1. Introduction
Enhanced knowledge of hypersonic turbulent boundary layers holds significant implications for various aspects of hypersonic vehicle design and performance. In contrast to subsonic/supersonic flight, the higher speed of hypersonic flight results in a significant conversion of kinetic energy into internal energy within the boundary layer. Consequently, it places a greater demand on thermal protection technology for the surfaces of vehicles. In subsonic/supersonic flight, the surface of the vehicle is approximately an adiabatic wall, while the wall temperature of the hypersonic vehicle is significantly lower than that of the adiabatic wall (Duan, Beekman & Martín Reference Duan, Beekman and Martín2010; Xu et al. Reference Xu, Wang, Wan, Yu, Li and Chen2021b; Ou, Wang & Chen Reference Ou, Wang and Chen2024). The accurate modelling of the wall-cooling and compressibility effects in a hypersonic turbulent boundary layer is of great significance for the prediction of the surface heat flux and the design of the thermal protection of the vehicle (Huang, Duan & Choudhari Reference Huang, Duan and Choudhari2022).
Direct numerical simulations (DNS) play a pivotal role in advancing the understanding of hypersonic boundary layers, which are crucial in the development of hypersonic technology. The DNS allow for highly accurate and detailed simulations, providing a wealth of three-dimensional flow field data. This enables researchers to investigate the complex flow phenomena comprehensively, including small-scale structures near the wall, which are otherwise challenging to access through experimental means. The high Mach numbers and low wall temperatures inherent in hypersonic boundary layers pose unique challenges, and DNS emerge as a powerful tool to tackle these complexities. By conducting DNS studies under various Mach numbers, Reynolds numbers and wall temperature conditions, researchers can gain profound insights into flow physics, turbulence characteristics and heat transfer mechanisms, as evidenced by the studies listed in table 1.
Table 1. Summary of DNS studies of supersonic and hypersonic turbulent boundary layers at different Mach numbers 
$Ma$, friction Reynolds numbers 
$Re_{\tau }$ and wall-to-recovery temperature ratios 
$T_{w}/T_{r}$ in recent years. The ‘
$\checkmark$’ indicates consideration of thermal or/and chemical non-equilibrium phenomena.
Duan, Beekman & Martín (Reference Duan, Beekman and Martín2011) investigated the effect of the Mach number (
$0.3\le Ma \le 12$) on the compressible turbulent flat plate boundary layer, and found that the compressibility of the fluid is enhanced with the increase of Mach number, while the scaling relations remain valid, such as Morkovin's scaling and the strong Reynolds analogy. Subsequently, Lagha et al. (Reference Lagha, Kim, Eldredge and Zhong2011) further performed DNS of the compressible turbulent boundary layer with Mach number up to 20, and found that the velocity dilatation varies significantly with increasing Mach number, but can still be normalized by considering the average density variation. Duan et al. (Reference Duan, Beekman and Martín2010) studied the effect of wall temperature (
$0.18\le T_{w} /T_{r} \le 1.00$) on the compressible turbulent flat plate boundary layer, and found that the compressibility of the fluid is enhanced by decreasing the wall temperature, while the scaling relations are still suitable. Zhang, Duan & Choudhari (Reference Zhang, Duan and Choudhari2018) further conducted DNS of compressible turbulent boundary layers with Mach numbers from 2.5 to 14 and wall-to-recovery temperature ratios from 0.18 to 1.0. They evaluated the applicability of the compressibility transformations, and verified the validity of several new scaling laws considering wall heat flux. The results of compressible turbulent boundary layer with Mach numbers from 11 to 14 and wall-to-recovery temperature ratio 0.2 by Huang et al. (Reference Huang, Nicholson, Duan, Choudhari and Bowersox2020) showed that the algebraic energy flux model can predict the streamwise turbulent heat flux better than the commonly constant turbulent Prandtl number model, based on the good prediction of statistics such as wall-normal turbulent heat flux. For the effect of wall cooling on pressure fluctuations, Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022) found that the cold-wall effect suppressed fast and slow pressures, resulting in a decrease in pressure fluctuations in the subsonic/supersonic case, while enhancing the compressible pressure, especially in the hypersonic case, caused an increase in pressure fluctuations. Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) studied the effect of high Reynolds number and wall cooling on the supersonic and hypersonic zero pressure gradient turbulent boundary layer, and found that uniform momentum spaces for the velocity and temperature fields exist in the high Reynolds number high-speed turbulent boundary layer.
Duan & Martín (Reference Duan and Martín2011) were the first to study the impact of high enthalpy effects on temporally evolving compressible turbulent boundary layers using DNS, and found that many scaling relations for low enthalpy boundary layers are still valid in high enthalpy boundary layers, and proposed a modified Crocco relation that is suitable for non-adiabatic cold walls and real gas effects. Subsequently, Passiatore et al. (Reference Passiatore, Sciacovelli, Cinnella and Pascazio2021) and Renzo & Urzay (Reference Renzo and Urzay2021) studied the effect of chemical non-equilibrium effects on the spatially evolving hypersonic boundary layer under quasi-adiabatic and wall-cooling conditions, respectively. Passiatore et al. (Reference Passiatore, Sciacovelli, Cinnella and Pascazio2022) further investigated the effect of thermochemical non-equilibrium effects on the wall-cooling spatially evolving hypersonic turbulent boundary layer. All of those studies reveal that the turbulence intensity and the fluctuations of thermodynamic quantities in the near-wall region of the hypersonic turbulent boundary layer are intensified at high Mach numbers and low wall temperatures. These findings suggest an enhancement in the compressibility of the turbulent boundary layer.
According to the Stokes assumption, the bulk viscosity coefficient in the Navier–Stokes equation is typically considered small compared to the shear viscosity coefficient, allowing it to be ignored in most numerical solutions of the Navier–Stokes equation (Stokes Reference Stokes1851). However, this assumption does not hold for highly compressible fluids or polyatomic molecules with significant internal energy excitation. In such cases, the effect of bulk viscosity becomes non-negligible. Particularly in the context of the hypersonic compressible turbulent boundary layer, where strong compressibility is a defining characteristic, the impact of bulk viscosity becomes notably important. Under these conditions, accounting for bulk viscosity is crucial for capturing accurately the behaviour and characteristics of the turbulent boundary layer in hypersonic flows.
More precisely, the viscous stress 
$\tau _{ij}$ with bulk viscosity in momentum equation is expressed as
\begin{equation} \tau_{ij} =2\mu S_{ij}^{(d) } +\mu_{b} u_{k,k} \delta_{ij}, \end{equation}
where 
$\mu$ is the shear viscosity coefficient, 
$\mu _{b}$ is the bulk viscosity coefficient, 
$S_{ij}^{ ( d ) }= S_{ij} - u_{k,k}\delta _{ij} /3$ represents the deviatoric part of the strain rate tensor, 
$S_{ij}$ denotes the strain rate tensor, 
$u_{k,k}$ denotes the velocity divergence, and 
$\delta _{ij}$ is the Kronecker symbol. Then the related viscous dissipation is expressed as (Lele Reference Lele1994)
\begin{equation} \varPhi = \tau_{ij}S_{ij}= 2 \mu S_{ij}^{(d) }S_{ij}^{(d) }+\mu_{b} u_{k,k} ^{2}, \end{equation}
where 
$2 \mu S_{ij}^{ ( d ) }S_{ij}^{ ( d ) }$ is the shear dissipation, and 
$\mu _{b} u_{k,k}^{2}$ is the dilatational dissipation. The dilatational dissipation is small compared to the shear dissipation, but many studies have shown that dilatational dissipation plays an important role in compressible turbulence, which is closely related to bulk viscosity (Sarkar et al. Reference Sarkar, Erlebacher, Hussaini and Kreiss1991; Lele Reference Lele1994; Zhu et al. Reference Zhu, Zhang, Chen, Yuan, Wu, Chen, Lee and Gad-el Hak2016).
Furthermore, the total stress can be expressed as
\begin{equation} \sigma_{ij}={-}p \delta_{ij}+ 2 \mu S_{ij}^{(d) }+\mu_{b}u_{k,k} \delta_{ij}, \end{equation}
according to the definition of mechanical pressure 
$p_{m}=-\sigma _{ii}/3$, using the above equation, the relation between mechanical pressure and thermodynamic pressure can be obtained as
\begin{equation} p_{m}=p-\mu_{b}u_{k,k}, \end{equation}where the mechanical pressure is related to the translational energy of gas molecules, and the thermodynamic pressure corresponds typically to the internal energy of molecules, including molecular translational, rotational and vibrational energies of diatomic and polyatomic gases. Classically, the bulk viscosity is employed to characterize relaxation phenomena of these energy modes, noting that the non-zero bulk viscosity of monatomic gases was reported in the literature (DeGottardi & Matveev Reference DeGottardi and Matveev2023; Sharma, Pareek & Kumar Reference Sharma, Pareek and Kumar2023), but very small. In the near-wall region of the hypersonic turbulent boundary layer, the effects of compression or dilatation are particularly significant, leading to intensified fluctuations of thermodynamic quantities. The rapid changes in the thermodynamic state of the gas give rise to non-equilibrium phenomena of energy in this region. The role of bulk viscosity becomes crucial in the re-equilibrium process of energy, underscoring its significance in the hypersonic turbulent boundary layer.
The bulk viscosity of a fluid can be determined by experimental, theoretical and numerical methods. As shown in table 2, the experimental methods include mainly acoustic wave absorption and dispersion experiments, and Rayleigh–Brillouin scattering experiments (Eu & Ohr Reference Eu and Ohr2001; Vieitez et al. Reference Vieitez, van Duijn, Ubachs, Witschas, Meijer, De Wijn, Dam and van de Water2010). The acoustic absorption and scattering experiments and Rayleigh–Brillouin scattering experiments measure the bulk viscosity at different frequencies, which leads to a significant difference in the bulk viscosity coefficients obtained from the two methods (Prangsma, Alberga & Beenakker Reference Prangsma, Alberga and Beenakker1973; Pan, Shneider & Miles Reference Pan, Shneider and Miles2005). In the area of theoretical research, Tisza (Reference Tisza1942) investigated the bulk viscosity coefficient of an ideal gas by theoretical methods and concluded that the bulk-to-shear viscosity ratio of 
${\rm CO}_{2}$ gas at room temperature and pressure is of the order of 
$10^{3}$. Zuckerwar & Ash (Reference Zuckerwar and Ash2006) developed an analytical formulation with multiple dissipative processes based on the variational principle of Hamilton, and found that the bulk-to-shear viscosity ratio of air is greater than 
$1.6\times 10^4$. Subsequently, Cramer (Reference Cramer2012) studied the variation of bulk viscosity with temperature and pressure for a variety of gases, and found that the bulk viscosities for a variety of fluids, including common polyatomic gases, are hundreds or thousands of times higher than the shear viscosity. However, Kustova, Mekhonoshina & Kosareva (Reference Kustova, Mekhonoshina and Kosareva2019) proposed a new bulk viscosity theory using the Chapman–Enskog method, which suggests that the 
${\rm CO}_{2}$ bulk viscosity and shear viscosity coefficients are of the same order at room temperature. In addition, many researchers have developed different theoretical models that include temperature-dependent bulk viscosity in recent years, such as the variable specific heat two-temperature Navier–Stokes equation (Kosuge & Aoki Reference Kosuge and Aoki2022), a state-to-state model suitable for mixtures of gases (Bruno & Giovangigli Reference Bruno and Giovangigli2022), and a kinetic model with temperature-dependent vibrational degrees of freedom (Li & Wu Reference Li and Wu2022). Despite the many methods of evaluating the bulk viscosity coefficients, there are still large uncertainties in the bulk viscosity coefficients of common gases such as air, 
${\rm N}_2$ and 
${\rm CO}_2$ (Graves & Argrow Reference Graves and Argrow1999; Vieitez et al. Reference Vieitez, van Duijn, Ubachs, Witschas, Meijer, De Wijn, Dam and van de Water2010; Jaeger, Matar & Müller Reference Jaeger, Matar and Müller2018; Sharma & Kumar Reference Sharma and Kumar2023). The Navier–Stokes equation with a bulk viscosity term holds if the local thermodynamic equilibrium condition is satisfied (Vincenti & Kruger Reference Vincenti and Kruger1965). When the bulk viscosity is large, the maximum energy relaxation time of the molecules is much smaller than the time scale of the compressible turbulent boundary layer, and the local thermodynamic equilibrium condition is satisfied.
Table 2. The bulk-to-shear viscosity ratios (
$\mu _b/\mu$) of different gases determined by experimental methods. In the table, SA denotes acoustic absorption method, and CRBS and SRBS indicate coherent and spontaneous Rayleigh–Brillouin scattering, respectively. Also, 
$T_g$ denotes the temperature of the gas during the experimental measurements.
At present, research on the effect of bulk viscosity on compressible turbulence is relatively scarce, and the main focus is on the effect of bulk viscosity on compressible homogeneous isotropic turbulence (Liao, Peng & Luo Reference Liao, Peng and Luo2009; Cramer & Bahmani Reference Cramer and Bahmani2014; Boukharfane et al. Reference Boukharfane, Ferrer, Mura and Giovangigli2019; Touber Reference Touber2019). Emanuel (Reference Emanuel1992) first studied the effect of bulk viscosity on hypersonic flow. The influence of bulk viscosity on the hypersonic laminar boundary layer at large Reynolds number is studied by presenting a first-order boundary layer equation including bulk viscosity at high Reynolds number. Pan & Johnsen (Reference Pan and Johnsen2017) studied the decay of compressible homogeneous isotropic turbulence using DNS in the bulk-to-shear viscosity ratio range 0–1000, investigating mainly the effect of bulk viscosity on turbulent kinetic energy dissipation. Chen et al. (Reference Chen, Wang, Wang, Wan and Chen2019) studied the effects of bulk viscosity on Mach number scaling laws and small-scale structures of homogeneous isotropic turbulence and homogeneous shear turbulence with turbulent Mach numbers ranging from 0.1 to 0.6, and bulk-to-shear viscosity ratios 0, 10 and 30, using numerical simulation methods. However, no one has systematically investigated the effect of bulk viscosity on the hypersonic compressible turbulent boundary layer. Because of the strong compressibility of the hypersonic turbulent boundary layer and the rapid change of the thermodynamic state of the gas near the wall, bulk viscosity is of great significance in the study of the hypersonic turbulent boundary layer.
Based on the importance of bulk viscosity to the hypersonic compressible flat plate turbulent boundary layer and the relevant findings from DNS, this paper aims to investigate the effect of bulk-to-shear viscosity ratio 
$\mu _{b}/\mu =100$ on the hypersonic compressible turbulent boundary layer using the DNS method. The rest of the paper is organized as follows. The governing equations, cases and parameters used in DNS are given in § 2. The statistics related to velocity are introduced in § 3, including transformed mean velocity profiles, Reynolds stresses and turbulent kinetic energy budgets. Section 4 presents the thermodynamic-related statistics, such as mean and fluctuating thermodynamic variables, strong Reynolds analogy and heat flux. Furthermore, the large-scale turbulent structures and small-scale properties are analysed in §§ 5 and 6, respectively. Additionally, the physical mechanism of the impact of bulk viscosity on turbulent boundary layers is discussed in § 7. Finally, the conclusions of this paper are summarized in § 8.
2. Numerical set-up
The three-dimensional compressible Navier–Stokes equations in conservation form of the calorimetrically perfect gas are
$$\begin{gather} \frac{\partial \rho}{\partial t} +\frac{\partial (\rho u_{j} ) }{\partial x_{j} }=0 , \end{gather}$$
$$\begin{gather}\frac{\partial ( \rho u_{i} ) }{\partial t} +\frac{\partial ( \rho u_{i} u_{j}+p \delta _{ij} )}{\partial x_{j} }=\frac{\partial \sigma_{ij} }{\partial x_{j}}, \end{gather}$$
$$\begin{gather}\frac{\partial E }{\partial t} +\frac{\partial [ ( E+p )u_{j} ]}{\partial x_{j} }=\frac{\partial }{\partial x_{j}}\left ( k\,\frac{\partial T}{\partial x_{j} } \right ) +\frac{\partial ( \sigma_{ij} u_{i}) }{\partial x_{j}} , \end{gather}$$
where 
$i,j=x,y,z$ represent streamwise, wall-normal and spanwise coordinates, respectively. Here, 
$\rho$ is density, 
$u_{j}$ represents velocity in three directions, and 
$p$ is the thermodynamic pressure. Also, 
$E=p/ ( \gamma -1 ) +1/2 \rho u_{k} u_{k}$ denotes total energy per unit volume, where 
$\gamma =1.4$, and 
$k=c_{p} \mu /Pr$ is the thermal conductivity coefficient, with Prandtl number 
$Pr=0.7$ and shear viscosity coefficient 
$\mu$. When the effect of bulk viscosity is considered, the viscous stress tensor 
$\sigma _{ij}$ is expressed as
\begin{equation} \sigma_{ij}=\mu \left ( \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}} -\frac{2}{3}\,\frac{\partial u_{k} }{\partial x_{k}}\,\delta_{ij} \right )+ \mu_{b}\,\frac{\partial u_{k} }{\partial x_{k}}\,\delta_{ij} , \end{equation}
where 
$\mu _{b}$ denotes the bulk viscosity coefficient.
The compressible Navier–Stokes equations are solved by the high-order finite-difference method (Hou et al. Reference Hou, Jin, Feng and Zheng2023). A hybrid scheme of sixth-order central finite difference combined with a fifth-order weighted essentially non-oscillatory method (Jiang & Shu Reference Jiang and Shu1996) is used for the convective terms to ensure that shock-capturing can be performed and to ensure the stability of the numerical simulation, and sixth-order central finite difference is used for the viscous terms. The system is advanced in time using a third-order Runge–Kutta scheme. Inflow turbulence boundary conditions are established by the digital filtering method, non-reflecting boundary conditions for the upper boundary and outflow, no-slip isothermal boundary conditions for the wall, and periodic boundary conditions for the spanwise direction, where the fluid is considered to be statistically homogeneous.
To investigate the influence of bulk viscosity on the hypersonic compressible turbulent boundary layer, six cases are selected carefully for DNS, as shown in table 3. Using the base case with parameters 
$Ma6T_{w}0.76$, the impact of wall cooling is analysed systematically by comparing the results with those of 
$Ma6T_{w}0.25$. Additionally, the effect of Mach number is investigated by comparing the results with those of 
$Ma8T_{w}0.48$. Numerous studies have been conducted for the cases where the bulk-to-shear viscosity coefficient ratio is 0 (
$\mu _{b}/\mu = 0$). Due to the uncertainty of the bulk viscosity coefficient (Vieitez et al. Reference Vieitez, van Duijn, Ubachs, Witschas, Meijer, De Wijn, Dam and van de Water2010; Kustova et al. Reference Kustova, Mekhonoshina and Kosareva2019) and the impact of the large bulk viscosity coefficient (
$\mu _{b}/\mu = O(Re^{1/2})$) on the boundary layer (Cramer & Bahmani Reference Cramer and Bahmani2014), the bulk-to-shear viscosity coefficient ratios 
$\mu _{b}/\mu$ are chosen as 0 and 100, which are of the same magnitude as the square root of inflow Reynolds number (
$Re^{1/2}_{\infty }$). Building upon this established foundation, the DNS are performed on the corresponding cases where the bulk viscosity (
$\mu _{b}/\mu = 100$) is taken into account. In summary, this paper investigates 
$Ma6T_{w}0.76$, 
$Ma8T_{w}0.48$ and 
$Ma6T_{w}0.25$, with and without bulk viscosity, for a total of six DNS cases.
Table 3. The cases and parameters of DNS. The parameters with symbol 
$\infty$ refer to the inflow parameters. The parameters with subscript 
$w$ indicate the wall parameters. The parameters with subscript 
$r$ refer to the recovery parameters. Here, 
$Re_{\tau } = \rho _{w} u_{\tau } \delta / \mu _{w}$ is the friction Reynolds number, 
$Re_{\tau }^{*} = \sqrt {\rho _{\infty } \tau _w}\,\delta / \mu _{\infty }$ is the semi-local friction Reynolds number, 
$Re_{\infty } = \rho _{\infty } u_{\infty } \delta / \mu _{\infty }$ is the inflow Reynolds number, and 
$\mu _{b}$ is the bulk viscosity coefficient.
The main parameters required for DNS are also shown in table 3, including Mach number 
$Ma_{\infty }$, velocity 
$U_{\infty }$, density 
$\rho _{\infty }$, temperature 
$T_{\infty }$ and 
$T_{w}$, where 
$\infty$ denotes the inflow value and 
$w$ denotes the value at the wall. Also, 
$T_{w} /T_{r}$ is the temperature ratio, where 
$T_{r} =T_{\infty } ( 1+0.5r ( \gamma -1 )\,Ma_{\infty }^{2} )$ denotes the recovery temperature, and 
$r=0.89$ is the recovery factor. The friction Reynolds number is defined as 
$Re_{\tau } = \rho _{w} u_{\tau } \delta / \mu _{w}$, where 
$u_{\tau }=\sqrt {\tau _{w}/\rho _{w}}$ is the friction velocity, 
$\tau _{w}=\mu ( \partial u / \partial y )$ represents the shear stress at the wall, 
$\delta$ is the boundary layer thickness based on 
$99\,\%$ of the inflow velocity, and the dynamic viscosity coefficient 
$\mu$ is calculated using the Sutherland formula 
$\mu =T^{3/2}( 1+110.4/T_{\infty } )/( T+110.4/T_{\infty } )$. Also, 
$Re_{\tau }^{*} = \rho _{\infty } \sqrt { \tau _w /\rho _{\infty }}\, \delta / \mu _{\infty }$ is the semi-local friction Reynolds number, and 
$Re_{\infty } = \rho _{\infty } u_{\infty } \delta / \mu _{\infty }$ is the inflow Reynolds number.
In order to ensure that the DNS results remain unaffected by the inflow conditions, a sufficiently long streamwise computational domain is employed. Additionally, to capture accurately the small-scale structures in the near-wall region within the boundary layer, a stretched grid is used in the wall-normal direction, where a minimum grid height 
$\Delta y_{\min }^{+}$ equals 0.46 in the first layer of grid. The streamwise and spanwise directions, however, utilize a uniform grid. The computational domain 
$( L_{x} \times L_{y} \times L_{z} )$, grid size 
$(\Delta x^{+}\times \Delta y^{+} \times \Delta z^{+} )$ and number of grids 
$( N_{x} \times N_{y} \times N_{z} )$ are shown in table 4, where the superscript 
$+$ indicates the inner scaling, 
$y^{+}=y/y_{\tau }$, 
$u^{+}=u/u_{\tau }$, with 
$y_{\tau }=\nu _{w}/u_{\tau }$ and 
$\nu _{w}=\mu _{w}/\rho _{w}$.
Table 4. The parameters of the computational domain and mesh. Here, 
$\delta _{i}$ is the inflow boundary layer thickness. The subscripts 
$x$, 
$y$ and 
$z$ represent the streamwise, wall-normal and spanwise directions, respectively.
Figure 1 displays the outcomes of the van Driest transformed mean velocity, the root mean square (r.m.s.) streamwise, wall-normal and spanwise velocity fluctuations and Reynolds shear stress, the r.m.s. density fluctuations, and the r.m.s. temperature fluctuations, which are compared with the corresponding results from Zhang et al. (Reference Zhang, Duan and Choudhari2018). The excellent agreement observed in the figure demonstrates that the physical setting and numerical tool employed in this study can successfully achieve the required accuracy for DNS.
Figure 1. Comparison between the DNS results without considering bulk viscosity and the results of Zhang et al. (Reference Zhang, Duan and Choudhari2018). Variation of (a) the van Driest (VD) transformed mean velocity, (b) the r.m.s. velocity fluctuations and Reynolds shear stress, (c) the r.m.s. density fluctuations, and (d) the r.m.s. temperature fluctuations along the wall-normal distance in the inner scaling.
It is worth noting here that both standard Reynolds averages and density-weighted (Favre) averages are utilized to characterize turbulence statistics. For a variable 
$f$, the mean component of the Reynolds averages is denoted as 
$\bar {f}$, and the fluctuating component is represented by 
$f^{\prime }$. Consequently, the variable 
$f$ can be expressed as 
$f=\bar {f}+f^{\prime }$. On the other hand, the mean component of the Favre average is denoted as 
$\tilde {f}$, and the fluctuating component is indicated by 
$f^{\prime \prime }$. The Favre average is computed as 
$\tilde {f}=\overline {\rho f}/\bar {\rho }$, where 
$\overline {\rho f}$ represents the density-weighted average of the variable 
$f$, and 
$\bar {\rho }$ denotes the mean density. Hence the variable 
$f$ can be expressed as 
$f=\tilde {f}+f^{\prime \prime }$.
3. Velocity-related statistics
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
$$\begin{gather} u^{+}=y^{+} , \end{gather}$$
$$\begin{gather}u^{+}=( 1/k )\log(\kern1.5pt y^{+} ) +C, \end{gather}$$
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
\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
$$\begin{gather} u_{TL}^{+}=\int_{0}^{{u} ^{+} }\sqrt{\frac{\bar{\rho}}{\rho_{w}}}\left [ 1+\frac{1}{2}\,\frac{1}{\bar{\rho}}\,\frac{{\rm d}\bar{\rho}}{{{\rm d}}y}\,y-\frac{1}{\bar{\mu}}\,\frac{\mathrm{d} \bar{\mu}}{\mathrm{d} y}\,y \right ] {{\rm d}}\bar{u}^{+} , \end{gather}$$
$$\begin{gather}y^{*}=\frac{\bar{\rho}( \tau_{w}/\bar{\rho} )^{1/2}y }{\bar{\mu}} . \end{gather}$$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
$$\begin{gather} u_{V}^{+}=\int_{0}^{{u} ^{+} }\frac{( \bar{\rho}/{\rho_{w}} )^{1/2} }{ (\bar{\mu}/{\mu_{w}} )^{1/2}}\, {{\rm d}}\bar{u}^{+} , \end{gather}$$
$$\begin{gather}y_{V}^{+}=\int_{0}^{{y} ^{+} }\frac{( \bar{\rho}/{\rho_{w}})^{1/2} }{ (\bar{\mu}/{\mu_{w}} )^{3/2}}\, {{\rm d}}\bar{y}^{+} . \end{gather}$$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
\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
\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
\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
$$\begin{gather} P={-}\overline{\rho u_{i}^{\prime \prime } v^{\prime \prime }}\,\frac{\partial \widetilde{u_{i}} }{\partial y } , \end{gather}$$
$$\begin{gather}T={-}\frac{1}{2}\,\frac{\partial}{\partial y } \overline{\rho u_{i}^{\prime \prime } u_{i}^{\prime \prime } v^{\prime \prime }} , \end{gather}$$
$$\begin{gather}\varPi =\varPi_{t} +\varPi_{d}={-} \frac{\partial}{\partial y } \overline{ v^{\prime \prime } p^{\prime}}+\overline{ p^{\prime}\,\frac{\partial u_{i}^{\prime \prime } }{\partial x_{i}}}, \end{gather}$$
$$\begin{gather}\varPhi_{dif} = \frac{\partial}{\partial y } \overline{u_{i}^{\prime \prime } \sigma_{iy}^{\prime}}, \end{gather}$$
$$\begin{gather}\varPhi_{dis} ={-} \overline{ \sigma_{ij}^{\prime}\,\frac{\partial u_{i}^{\prime \prime } }{\partial x_{j} }} , \end{gather}$$
$$\begin{gather}ST ={-} \overline{v^{\prime \prime }}\,\frac{\partial \bar{p} }{\partial y } +\overline{u_{i} ^{\prime \prime }}\,\frac{\partial \overline{\sigma_{ij}} }{\partial x_{j} }-\bar{\rho}\tilde{k}\,\frac{\partial \tilde{v} }{\partial y} . \end{gather}$$ 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. Thermodynamic-related statistics
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
\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:
\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
$$\begin{gather} Pr_{t}=\frac{\overline{\rho u^{\prime \prime }v^{\prime \prime }}\,\partial \tilde{T}/\partial y}{\overline{\rho v^{\prime \prime }T^{\prime \prime }}\,\partial \tilde{u}/\partial y} \approx 1, \end{gather}$$
$$\begin{gather}R_{u^{\prime \prime }T^{\prime \prime }}=\frac{\widetilde{uT}-\tilde{u} \tilde{T}}{u_{rms}^{\prime \prime }T_{rms}^{\prime \prime }}. \end{gather}$$ 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:
\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
\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 
$\overline {\rho v^{\prime \prime } T^{\prime \prime } }$.
5. Turbulent structures
In order to reveal further the effect of bulk viscosity on hypersonic boundary layer flow, the turbulent structures are analysed in this section. Figure 12 reports the instantaneous vortical structures of the hypersonic turbulent boundary layer based on the Q-criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). The large-scale vortex structures in the outer layer become richer when the Mach number increases (figures 12a,c), while the spanwise spacing of the large-scale vortex structures becomes larger as the wall temperature decreases (figures 12a,e), which is consistent with the conclusion of Huang et al. (Reference Huang, Duan and Choudhari2022). When considering bulk viscosity, the large-scale vortex structure within the boundary layer becomes fragmented, and the small-scale vortex structure becomes richer by comparing figures 12(g,h).
Figure 12. Snapshots of instantaneous vortical structures based on the Q-criterion for (a,b) 
$Ma6T_{w}0.76$, (c,d) 
$Ma8T_{w}0.48$ and (e,f) 
$Ma6T_{w}0.25$ for 
$Q=20$, and (g,h) 
$Ma6T_{w}0.76$ for 
$Q=300$ – (b,d,f,h) with and (a,c,e,g) without bulk viscosity. The colour of the vortex structure changes from red to blue along the wall-normal directions. The size of the box is 
$\Delta x \times \Delta y \times \Delta z =8\delta _{i} \times 3\delta _{i} \times 3\delta _{i}$.
Figure 13 displays the instantaneous fields of normalized streamwise velocity fluctuations 
$\sqrt {\rho }\,u^{\prime \prime }/\sqrt {\tau _w}$ in a wall-parallel slice at 
$y^{+}=15$ at the selected stations. The alternating distribution of high and low velocity momentum forms the velocity striping structure in this figure, which is related to the ejection and sweep events. Those structures have been observed widely in both incompressible (Jiménez Reference Jiménez2013) and compressible (Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011; Bross, Scharnowski & Khler Reference Bross, Scharnowski and Khler2021; Xu et al. Reference Xu, Wang and Chen2022) turbulence. Those large-scale coherent structures can reach several boundary layer thicknesses in the streamwise direction, and contain a large part of the turbulent kinetic energy (Bross et al. Reference Bross, Scharnowski and Khler2021). The near-wall streaks are similar for different Mach numbers, such as figures 13(a,c), which is consistent with the findings of Duan et al. (Reference Duan, Beekman and Martín2011). As shown in figures 13(a) and 13(e), the near-wall streaks marked by the rectangular boxes become thicker and more coherent as the wall temperature decreases, in agreement with the observation of Duan et al. (Reference Duan, Beekman and Martín2010), Lagha et al. (Reference Lagha, Kim, Eldredge and Zhong2011), Huang et al. (Reference Huang, Duan and Choudhari2022) and Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022). The effect of bulk viscosity on the instantaneous fields of streamwise fluctuating velocity 
$\sqrt{\rho u^{\prime\prime}}/\sqrt{\tau_w}$ is negligible.
Figure 13. Instantaneous streamwise velocity fluctuations 
$u^{\prime \prime }$ normalized by the friction velocity 
$u^{\ast }_{\tau }=\sqrt {\tau _{w}/\rho }$ in a wall-parallel slice at 
$y^{+}=15$ for different cases: (a) 
$Ma6T_{w}0.76$, (b) 
$Ma6T_{w}0.76\mu _{b}$, (c) 
$Ma8T_{w}0.48$, (d) 
$Ma8T_{w}0.48\mu _{b}$, (e) 
$Ma6T_{w}0.25$, (f) 
$Ma6T_{w}0.25\mu _{b}$ – (a,c,e) without bulk viscosity (
$\mu _b/\mu =0$), and (b,d,f) with bulk viscosity (
$\mu _b/\mu =100$).
Figures 14 and 15 present the instantaneous fields of normalized density fluctuations 
$\rho ^{\prime }/\bar {\rho }$ and normalized temperature fluctuations 
$T^{\prime }/\bar {T}$ in a wall-parallel slice at 
$y^{+}=15$, respectively. It can be observed from the figures that the instantaneous fields of normalized density fluctuations 
$\rho ^{\prime }/\bar {\rho }$ and normalized temperature fluctuations 
$T^{\prime }/\bar {T}$ exhibit travelling-wave-like alternating positive and negative structures (TAPNS), which is similar to the fluctuating velocity streaks (Xu et al. Reference Xu, Wang and Chen2023). In addition, the similar near-wall streak structures of the fluctuating temperature and fluctuating velocity instantaneous fields can be corroborated with the temperature and velocity fluctuations correlation approximating to 1, which was also reported by Xu et al. (Reference Xu, Wang, Wan, Yu and Chen2021a, Reference Xu, Wang and Chen2022) and Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022). Moreover, comparing the figures with and without bulk viscosity, the density fluctuations 
$\rho ^{\prime }/\bar {\rho }$ and temperature fluctuations 
$T^{\prime }/\bar {T}$ in the near-wall region decrease slightly after considering the bulk viscosity, which is consistent with the results in figure 9. Additionally, the effect of bulk viscosity on TAPNS of normalized density and temperature fluctuations is negligible in the compressible turbulent boundary layer.
Figure 14. Instantaneous density fluctuations 
$\rho ^{\prime }$ normalized by the mean density 
$\bar {\rho }$ in a wall-parallel slice at 
$y^{+}=15$ for different cases: (a) 
$Ma6T_{w}0.76$, (b) 
$Ma6T_{w}0.76\mu _{b}$, (c) 
$Ma8T_{w}0.48$, (d) 
$Ma8T_{w}0.48\mu _{b}$, (e) 
$Ma6T_{w}0.25$, (f) 
$Ma6T_{w}0.25\mu _{b}$ – (a,c,e) without bulk viscosity (
$\mu _b/\mu =0$), and (b,d,f) with bulk viscosity (
$\mu _b/\mu =100$).
Figure 15. Instantaneous temperature fluctuations 
$T^{\prime }$ normalized by the mean temperature 
$\bar {T}$ in a wall-parallel slice at 
$y^{+}=15$ for different cases: (a) 
$Ma6T_{w}0.76$, (b) 
$Ma6T_{w}0.76\mu _{b}$, (c) 
$Ma8T_{w}0.48$, (d) 
$Ma8T_{w}0.48\mu _{b}$, (e) 
$Ma6T_{w}0.25$, (f) 
$Ma6T_{w}0.25\mu _{b}$ – (a,c,e) without bulk viscosity (
$\mu _b/\mu =0$), and (b,d,f) with bulk viscosity (
$\mu _b/\mu =100$).
Figure 16 shows the instantaneous fields of normalized pressure fluctuations 
$p^{\prime }/\bar {p}$ in a wall-parallel slice at 
$y^{+}=5$. It can be observed from the figure that the instantaneous fields of normalized pressure fluctuations 
$p^{\prime }/\bar {p}$ present the TAPNS, which were also reported by Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022) and Xu et al. (Reference Xu, Wang and Chen2023). As shown in figures 16(a) and 16(e), the intensity of pressure fluctuations 
$p^{\prime }/\bar {p}$ and the TAPNS are increased as the wall temperature decreases, indicating that wall cooling enhances the compressibility of the fluid in the near-wall region, which is consistent with the conclusion of Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022) and Xu et al. (Reference Xu, Wang and Chen2023). Moreover, comparing the figures with and without bulk viscosity, the intensity of pressure fluctuations 
$p^{\prime }/\bar {p}$ and the TAPNS marked by the rectangular boxes in the near-wall region decrease distinctly after considering the bulk viscosity, indicating that the bulk viscosity reduces the compressibility of fluid in the near-wall region.
Figure 16. Instantaneous pressure fluctuations 
$p^{\prime }$ normalized by the mean pressure 
$\bar {p}$ in a wall-parallel slice at 
$y^{+}=5$ for different cases: (a) 
$Ma6T_{w}0.76$, (b) 
$Ma6T_{w}0.76\mu _{b}$, (c) 
$Ma8T_{w}0.48$, (d) 
$Ma8T_{w}0.48\mu _{b}$, (e) 
$Ma6T_{w}0.25$, (f) 
$Ma6T_{w}0.25\mu _{b}$ – (a,c,e) without bulk viscosity (
$\mu _b/\mu =0$), and (b,d,f) with bulk viscosity (
$\mu _b/\mu =100$).
6. Small-scale properties
The role of bulk viscosity on small-scale properties is studied in this section. Figures 17 and 18 plot the probability density function (PDF) of the normalized derivative of the streamwise velocity fluctuations 
$u^{\prime }$ with respect to streamwise coordinate 
$x$ and wall-normal coordinate 
$y$ in the different part of turbulent boundary layer, respectively. As can be seen from figure 17, for the absolute value of the fluctuating velocity derivative with respect to streamwise coordinate 
$| \partial u^{\prime } / \partial x |$ greater than 5 in the viscous sublayer, the PDF of fluctuating velocity derivative 
$\partial u^{\prime } / \partial x$ of the compressible turbulent boundary layer is significantly larger than that of the compressible homogeneous isotropic turbulence (HIT) and homogeneous shear turbulence (HST), which indicates qualitatively that the fluctuation intensity of the viscous sublayer of the compressible turbulent boundary layer is stronger than that of HIT and HST. Meanwhile, in the inner layer of the boundary layer, the probability density distribution of the fluctuating velocity derivative is more similar to that of HST, indicating a stronger shear effect in the compressible turbulent boundary layer. As the wall-normal distance increases, the probability density distribution of the fluctuating velocity derivative in the outer layer of the boundary layer approaches that of HIT, indicating that the outer layer of the boundary layer is nearly isotropic. The bulk viscosity decreases the fluctuating velocity derivative 
$\partial u^{\prime } / \partial x$ within the viscous sublayer of the compressible turbulent boundary layer, and the PDF for the absolute value of the fluctuating velocity derivative greater than 5 decreases, while in the buffer, log and outer layers of the boundary layer, the effect of bulk viscosity on the fluctuating velocity derivative can be neglected. As can be seen in figure 18, the maximum negative values of the fluctuating velocity derivative with respect to wall-normal coordinate 
$\partial u^{\prime } / \partial y$ in the viscous sublayer and buffer layer are smaller comparing with its maximum positive values due to the effect of the wall of the compressible turbulent boundary layer. In the log layer and outer layer of the boundary layer, the distribution of the PDF of the fluctuating velocity derivative with respect to wall-normal coordinate is similar to that of HIT and HST. The effect of bulk viscosity on the fluctuating velocity derivative with respect to wall-normal coordinate 
$\partial u^{\prime } / \partial y$ in the compressible turbulent boundary layer is negligible, which is consistent with the conclusions of Chen et al. (Reference Chen, Wang, Wang, Wan and Chen2019).
Figure 17. The PDF of the normalized derivative of the streamwise velocity fluctuations 
$u^{\prime }$ with respect to streamwise coordinate 
$x$ in the different parts of the turbulent boundary layer: (a) viscous sublayer, (b) buffer layer, (c) log layer, and (d) outer layer. Symbols from Chen et al. (Reference Chen, Wang, Wang, Wan and Chen2019): 
$\bigcirc$, HIT without bulk viscosity; 
$\triangle$, HIT with bulk-to-shear viscosity ratio 30; 
$\bullet$, HST without bulk viscosity; 
$\blacktriangle$, HST with bulk-to-shear viscosity ratio 30.
Figure 18. The PDF of the normalized derivative of the streamwise velocity fluctuations 
$u^{\prime }$ with respect to wall-normal coordinate 
$y$ in the different parts of the turbulent boundary layer: (a) viscous sublayer, (b) buffer layer, (c) log layer, and (d) outer layer. Symbols from Chen et al. (Reference Chen, Wang, Wang, Wan and Chen2019): 
$\bigcirc$, HIT without bulk viscosity; 
$\triangle$, HIT with bulk-to-shear viscosity ratio 30; 
$\bullet$, HST without bulk viscosity; 
$\blacktriangle$, HST with bulk-to-shear viscosity ratio 30.
The PDF of the dilatation 
$\theta = \partial u /\partial x + \partial v /\partial y + \partial w /\partial z$ in four different parts of the turbulent boundary layer are shown in figure 19 to assess the compressibility of fluids. The compression (
$\theta <0$) and expansion (
$\theta >0$) of the fluids are enhanced significantly with increasing Mach number and increasing wall temperature by comparing the dashed lines in the figure, as reported by Lagha et al. (Reference Lagha, Kim, Eldredge and Zhong2011). However, the compression and expansion effects of the fluids gradually diminish with increasing wall-normal distance. The compression of the fluids is weakened significantly in the turbulent boundary layer when bulk viscosity is considered, which is consistent with the conclusions of Pan & Johnsen (Reference Pan and Johnsen2017) and Chen et al. (Reference Chen, Wang, Wang, Wan and Chen2019).
Figure 19. The PDF of the dilatation 
$\theta$ for different cases in different parts of the turbulent boundary layer: (a) viscous sublayer, (b) buffer layer, (c) log layer and (d) outer layer.
The shear stress budget in the near-wall region of the turbulent boundary layer is of great significance for turbulence modelling and understanding the interaction of small-scale structures in the inner layer. Meanwhile, the total stress is approximately equal to the sum of the viscous shear stress and Reynolds shear stress in the near-wall region, which is the physical basis for the modelling of the mean velocity transformations of the viscous sublayer and the buffer layer. Recently, Lee et al. (Reference Lee, Helm, Martín and Williams2023) conducted a detailed analysis of the effects of density and viscosity fluctuations on the viscous, Reynolds and total shear stresses in the near-wall region, and found that the sum of generalized viscous shear stress and Reynolds shear stress can better approximate 1 in the viscous sublayer. The equation of the generalized shear stress budget in the near-wall region of turbulent boundary layer is
\begin{equation} 1=\left [ \frac{\bar{\mu}\,\dfrac{\partial \tilde{u} }{\partial y} }{\tau_{w}} + \left ( \frac{\dfrac{\overline{\bar{\mu}\,\partial u''} }{\partial y}+ \overline{ \mu'\, \dfrac{\partial u'' }{\partial y}}}{\tau_{w}} \right ) \right ] - \frac{\bar{\rho}\, \widetilde{u''v''} }{\tau_{w}} =[ \tau_{V}^{+}+\zeta_{\mu}^{+} ] + \tau_{R}^{+} , \end{equation}
where 
$\tau _{V}^{+}$ is the Favre-averaged viscous shear stress, 
$\zeta _{\mu }^{+}$ is the influence arising from viscosity fluctuations, the sum of 
$\tau _{V}^{+}$ and 
$\zeta _{\mu }^{+}$ is defined as the generalized viscous shear stress 
$\tau _{VG}^{+}$, and 
$\tau _{R}^{+}$ is the Favre-averaged Reynolds shear stress.
Figure 20 shows the wall-normal distributions of the generalized shear stress budget in the semi-local scaling. As shown in figure 20(a), due to the consideration of bulk viscosity, the viscosity fluctuations in the near-wall region are enhanced, so that the fluctuating viscosity term 
$\zeta _{\mu }^{+}$ increases in the viscous sublayer. And the generalized viscous shear stress 
$\tau _{VG}^{+}$ also increases slightly in the viscous sublayer, as shown in figure 20(b). The Reynolds shear stress shown in figure 20(c) is the same as that in figure 4(b). After considering the bulk viscosity, the compressibility of turbulent boundary layer decreases, so the peak of Reynolds shear stress 
$\tau _{R}^{+}$ is reduced slightly in the log layer. Figure 20(d) represents the generalized total stress budget taking into account the effect of density and viscosity fluctuations. Combining the effect of bulk viscosity on the generalized viscous and Reynolds shear stresses, it can be found that the generalized total stress 
$\tau ^{+}$ increases slightly in the viscous sublayer by considering the bulk viscosity, but the effect of the bulk viscosity on the total stress budget is kept in a negligible range, which is why the mean velocity transformation is still valid after considering the bulk viscosity.
Figure 20. The shear stress budget in the compressible turbulent boundary layer for different cases: (a) viscosity fluctuation term 
$\zeta _{\mu }^{+}$, (b) generalized viscous shear stress 
$\tau _{VG}^{+} =\tau _{V}^{+}+\zeta _{\mu }^{+}$, (c) Reynolds shear stress 
$\tau _{R}^{+}$ and (d) total stress 
$\tau ^{+}$ along the wall-normal distance in the semi-local scaling.
7. Discussion of results
Throughout the entire paper, the influence of bulk viscosity on compressible turbulent boundary layers is characterized mainly by a notable decrease in wall-normal velocity fluctuations, thermodynamic fluctuations, and related quantities such as the pressure dilatation and pressure diffusion terms in the turbulent kinetic energy budget. This implies a substantial reduction in the compressibility of the turbulent boundary layer due to the presence of bulk viscosity. Subsequently, we try to analyse the physical mechanism of how bulk viscosity induces a reduction in the wall-normal velocity fluctuations and thermodynamic fluctuations in the near-wall region of turbulent boundary layer.
In order to reveal the physical mechanism of the effect of bulk viscosity on the wall-normal velocity fluctuations, this paper further investigates the wall-normal Reynolds stress budgets in compressible turbulent boundary layers. The Favre-averaged Reynolds stress tensor is defined as 
$\tau _{ij}=\overline {\rho u_{i}^{\prime \prime } u_{j}^{\prime \prime }}/\bar {\rho }$, and its transport equation is
\begin{equation} \frac{\partial (\bar{\rho}\tau_{ij})}{\partial t} + \frac{\partial (\bar{\rho}\tau_{ij} \widetilde{u_k} )}{\partial x_k} =P_{ij}+T_{ij}+\varPi_{ij}-\bar{\rho}\epsilon_{ij}+D_{ij}+M_{ij}, \end{equation}where
$$\begin{gather} P_{ij}={-}\left ( \overline{\rho u_{i}^{\prime \prime} u_{k}^{\prime \prime} }\, \frac{\partial \widetilde{u_j}}{\partial x_k}+ \overline{\rho u_{j}^{\prime \prime} u_{k}^{\prime \prime} }\,\frac{\partial \widetilde{u_i}}{\partial x_k} \right), \end{gather}$$
$$\begin{gather}T_{ij}={-}\frac{\partial }{\partial x_k} ( \overline{\rho u_{i}^{\prime \prime} u_{j}^{\prime \prime} u_{k}^{\prime \prime} } ), \end{gather}$$
$$\begin{gather}\varPi_{ij}={-}\Bigg(\overline{ u_{i}^{\prime \prime}\,\frac{\partial p^{\prime} }{\partial x_j} +u_{j}^{\prime \prime}\,\frac{\partial p^{\prime} }{\partial x_i}} \Bigg), \end{gather}$$
$$\begin{gather}\bar{\rho} \epsilon_{ij}=\overline{\sigma_{ik}^{\prime}\,\frac{\partial u_{j}^{\prime \prime} }{\partial x_k}} + \overline{ \sigma_{jk}^{\prime}\,\frac{\partial u_{i}^{\prime \prime} }{\partial x_k} }, \end{gather}$$
$$\begin{gather}D_{ij}=\frac{\partial }{\partial x_k} ( \overline{\sigma_{ik}^{\prime} u_{j}^{\prime \prime} + \sigma_{jk}^{\prime} u_{i}^{\prime \prime}} ), \end{gather}$$
$$\begin{gather}M_{ij}= \overline{ u_{i}^{\prime \prime}}\left ( \frac{\partial \overline{\sigma_{kj}}}{\partial x_k} - \frac{\partial \bar{p}}{\partial x_j}\right ) + \overline{ u_{j}^{\prime \prime}}\left ( \frac{\partial \overline{\sigma_{ki}}}{\partial x_k} - \frac{\partial \bar{p}}{\partial x_i}\right ). \end{gather}$$ The transport terms of Reynolds stress include the production term 
$P_{ij}$, turbulent diffusion term 
$T_{ij}$, velocity pressure-gradient term 
$\varPi _{ij}$, viscous dissipation term 
$-\bar {\rho }\epsilon _{ij}$, viscous diffusion term 
$D_{ij}$, and mass flux contribution term 
$M_{ij}$ (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988).
Figure 21 shows the wall-normal Reynolds stress budgets 
$\tau _{yy}=\overline {\rho v^{\prime \prime } v^{\prime \prime }}/\bar {\rho }$. It can be observed that the transport of wall-normal Reynolds stress 
$\tau _{yy}$ in the buffer layer and log layer is determined mainly by the turbulent diffusion term 
$T_{yy}$, velocity pressure-gradient term 
$\varPi _{yy}$, and viscous dissipation term 
$-\bar {\rho }\epsilon _{yy}$ in the turbulent boundary layer. However, the production term 
$P_{yy}$, viscous diffusion term 
$D_{yy}$, and mass flux contribution term 
$M_{yy}$ have little effect on the wall-normal Reynolds stress budgets, which was also reported by Smits et al. (Reference Smits, Hultmark, Lee, Pirozzoli and Wu2021) and Nicholson et al. (Reference Nicholson, Huang, Duan, Choudhari, Morreale and Bowersox2022). The influence of bulk viscosity on the transport terms of wall-normal Reynolds stress can be ignored, except for the viscous dissipation term 
$-\bar {\rho }\epsilon _{yy}$. The bulk viscosity enhances mainly the viscous dissipation term 
$-\bar {\rho }\epsilon _{yy}$, thereby reducing significantly the wall-normal velocity fluctuations 
$v_{rms}^{\prime \prime }$ in the near-wall region.
Figure 21. The wall-normal Reynolds stress budgets 
$\tau _{yy}=\overline {\rho v^{\prime \prime } v^{\prime \prime }}/\bar {\rho }$. The wall-normal distributions of (a) production 
$P_{yy}$, (b) turbulent diffusion 
$T_{yy}$, (c) velocity pressure-gradient 
$\varPi _{yy}$, (d) viscous dissipation 
$-\bar {\rho }\epsilon _{yy}$, (e) viscous diffusion 
$D_{yy}$ and (f) mass flux contribution 
$M_{yy}$, normalized by 
$\rho u_{\tau }^{3}/y_{\tau }$ in the inner scaling. Solid lines are for 
$\mu _{b}/\mu =100$; dashed lines are for 
$\mu _{b}/\mu =0$.
In order to provide more physical explanations, we decompose the pressure fluctuations 
$p_{rms}^{+}$ into four components. The equation of the pressure fluctuation is (Yu, Xu & Pirozzoli Reference Yu, Xu and Pirozzoli2020)
\begin{align} \frac{\partial^2 p^{\prime} }{\partial x_i\,\partial x_i}=\frac{\partial^2 \sigma^{\prime}_{ij} }{\partial x_i\,\partial x_j}-\frac{\partial^2 }{\partial x_i\,\partial x_j} ( 2\rho \tilde{u}_{i} u^{\prime\prime}_{j}+\rho^{\prime} \tilde{u}_{i} \tilde{u}_{j} )- \frac{\partial^2 }{\partial x_i\,\partial x_j}( \rho u^{\prime\prime}_{i} u^{\prime\prime}_{j}- \overline{\rho u^{\prime\prime}_{i} u^{\prime\prime}_{j} } )+\frac{\partial^2 \rho^{\prime} }{\partial t^2}. \end{align}Based on the characterization of the right-hand-side terms of the pressure equation, the pressure is decomposed into (Tang et al. Reference Tang, Zhao, Wan and Liu2020)
\begin{equation} p^{\prime}=p_r+p_s+p_{\sigma}+p_c, \end{equation}
where 
$p_r$ is the rapid pressure characterizing the linear mean shear–turbulence interactions, 
$p_s$ is the slow pressure characterizing the nonlinear turbulence–turbulence interactions, 
$p_{\sigma }$ represents the viscous pressure characterizing the contribution of viscous stresses to the pressure fluctuations, and 
$p_c$ denotes the compressible pressure characterizing the contribution of the compressibility to the pressure fluctuations.
By comparing (7.3) and (7.4), the following four equations for fluctuating pressure components can be obtained, respectively:
$$\begin{gather} \frac{\partial^2 p_{r} }{\partial x_i\,\partial x_i}={-}2\,\frac{\partial \tilde{u}_{i} }{\partial x_j}\,\frac{\partial \rho u^{\prime\prime}_{j} }{\partial x_i}, \end{gather}$$
$$\begin{gather}\frac{\partial^2 p_{s} }{\partial x_i\,\partial x_i}={-} \frac{\partial^2 }{\partial x_i\,\partial x_j}(\rho u^{\prime\prime}_{i} u^{\prime\prime}_{j}- \overline{\rho u^{\prime\prime}_{i} u^{\prime\prime}_{j}}), \end{gather}$$
$$\begin{gather}\frac{\partial^2 p_{\sigma} }{\partial x_i\,\partial x_i}=\frac{\partial^2 \sigma^{\prime}_{ij} }{\partial x_i\,\partial x_j}, \end{gather}$$
$$\begin{gather}\frac{\partial^2 p_c }{\partial x_i\,\partial x_i}=\frac{\partial^2 \rho^{\prime} }{\partial t^2}-\frac{\partial^2 }{\partial x_i\,\partial x_j}(2\rho\tilde{u}_{i}u^{\prime\prime}_{j}+ \rho^{\prime} \tilde{u}_{i} \tilde{u}_{j} )+2\,\frac{\partial \tilde{u}_{i} }{\partial x_j}\,\frac{\partial \rho u^{\prime\prime}_{j} }{\partial x_i} . \end{gather}$$These fluctuating pressure component equations are solved by performing second-order central difference in the streamwise and wall-normal directions, and by using the Fourier–Galerkin method in the spanwise direction, with boundary conditions and other details that can be found in Zhang et al. (Reference Zhang, Wan, Liu, Sun and Lu2022).
To provide a quantitative explanation of the influence mechanism of bulk viscosity on fluctuating pressure, a comparative analysis of the r.m.s. pressure fluctuations 
$p_{rms}^{+}$ and its components was conducted across different cases, as shown in figure 22. The reduction of both rapid pressure 
$p_{r,rms}^+$ and slow pressure 
$p_{s,rms}^+$ in the near-wall region is attributed to the influence of bulk viscosity, indicating that the bulk viscosity suppresses both linear mean shear–turbulence interactions and nonlinear turbulence–turbulence interactions, while the impact of bulk viscosity appears slightly more pronounced on the rapid pressure 
$p_{r,rms}^+$ than on the slow pressure 
$p_{s,rms}^+$. Due to its ability to mitigate the compressibility of the hypersonic turbulent boundary layer, bulk viscosity induces a substantial reduction in density fluctuations within the viscous sublayer and the buffer layer, consequently leading to a notable decrease in compressible pressure 
$p_{c,rms}^{+}$. With the Mach number increase and the wall temperature decrease, the compressibility of the turbulent boundary layer is enhanced, and the influence of bulk viscosity is more significant. In addition, the viscous pressure 
$p_{\sigma,rms}^{+}$ also decreases due to the consideration of bulk viscosity. In the comprehensive analysis, while acknowledging the suppressive impact of bulk viscosity on incompressible pressure, it is essential to recognize that the compressible component of pressure fluctuations 
$p_{c,rms}^{+}$ experiences a pronounced reduction within the viscous sublayer and the buffer layer, which is attributed to the impact of bulk viscosity on the compressibility of the turbulent boundary layer.
Figure 22. Wall-normal distributions of the r.m.s. pressure fluctuations 
$p_{rms}^{+}$ and its components obtained from (7.5), which are normalized by 
$\tau _w$ in the inner scaling, for cases (a) 
$Ma6T_{w}0.76$, (b) 
$Ma8T_{w}0.48$ and (c) 
$Ma6T_{w}0.25$, with and without bulk viscosity. As shown in the legend, the dashed lines denote the cases without bulk viscosity (
$\mu _{b}/\mu =0$), while the solid lines denote the cases with bulk viscosity (
$\mu _{b}/\mu =100$). And the lines with symbols represent the pressure fluctuations 
$p_{rms}^{+}$, which is consistent with figure 9(c), while the lines without symbols represent its components.
8. Conclusion
In this paper, direct numerical simulations are performed for six cases with Mach numbers 
$Ma =6, 8$, temperature ratios 
$T_{w} /T_{r}= 0.25, 0.48, 0.76$, and bulk-to-shear viscosity coefficient ratios 
$\mu _{b} /\mu = 0, 100$, respectively. We analyse comprehensively the effects of bulk viscosity on velocity-related variables, thermodynamic-related statistics, large-scale structures, and small-scale properties in the hypersonic compressible turbulent boundary layers.
The impact of bulk viscosity on mean statistics is found to be relatively small, with negligible effects on mean density and mean temperature. The transformed mean velocity profiles and Walz equation are minimally affected by bulk viscosity.
In contrast, the effect of bulk viscosity on fluctuating statistics is relatively significant. The wall-normal velocity fluctuation 
$v_{rms}^{\prime \prime }$ is reduced significantly in the viscous sublayer by the enhanced viscous dissipation 
$-\bar {\rho } \epsilon _{yy}$ related to bulk viscosity in the wall-normal Reynolds stress budgets. The intensity of pressure fluctuations 
$p_{rms}^{\prime }$ in the near-wall region decreases distinctly after considering the bulk viscosity, which is attributed mainly to the reduction of compressible pressure fluctuations 
$p_{c,rms}^{+}$. The bulk viscosity leads to a slight decrease in the value of the turbulent Mach number, indicating that the bulk viscosity reduces the effect of the turbulence compressibility.
Regarding the turbulent kinetic energy budget, the pressure diffusion and pressure dilatation terms decrease significantly near the wall when considering bulk viscosity. The effect is attributed to the impact of bulk viscosity on decreasing fluid compressibility and density fluctuations near the wall.
The strong Reynolds analogy is mostly unaffected by bulk viscosity, with minimal influence on the correlation between temperature fluctuations and velocity fluctuations. The turbulent heat flux 
$\overline {\rho v^{\prime \prime } T^{\prime \prime }}$ decreases slightly in the viscous sublayer by considering bulk viscosity. However, it exhibits little effect on the mean temperature and conductive heat flux 
$-k\,{\partial T}/{\partial y}$. As a consequence, the total heat flux in the viscous sublayer is decreased by bulk viscosity.
When considering bulk viscosity, the large-scale vortex structure based on the Q-criterion becomes fragmented in the outer layer of the turbulent boundary layer, while the small-scale vortex structure becomes richer. Additionally, the intensity of the travelling-wave-like alternating positive and negative structures of instantaneous pressure field 
$p^{\prime }/\bar {p}$ in the near-wall region decreases distinctly.
In terms of small-scale properties, bulk viscosity gives a significant impact on the near-wall region of the turbulent boundary layer. The intensity of turbulent fluctuations and shear effects in the viscous sublayer is stronger compared to compressible homogeneous isotropic turbulence and homogeneous shear turbulence. Furthermore, the bulk viscosity decreases the fluctuating velocity derivative with respect to the streamwise coordinate 
$\partial u^{\prime } / \partial x$ within the viscous sublayer, and the probability density function (PDF) for the absolute value of the fluctuating velocity derivative 
$| \partial u^{\prime } / \partial x |$ greater than 5 decreases. However, the effect of bulk viscosity on the fluctuating velocity derivative with respect to the wall-normal coordinate 
$\partial u^{\prime } / \partial y$ is negligible. Moreover, the PDF of the velocity divergence 
$\theta =\partial u_{i} / \partial x_{i}$ decreases in the turbulent boundary layer, indicating weakened fluid compressibility when considering bulk viscosity.
The compressibility of the hypersonic turbulent boundary layer is weakened in the near-wall region by considering bulk viscosity, which is manifested most clearly by a reduction of the pressure fluctuations 
$p_{rms}^{\prime }$ by approximately 25 
$\%$ in the viscous sublayer. And this effect becomes more significant as the Mach number increases and the wall temperature decreases. By analysing comprehensively the existing results, this paper concludes that when the bulk-to-shear viscosity ratio of the gas reaches a few hundred levels (
$\mu _b/\mu =O(10^2)$), and the mechanical behaviour of the near-wall region (
$\kern1.5pt y^+\le 30$) of the turbulent boundary layer is of greater interest, the impact of bulk viscosity on the hypersonic cold-wall turbulent boundary layer may not be negligible.
Acknowledgements
The authors gratefully acknowledge the Institute of Extreme Mechanics of Northwestern Polytechnical University and computing resources provided by Hefei Advanced Computing Center. And the authors are particularly grateful to Dr Peng-jun-yi Zhang from the University of Science and Technology of China, who provided us with the code for the fluctuating pressure decomposition.
Funding
This research is supported by the National Natural Science Foundation of China (nos 12388101, 92371103 and 11972272).
Declaration of interests
The authors report no conflict of interest.
Data availability statement
Data available on request from the authors.
