ in the log layer is greater than 1 in the semi-local scaling, while the ratio of the turbulent kinetic energy production term $P$ to the sum of the viscous dissipation term $\varPhi _{dis}$ and the turbulent diffusion term $T$ is approximately 1, which is consistent with the findings of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021) and Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). As shown in the figures, the bulk viscosity significantly reduces the pressure diffusion term $\varPi _{t}$ and pressure dilatation term $\varPi _{d}$ near the wall. The wall-normal velocity fluctuations in the pressure diffusion term are much smaller than the pressure fluctuations, so the pressure diffusion term $\varPi _{t}$ near the wall is determined by the gradient of the wall-normal velocity fluctuations. And the bulk viscosity will significantly reduce the gradient of the wall-normal velocity fluctuations near the wall, as shown in figures 3(c) and 3(d). Therefore, the pressure diffusion term $\varPi _{t}$ near the wall experiences a notable reduction. Similarly, the pressure dilatation term $\varPi _{d}$ represents the influence of the fluctuating density generated by the compressibility effect on the growth rate of turbulent kinetic energy. The presence of bulk viscosity causes a decrease in fluid compressibility and density fluctuations near the wall, resulting in a significant reduction of the pressure dilatation term $\varPi _{d}$.

Figure 6.

Turbulent kinetic energy (TKE) budget. Wall-normal distribution of (a) $Ma6T_{w}0.76$, (b) $Ma8T_{w}0.48$ and (c) $Ma6T_{w}0.25$ with (solid lines) and without (dashed lines) bulk viscosity in inner scaling, and (d) turbulent kinetic energy budget for different cases in semi-local scaling. The turbulent kinetic energy budget consists of the production $P$ (red lines), turbulent diffusion $T$ (green lines), pressure diffusion $\varPi _{t}$ (orange lines), pressure dilatation $\varPi _{d}$ (magenta lines), viscous diffusion $\varPhi _{dif}$ (blue lines) and viscous dissipation $\varPhi _{dis}$ (black lines) in (a–c). Plot (d) consists of the turbulent kinetic energy budget of $Ma6T_{w}0.76$ (black lines), $Ma8T_{w}0.48$ (red lines) and $Ma6T_{w}0.25$ (blue lines) with and without bulk viscosity. Solid lines are for $\mu _{b}/\mu =100$; dashed lines are for $\mu _{b}/\mu =0$.

4.1. Mean thermodynamic statistics

Figure 7 displays mean density and mean temperature for different cases. The mean density and mean temperature considering the bulk viscosity are in good agreement with the results without considering bulk viscosity, so the influence of bulk viscosity on the mean density and mean temperature can be ignored. The effects of Mach number and wall cooling on statistics, such as mean density, are consistent with the results of Duan et al. (Reference Duan, Beekman and Martín2010) and Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022). As the wall-normal distance increases, the mean density in the boundary layer decreases slightly in the viscous sublayer, then increases linearly. The mean density continues to increase linearly in the log layer and the outer layer of the boundary layer with a slope smaller than that of the viscous sublayer. The mean temperature increases slightly in the near-wall region, and this phenomenon becomes more significant as the wall temperature decreases. As the Mach number increases, the velocity gradient near the wall increases, resulting in an increase in wall shear stress and an increase in temperature gradient near the wall. As the wall temperature decreases, the viscosity near the wall increases, and consequently, the temperature gradient near the wall also intensifies. However, with an increase in the wall-normal distance, the temperature gradient gradually decreases. Moreover, higher Mach numbers lead to an increase in the mean temperature within the boundary layer, while decreasing wall temperature results in a reduction of the mean temperature, causing the peak value of the mean temperature to move away from the wall.

Figure 7.

Wall-normal distributions of (a) mean density and (b) mean temperature in outer scaling.

Figure 8 plots the relation between mean temperature and mean velocity comparing with the Walz (Reference Walz1969) equation and the modified mean temperature–velocity relation of Zhang et al. (Reference Zhang, Bi, Hussain and She2014), respectively. The classical Walz (Reference Walz1969) equation is

(4.1)\begin{equation} \frac{T}{T_{\infty}} =\frac{T_{w}}{T_{\infty}}+\frac{T_{r}-T_{w}}{T_{\infty}}\left ( \frac{u}{u_{\infty}} \right ) +\frac{T_{\infty}-T_{r}}{T_{\infty}}\left ( \frac{u}{u_{\infty}} \right )^{2} , \end{equation}

where $T_{w}$ is wall temperature, $T_{r} =T_{\infty } ( 1+0.5r ( \gamma -1 )Ma_{\infty }^{2} )$ denotes the recovery temperature, and $r=0.89$ is the recovery factor.

Figure 8.

Relation between mean temperature and mean velocity: (a) equation of Walz (Reference Walz1969); (b) the modified equation proposed by Zhang et al. (Reference Zhang, Bi, Hussain and She2014). Symbols indicate theory results; lines indicate DNS results.

Zhang et al. (Reference Zhang, Bi, Hussain and She2014) modified the relation between mean temperature and mean velocity by considering the variation of heat flux at the wall:

(4.2)\begin{equation} \frac{T}{T_{\infty}} =\frac{T_{w}}{T_{\infty}}+\frac{T_{rg}-T_{w}}{T_{\infty}}\left ( \frac{u}{u_{\infty}} \right ) +\frac{T_{\infty}-T_{rg}}{T_{\infty}}\left ( \frac{u}{u_{\infty}} \right )^{2} , \end{equation}

where $T_{rg}=T_{\infty }+r_{g}u_{\infty }^{2}/2c_{p}$, $r_{g}=2c_{p} ( T_{w}-T_{\infty } )/ u_{\infty }^{2} -2p_{r} q_{w}/ ( u_{\infty }\tau _{w} )$ and $q_{w} = - ( k\,\partial T / \partial y )_{w}$.

As shown in the figure, the mean temperature–velocity relation modified by Zhang et al. (Reference Zhang, Bi, Hussain and She2014) is more accurate than that predicted by the Walz equation, especially as the Mach number increases and the wall temperature decreases, which is consistent with the results obtained by Zhang et al. (Reference Zhang, Duan and Choudhari2018), Huang et al. (Reference Huang, Duan and Choudhari2022) and Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). However, the curves for $\mu _b/\mu =0$ and $\mu _b/\mu =100$ overlap approximately, indicating that bulk viscosity has almost no effect on the relation between mean temperature and mean velocity, which can be inferred from the results in figure 7(b).

4.2. Fluctuating thermodynamic statistics

Figure 9 shows the variation of thermodynamic fluctuations along the wall-normal distance, such as $\rho ^{\prime }_{rms}/\bar {\rho }$ in the inner scaling, $T^{\prime }_{rms}/\bar {T}$ in the inner scaling, and $p^{\prime }_{rms}/\tau _w$ in the inner scaling and outer scaling, respectively. As the Mach number increases, the peak of r.m.s. density fluctuations $\rho _{rms}^{\prime }$ and r.m.s. temperature fluctuations $T_{rms}^{\prime }$ at the boundary layer edge increases, and as the wall temperature decreases, a secondary peak of temperature fluctuations occurs in the buffer region due to the presence of a maximum value of the mean temperature $\bar {T}$ in the near-wall region at low wall temperatures, which enhances the contribution of the coupling of wall-normal velocity fluctuations $v_{rms}^{\prime \prime }$ and mean temperature $\bar {T}$ to the r.m.s. temperature fluctuations, in agreement with the observations of Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). In the hypersonic turbulent boundary layer, the intensity of the r.m.s. pressure fluctuations $p_{rms}^{\prime }$ is enhanced as the Mach number increases and wall temperature decreases, which is consistent with the findings of Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022) and Xu, Wang & Chen (Reference Xu, Wang and Chen2023). The bulk viscosity decreases the r.m.s. density fluctuations $\rho _{rms}^{\prime }$, r.m.s. temperature fluctuations $T_{rms}^{\prime }$ and r.m.s. pressure fluctuations $p_{rms}^{\prime }$ in the inner layer of the boundary layer, especially in the viscous sublayer, where the reduction of thermodynamic fluctuations is more significant. This is because the rapid change of the thermodynamic state of the gas in the near-wall region of the boundary layer causes a non-equilibrium between the thermodynamic and mechanical pressures, and the bulk viscosity rebalances the two pressures by reducing the fluctuations of the thermodynamic quantities of the fluid in the boundary layer. As the Mach number increases or the wall temperature decreases, the effect of bulk viscosity becomes more significant.

Figure 9.

Profiles of the r.m.s. thermodynamic fluctuations for different cases along the wall-normal distance: (a) the r.m.s. density fluctuations scaled by mean density $\rho ^{\prime }_{rms}/\bar {\rho }$ in the inner scaling $y^+$; (b) the r.m.s. temperature fluctuations scaled by mean temperature $T^{\prime }_{rms}/\bar {T}$ in the inner scaling $y^+$; and the r.m.s. pressure fluctuations scaled by wall shear stress $p^{\prime }_{rms}/\tau _w$ in (c) the inner scaling $y^+$ and (d) the outer scaling $y/\delta$.

4.3. The strong Reynolds analogy

Next, the impact of bulk viscosity on the relation between velocity fluctuations and temperature fluctuations is investigated. The turbulent Prandtl number and the correlation of velocity fluctuations and temperature fluctuations (Morkovin Reference Morkovin1962) are

(4.3)$$\begin{gather} Pr_{t}=\frac{\overline{\rho u^{\prime \prime }v^{\prime \prime }}\,\partial \tilde{T}/\partial y}{\overline{\rho v^{\prime \prime }T^{\prime \prime }}\,\partial \tilde{u}/\partial y} \approx 1, \end{gather}$$

(4.4)$$\begin{gather}R_{u^{\prime \prime }T^{\prime \prime }}=\frac{\widetilde{uT}-\tilde{u} \tilde{T}}{u_{rms}^{\prime \prime }T_{rms}^{\prime \prime }}. \end{gather}$$

As shown in figure 10(a), the effect of bulk viscosity on the turbulent Prandtl number $Pr_{t}$ can be neglected. The turbulent Prandtl number $Pr_{t}$ is approximately 1 in the whole boundary layer, indicating that the turbulent Prandtl number is not affected by the variation of Mach number, wall temperature and other conditions. As shown in figure 10(b), $-R_{u^{\prime \prime }T^{\prime \prime }}$ remains at approximately 0.6 over most of the boundary layer, in agreement with the results obtained by Guarini et al. (Reference Guarini, Moser, Shariff and Wray2000) and Huang et al. (Reference Huang, Duan and Choudhari2022). Since the mean temperature maximum will shift to the right when the wall temperature decreases, as shown in figure 7(c), $-R_{u^{\prime \prime }T^{\prime \prime }}$ will shift to the right, which is consistent with Coleman, Kim & Moser (Reference Coleman, Kim and Moser1995), Duan et al. (Reference Duan, Beekman and Martín2010) and Huang et al. (Reference Huang, Duan and Choudhari2022). Notably, the mean and fluctuating streamwise velocity and temperature are almost unaffected by the bulk viscosity in the log region and the outer layer. Therefore, the bulk viscosity has little effect on the correlation between velocity fluctuations and temperature fluctuations within most of the boundary layers.

Figure 10.

Relation between temperature fluctuations and velocity fluctuations: (a) turbulent Prandtl number $Pr_{t}$; (b) temperature–velocity correlation coefficient $-R_{u^{\prime \prime }T^{\prime \prime }}$; (c) the SRA relations; and (d) the modified SRA proposed by Huang (HSRA).

Figure 10(c) shows the strong Reynolds analogy (SRA) theory for different cases. The SRA theory was proposed by Morkovin (Reference Morkovin1962) for analysing the correlation between temperature fluctuations and velocity fluctuations as follows:

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

By considering the variation of wall heat flux and eliminating wall temperature dependence, Gaviglio (Reference Gaviglio1987), Rubesin (Reference Rubesin1990) and Huang et al. (Reference Huang, Coleman and Bradshaw1995) proposed a modified SRA (GSRA, RSRA, HSRA, respectively) of the form

(4.6)\begin{equation} \frac{T_{rms}^{\prime \prime }/\tilde{T} }{( \gamma -1 )\,Ma^{2}\,( u_{rms}^{\prime \prime } /\tilde{u} )} \approx \frac{1}{c( 1-\partial \widetilde{T_{t} } /\partial \widetilde{T } ) }, \end{equation}

where $c$ is $1.0$, $1.34$ and $Pr_{t}$, respectively. It can be found from figure 10(c) that the effect of bulk viscosity on the SRA theory can be neglected. Looking at the left-hand side of the above equation, it can be seen that it is a function of temperature and velocity fluctuations. From the previous results, it can be seen that the temperature fluctuations and velocity fluctuations are affected by the wall temperature, therefore, when the wall temperature changes, the SRA theory is no longer applicable, which is consistent with the results of Martin (Reference Martin2007) and Duan et al. (Reference Duan, Beekman and Martín2011). As shown in figure 10(d), by considering the effect of the wall heat flux, the generalized Reynolds analogy theory modified by Huang et al. (Reference Huang, Coleman and Bradshaw1995) (HSRA) is able to well describe the relation between temperature fluctuations and velocity fluctuations for wall-cooling cases, so that the HSRA of the cases under different working conditions is close to 1. Additionally, bulk viscosity has almost no effect on the HSRA, and this result is not affected by the variations of Mach number and wall temperature.

4.4. Heat flux distribution

Figure 11 shows the distributions of the heat flux and skin friction coefficient. Figures 11(a,b) plot the distributions of the skin friction coefficient $C_f$ and wall conductive heat flux $-k\,{\partial T}/{\partial y}$ with the relative streamwise distance $x/L_x$, respectively. In order to make it easier to compare the statistics of different cases in the same figure, the streamwise location $x$ is normalized by the streamwise length of computational domain $L_x$ (Xu et al. Reference Xu, Wang and Chen2022). As shown in the figures, the skin friction coefficient $C_f$ and wall conductive heat flux $-k\,{\partial T}/{\partial y}$ will not vary with the streamwise distance when the turbulent boundary layer is sufficiently evolved. And the skin friction coefficient $C_f$ increases as the wall temperature decreases, which is consistent with the conclusion of Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022). However, after considering bulk viscosity, the skin friction coefficient $C_f$ decreases slightly as the wall temperature decreases. As the Mach number increases and the wall temperature decreases, the wall conductive heat flux $-k\,{\partial T}/{\partial y}$ increases. If the influence of thermal conductivity coefficient $k$ is not considered temporarily, then it indicates that the wall-normal temperature gradient ${\partial T}/{\partial y}$ increases. This conclusion is consistent with what we obtained in § 4.1. In addition, the effect of bulk viscosity on wall conductive heat flux $-k{\partial T}/{\partial y}$ is negligible. Figures 11(c,d) plot the variation of the conductive heat flux $-k\,{\partial T}/{\partial y}$ and turbulent heat flux $\overline {\rho v^{\prime \prime } T^{\prime \prime } }$ with the wall-normal distance in inner scaling $y^{+}$, respectively. From the figures, it can be seen that the conductive heat flux $-k\,{\partial T}/{\partial y}$ is dominant, and the turbulent heat flux $\overline {\rho v^{\prime \prime } T^{\prime \prime } }$ is very small in the near-wall region due to the velocity in the viscous sublayer being approximately 0. The total heat flux inside the viscous sublayer is determined mainly by the conductive heat flux $-k\,{\partial T}/{\partial y}$. In the log region, both the wall-normal velocity fluctuations $v^{\prime \prime }$ and the temperature fluctuations $T^{\prime \prime }$ reach their maximum values, while the mean temperature gradient ${\partial T}/{\partial y}$ decreases as the wall distance increases, so that the total heat flux magnitude in the log region depends mainly on the turbulent heat flux $\overline {\rho v^{\prime \prime } T^{\prime \prime } }$. Due to the effect of dilatational dissipation, the bulk viscosity weakens the compressibility of the hypersonic turbulent boundary layer, and the wall-normal velocity fluctuations and temperature fluctuations decrease in the viscous sublayer, so the turbulent heat flux $\overline {\rho v^{\prime \prime } T^{\prime \prime } }$ decreases slightly in both the viscous sublayer and the log layer of the boundary layer by considering bulk viscosity. In contrast, bulk viscosity has almost no effect on the mean temperature, so bulk viscosity has less effect on the thermal conductivity heat flux $-k\,{\partial T}/{\partial y}$, and in conclusion, bulk viscosity decreases slightly the total heat flux in the viscous sublayer and log layers. In addition, the wall-normal turbulent heat flux $\overline {\rho v^{\prime \prime } T^{\prime \prime } }$ increases significantly with increasing Mach number, which is consistent with the results of Huang et al. (Reference Huang, Nicholson, Duan, Choudhari and Bowersox2020, Reference Huang, Duan and Choudhari2022).

Figure 11.

Heat flux and skin friction coefficient. The distributions of (a) the skin friction coefficient $C_f$ and (b) the conductive heat flux $-k{\partial T}/{\partial y}$ with the relative streamwise coordinates $x/L_x$, where $L_x$ is the length of the computational domain in the streamwise direction. The wall-normal distributions of (c) the conductive heat flux $-k\,{\partial T}/{\partial y}$ and (d) the wall-normal turbulent heat flux $\overline {\rho v^{\prime \prime } T^{\prime \prime } }$.