Wall cooling effect on spectra and structures of thermodynamic variables in hypersonic turbulent boundary layers

The mechanisms of the hypersonic turbulent boundary layers are of great importance in aerospace industry due to the direct application to the hypersonic vehicles (Smits & Dussauge, 2006; Gatski & Bonnet, 2009). It has been widely observed that the cold wall can significantly enhance the compressibility effect near the wall in hypersonic turbulent boundary layers (Duan et al., 2010; Zhang et al., 2017, 2018; Xu et al., 2021a, b, 2022b, 2022c, 2022a; Huang et al., 2022). Therefore, the systematic investigations of the properties of the physical quantities in the cooled wall hypersonic turbulent boundary layers are extraordinarily critical to better understanding of the underlying mechanisms and more accurate physics-based modelling for this type of flows.

Most of the previous investigations about the hypersonic turbulent boundary layers were concentrated on the flow statistics of velocities (Duan et al., 2010, 2011; Lagha et al., 2011; Chu et al., 2013a; Zhang et al., 2018; Xu et al., 2021a, b, 2022b, 2022c, 2022a; Huang et al., 2022). Duan et al. (2010) performed direct numerical simulation (DNS) of hypersonic turbulent boundary layers at Mach number 5 with isothermal boundary condition. The ratio of the wall-to-edge temperature is ranging from 1.0 to 5.4. The effect of wall cooling on Morkovin${}^{{}^{\prime}}$s scaling, Walz${}^{{}^{\prime}}$s equation, the strong Reynolds analogy (SRA), turbulent kinetic energy budgets, compressibility effect and near-wall coherent structures were systematically investigated. They found that many scaling relations for the non-adiabatic hypersonic turbulent boundary layers are similar to those found in adiabatic wall cases, and the compressibility effect is insignificantly enhanced by wall cooling. Furthermore, Zhang et al. (2018) developed DNS databases of spatially evolving zero-pressure-gradient compressible turbulent boundary layers with nominal free-stream Mach number ranging from 2.5 to 14 and wall-to-recovery temperature ranging from 0.18 to 1.0. They assessed the performance of compressibility transformations, including the Morkovin${}^{{}^{\prime}}$s scaling and SRA, as well as the mean velocity and temperature scaling. Recently, a series of researches were aiming to reveal the effect of wall cooling on other complicated flow statistics beyond the well-observed compressibility transformations and the mean velocity and temperature scaling (Xu et al., 2021a, b, 2022b, 2022c, 2022a). Xu et al. (2021a, b) performed the DNS of hypersonic turbulent boundary layers at Mach numbers 6 and 8 with wall-to-recovery temperature ranging from 0.15 to 0.8. They used the Helmholtz decomposition to divide the fluctuating velocities into the solenoidal and dilatational components. They investigated the interactions among mean and fluctuating fields of kinetic and internal energy (Xu et al., 2021a) as well as the kinetic energy transfer across different scales (Xu et al., 2021b). Furthermore, the flow topology and its effect on the kinetic energy transfer across different scales were also systematically investigated in Xu et al. (2022b, c). In order to explain the possible reasons of the overshoot phenomena of the wall skin friction and wall heat transfer in transitional hypersonic boundary layers, Xu et al. (2022a) applied the decomposition method on the wall skin friction and heat transfer coefficients based on the two-fold repeated integration. Moreover, the effect of the wall cooling on the wall skin friction and heat transfer decomposition in hypersonic turbulent boundary layers was also discussed.

However, most of the previous studies were focused on the flow statistics and structures of velocities, while the mechanisms of the thermodynamic statistics in hypersonic turbulent boundary layers were less studied (Duan et al., 2016; Zhang et al., 2017; Ritos et al., 2019; Zhang et al., 2022; Cogo et al., 2022). Recently, Zhang et al. (2022) investigated the wall cooling effect on pressure fluctuations in compressible turbulent boundary layers. They explored the generating mechanisms of pressure fluctuations by dividing the pressure fluctuations into five components, among which the rapid pressure, slow pressure and compressible pressure are dominant. Furthermore, Cogo et al. (2022) investigated the high-Reynolds-number effect in hypersonic turbulent boundary layers. They studied the structural properties of the uniform streamwise momentum and uniform temperature regions in the high-speed regime. Furthermore, they also evaluated the accuracy of different compressibility transformations and temperature-velocity relations at moderate-high Reynolds numbers. A revised scaling for the characteristic length scales of the spanwise spectra of the fluctuating velocity and temperature at various wall distances was proposed based on the local mean shear. Nevertheless, the wall cooling effect on the multi-scale properties and the spatial structures of the pressure, density, temperature and entropy in hypersonic turbulent boundary layers need more systematic investigations, for the sake of a better understanding of the underlying mechanisms and more accurate physics-based modelling of the thermodynamic variables.

The goal of this study is to systematically explore the wall cooling effect on the spectra and structures of the thermodynamic variables in hypersonic turbulent boundary layers by direct numerical simulation. The fluctuating density and temperature are divided into the acoustic and entropic modes based on the Kovasznay decomposition (Kovasznay, 1953; Chassaing et al., 2002; Gauthier, 2017; Wang et al., 2019). The streamwise and spanwise spectra of the thermodynamic variables are systematically studied to figure out the multi-scale properties and spatial structures of thermodynamic variables. The streamwise and spanwise spectra of the fluctuating streamwise velocity are also investigated aiming for comparing with those of the thermodynamic variables. It is found that the wall cooling effect on the spectra and structures of the thermodynamic variables are much larger than those of the fluctuating streamwise velocity.

The remainder of the paper is organized as follows. The governing equations and simulation parameters are outlined in Section 2. The turbulent intensities of the streamwise velocity and thermodynamic variables are shown in Section 3. Section 4 presented the streamwise and spanwise spectra of the streamwise velocity and thermodynamic variables. Some discussions are given in Section 5. Finally, summary and conclusions are given in Section 6.

The compressible Navier-Stokes equations can be non-dimensionalised by a set of reference scales: the reference length $L_{\infty}$, free-stream density $\rho_{\infty}$, velocity $U_{\infty}$, temperature $T_{\infty}$, pressure $p_{\infty}=\rho_{\infty}U_{\infty}^{2}$, energy per unit volume $\rho_{\infty}U_{\infty}^{2}$, viscosity $\mu_{\infty}$ and thermal conductivity $\kappa_{\infty}$. Therefore, there are three non-dimensional governing parameters, namely the Reynolds number $Re=\rho_{\infty}U_{\infty}L_{\infty}/\mu_{\infty}$, Mach number $M=U_{\infty}/c_{\infty}$ and Prandtl number $Pr=\mu_{\infty}C_{p}/\kappa_{\infty}$. The ratio of specific heat at constant pressure $C_{p}$ to that at constant volume $C_{v}$ is defined as $\gamma=C_{p}/C_{v}=1.4$. The parameter $\alpha$ is defined as $\alpha=PrRe\left(\gamma-1\right)M^{2}$, where $Pr=0.7$.

The following compressible dimensionless Navier-Stokes equations in the conservative form are solved numerically (Liang & Li, 2015; Xu et al., 2021a, b, 2022b, 2022c, 2022a)

$$\frac{\partial\rho}{\partial t}+\frac{\partial\left(\rho u_{j}\right)}{\partial x_{j}}=0,$$

(1)

$$\frac{\partial\left(\rho u_{i}\right)}{\partial t}+\frac{\partial\left[\rho u_{i}u_{j}+p\delta_{ij}\right]}{\partial x_{j}}=\frac{1}{Re}\frac{\partial\sigma_{ij}}{\partial x_{j}},$$

(2)

$$\frac{\partial E}{\partial t}+\frac{\partial\left[\left(E+p\right)u_{j}\right]}{\partial x_{j}}=\frac{1}{\alpha}\frac{\partial}{\partial x_{j}}\left(\kappa\frac{\partial T}{\partial x_{j}}\right)+\frac{1}{Re}\frac{\partial\left(\sigma_{ij}u_{i}\right)}{\partial x_{j}},$$

(3)

$$p=\rho T/\left(\gamma M^{2}\right),$$

(4)

where $\rho$, $u_{i}$, $T$ and $p$ are the density, velocity component, temperature and pressure, respectively. The viscous stress $\sigma_{ij}$ is defined as

$$\sigma_{ij}=\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}\mu\theta\delta_{ij},$$

(5)

where $\theta=\partial u_{k}/\partial x_{k}$ is the velocity divergence, and the viscosity $\mu$ is determined by the Sutherland${}^{{}^{\prime}}$s law. The total energy per unit volume $E$ is

$$E=\frac{p}{\gamma-1}+\frac{1}{2}\rho\left(u_{j}u_{j}\right).$$

(6)

The convection terms of the compressible governing equations are discretized by a hybrid scheme. In order to judge on the local smoothness of the numerical solution, the modified Jameson sensor (Jameson et al., 1981) is used in the hybrid scheme, which can be given by (Dang et al., 2022)

$$\begin{split}&{\phi_{i}=\frac{\left|-p_{i-1}+2p_{i}-p_{i+1}\right|}{p_{i-1}+2p_{i}+p_{i+1}},}\\ &{\phi_{j}=\frac{\left|-p_{j-1}+2p_{j}-p_{j+1}\right|}{p_{j-1}+2p_{j}+p_{j+1}},}\\ &{\phi_{k}=\frac{\left|-p_{k-1}+2p_{k}-p_{k+1}\right|}{p_{k-1}+2p_{k}+p_{k+1}},}\end{split}$$

(7)

$${\Theta=\phi_{i}+\phi_{j}+\phi_{k}.}$$

(8)

The threshold $\Theta_{1}$ is set to 0.02 (Dang et al., 2022). When $\Theta\leq\Theta_{1}$, the eighth-order central difference scheme is used; when $\Theta>\Theta_{1}$, the seventh-order weighted essentially non-oscillatory scheme (Balsara & Shu, 2000) is applied. Furthermore, the viscous terms are approximated by an eighth-order central difference scheme. A third-order total variation diminishing type of Runge-Kutta method is utilized for time advancing (Shu & Osher, 1988). The compressible governing equations are numerically solved by the OPENCFD code, which has been widely validated in compressible transitional and turbulent wall-bounded flows (Liang & Li, 2015; Xu et al., 2021a, b, 2022b, 2022c, 2022a; Dang et al., 2022). The schematic of the hypersonic transitional and turbulent boundary layers is shown in figure 1. The spatially evolving hypersonic transitional and turbulent boundary layer is numerically simulated under the inflow and outflow boundary conditions, a wall boundary condition, an upper far-field boundary condition, and a periodic boundary condition in the spanwise direction. A time-independent laminar compressible boundary-layer similarity solution is applied at the inflow boundary. The laminar flow is disturbed by the wall blowing and suction region, and then transitioned to the fully developed turbulent state. Moreover, in order to inhibit the reflection of disturbance due to the numerical treatment of the outflow boundary condition, a progressively coarse grid is implemented in the streamwise direction near the outflow boundary condition. The non-slip and isothermal boundary conditions are applied for the wall boundary, and the non-reflecting boundary condition is imposed for the upper boundary. More detailed descriptions can refer to Pirozzoli et al. (2004); Liang & Li (2015); Xu et al. (2021a, b, 2022b, 2022c, 2022a).

In this study, $\overline{f}$ denotes the Reynolds average (spanwise and time average) of flow field $f$, and the fluctuating component of the Reynolds average is ${f}^{\prime}=f-\overline{f}$. Furthermore, $\widetilde{f}=\overline{\rho f}/\bar{\rho}$ represents the Favre average of $f$, and the fluctuating component is ${f}^{\prime\prime}=f-\widetilde{f}$.

The DNS of three hypersonic transitional and turbulent boundary layers at Mach number 8 with different wall temperatures are performed and the fundamental parameters of the database are listed in table 1. The free-stream temperature $T_{\infty}$ is prescribed to be $T_{\infty}=169.44K$. Temperature $T_{w}$ is the wall temperature, and the recovery temperature $T_{r}$ can be defined as $T_{r}=T_{\infty}\left(1+r\left(\left(\gamma-1\right)/2\right)M_{\infty}^{2}\right)$ with recovery factor $r=0.9$ (Duan et al., 2010; Xu et al., 2022a). The coordinates along the streamwise, wall-normal and spanwise directions are represented by $x$, $y$ and $z$ respectively. The computational domains $L_{x}$, $L_{y}$ and $L_{z}$ are nondimensionalized by the inflow boundary layer thickness $\delta_{in}$, and the symbols $N_{x}$, $N_{y}$ and $N_{z}$ represent the grid resolutions along the streamwise, wall-normal and spanwise directions respectively.

It should be pointed out that the no-ideal gas effect, as well as the non-equilibrium and radiative effects are neglected in the present DNS databases. The reasons are as follows. It is noted that the free-stream temperature $T_{\infty}$ is prescribed to be $T_{\infty}=169.44K$ in the present study. The largest temperature in the highest wall temperature case M8T08 is approximately 1700K. In the previous studies, the maximum temperatures of cases “M7”, “M8” and “M12” in Duan et al. (2011) and $M_{\infty}=7.5$, 10, 15 and 20 in Lagha et al. (2011) are much larger than that of M8T08, and the no-ideal gas effect, as well as the non-equilibrium and radiative effects were also neglected in their study. Therefore, the neglect of the no-ideal gas, non-equilibrium and heat radiative effects is a reasonable simplification for the present study of the hypersonic turbulent boundary layers. The influence of the no-ideal gas, non-equilibrium and radiative effects in the hypersonic turbulent boundary layers will be considered in the future.

Three sets of data in a small streamwise window of $\left[x_{a}-0.5\delta,x_{a}+0.5\delta\right]$ extracted from the fully developed region of the above three transitional and hypersonic turbulent boundary layers are used for following statistical analysis, where $x_{a}$ is the reference streamwise location selected for statistical analysis, and $\delta$ is the boundary layer thickness at the streamwise location $x_{a}$. It is noted that a similar technique has been used by the previous studies of Pirozzoli & Bernardini (2011), Zhang et al. (2018) and Huang et al. (2022), and the width of the streamwise window in this study is consistent with that of Huang et al. (2022). The fundamental parameters of the three sets of data are listed in table 2. The friction Reynolds number $Re_{\tau}$ is defined as $Re_{\tau}=\bar{\rho}_{w}u_{\tau}\delta/\bar{\mu}_{w}$, where $\bar{\rho}_{w}$ and $\bar{\mu}_{w}$ are the mean wall density and wall viscosity respectively, and $u_{\tau}=\sqrt{\tau_{w}/\bar{\rho}_{w}}$ and $\tau_{w}=\left(\mu\partial\bar{u}/\partial y\right)_{y=0}$ are the friction velocity and the wall shear stress respectively. Furthermore, $\Delta x^{+}=\Delta x/\delta_{\nu}$, $\Delta y_{w}^{+}=\Delta y_{w}/\delta_{\nu}$, $\Delta y_{e}^{+}=\Delta y_{e}/\delta_{\nu}$ and $\Delta z^{+}=\Delta z/\delta_{\nu}$ are the normalized spacing of the streamwise direction, the first point off the wall, the wall-normal grid at the edge of the boundary layer and the spanwise direction respectively, where $\delta_{\nu}=\bar{\mu}_{w}/(\bar{\rho}_{w}u_{\tau})$ is the viscous length scale. The semi-local lengthscale is defined as $\delta_{\nu}^{*}=\bar{\mu}/\left(\bar{\rho}u_{\tau}^{*}\right)$, where $u_{\tau}^{*}=\sqrt{\tau_{w}/\bar{\rho}}$ (Huang et al., 1995). The semi-local Reynolds number can be defined as $Re_{\tau}^{*}=\delta/\left(\delta_{\nu}^{*}\right)_{e}$. It is noted that the grid resolutions $\Delta x^{+}$, $\Delta y^{+}_{w}$, $\Delta y^{+}_{e}$ and $\Delta z^{+}$ in three cases are comparable and even smaller than many previous investigations including Duan et al. (2010), Pirozzoli & Bernardini (2013), Zhang et al. (2018) and Huang et al. (2022), indicating that the grid resolutions of the present DNS study are fine enough. Furthermore, the accuracy of the DNS cases in this study are validated via comparisons with the available DNS database in Zhang et al. (2018) in Appendix A.

The mean density and temperature profiles along wall-normal direction are shown in figure 2. Here the “turning points” marked by the green circles represent the minimum values of the mean density in figure 2 (a), (b) and (c), and the maximum values of the mean temperature in figure 2 (d), (e) and (f). The wall unit scaling is defined as $y^{+}=y/\delta_{\nu}$, and the semi-local scaling is defined as $y^{*}=y/\delta_{\nu}^{*}$. It is noted that the wall unit scaling $y^{+}$ and the semi-local scaling $y^{*}$ are the inner scaling in order to reveal the scaling relation in the near-wall region, while the outer scaling $y/\delta$ shows the statistical behaviour in the far-wall region.

The mean density and temperature are significantly influenced by the wall temperature. When the wall temperature is close to the recovery temperature $T_{r}$ (“M8T08” case), the mean temperature is nearly constant near the wall, and then decreases drastically when $y^{+}>4$. However, in “M8T04” and “M8T015”, the mean temperature initially increases near the wall. After reaching the maximum value at the turning point, the mean temperature then decreases as $y$ increases. It is also found that as the wall temperature decreases, the positive values of the wall-normal gradient of the mean temperature become larger below the wall-normal location of the turning point, while the negative values of the wall-normal gradient of the mean temperature become smaller above it. Furthermore, when the wall temperature becomes cooler, the maximum value of the mean temperature decreases, while the wall-normal location $y^{+}$ of the turning point increases under the wall unit scaling. The semi-local scaling $y^{*}$ can significantly decrease the discrepancy of the wall-normal locations of turning points in different wall temperature cases. It should be noted that the mean density profiles reveal the opposite variation trends compared with the mean temperature profiles.

In order to investigate the effect of wall temperature on the compressibility effect, the turbulent Mach number $M_{t}=\sqrt{\overline{{u}^{\prime\prime}_{i}{u}^{\prime\prime}_{i}}}/\bar{c}$ and the root mean square (r.m.s) values of the local Mach number based on the fluctuating velocities ${M}^{\prime}_{rms}$ are evaluated in figure 3. Here the local Mach number based on the fluctuating velocities $M^{\prime}$ is defined as ${M}^{\prime}=\sqrt{{u}^{\prime}_{i}{u}^{\prime}_{i}}/c$, and $c$ represents the local sound speed. As the wall temperature decreases, the peak values of $M_{t}$ and ${M}^{\prime}_{rms}$ increase, indicating that the cooling wall can enhance the compressibility effect, which is consistent with many previous studies including Zhang et al. (2018), Xu et al. (2021b) and Zhang et al. (2022). Moreover, it is shown in figure 3 (a) and (d) that the wall-normal locations $y^{+}$ of the peak values of $M_{t}$ and ${M}^{\prime}_{rms}$ increase under the wall unit scaling as the wall temperature decreases, while it is found in figure 3 (b) and (e) that the $M_{t}$ and ${M}^{\prime}_{rms}$ profiles against the semi-local scaling attain their peaks at almost the same values of $y^{*}$ in different wall temperature cases.

The normalised turbulent intensity of the streamwise velocity $u_{rms}^{\prime\prime}/u_{\tau}^{*}$ along wall-normal direction is shown in figure 4. It is found that $u_{rms}^{\prime\prime}/u_{\tau}^{*}$ attains its peak in the buffer layer (approximately $y^{*}\approx 18$). Furthermore, the peak values of $u_{rms}^{\prime\prime}/u_{\tau}^{*}$ are similar in “M8T04” and “M8T08” cases. However, the peak value is slightly larger in “M8T015”, which can be ascribed to the strongly colder wall temperature and slightly larger friction Reynolds number $Re_{\tau}$.

The normalised turbulent intensity of the pressure $p_{rms}^{\prime}/\bar{p}$ along wall-normal direction is plotted in figure 5. It is found that the $p_{rms}^{\prime}/\bar{p}$ profile in nearly adiabatic wall case (“M8T08” case) reaches the maximum value at $y^{*}\approx 60$, and then decreases rapidly among the boundary layer to a flat platform. As the wall temperature decreases, the intensity of pressure $p_{rms}^{\prime}/\bar{p}$ significantly increases among the whole boundary layer. The profiles of $p_{rms}^{\prime}/\bar{p}$ in “M8T04” and “M8T015” have secondary peaks at $y^{*}\approx 60$, which are consistent with the wall-normal location of the primary peak in “M8T08”. A special phenomenon is observed that the intensities of $p_{rms}^{\prime}/\bar{p}$ are significantly enhanced near the wall when the wall is strongly cooled, which further result in the fact that the $p_{rms}^{\prime}/\bar{p}$ profiles attain their primary peaks at the wall in “M8T04” and “M8T015”. The significant enhancement of $p_{rms}^{\prime}/\bar{p}$ near the wall can be ascribed to the appearance of a special acoustic structure (“the travelling-wave-like alternating positive and negative structures” or TAPNS) in the near-wall region, which will be specifically discussed in Section 4.

The dimensionless entropy per unit mass $s$ can be defined as $s=C_{v}log\left(T/\rho^{\gamma-1}\right)$ (Gerolymos & Vallet, 2014; Wang et al., 2019). The normalised turbulent intensities $s_{rms}^{\prime}\gamma M^{2}$, $\rho_{rms}^{\prime}/\bar{\rho}$ and $T_{rms}^{\prime}/\bar{T}$ along wall-normal direction are shown in figure 6. It is found in figure 6 (a) and (d) that the normalised turbulent intensity of the entropy $s_{rms}^{\prime}\gamma M^{2}$ attains its primary peak near the edge of the boundary layer ($y/\delta\approx 0.7$). Furthermore, a secondary peak is observed at $y^{*}\approx 50$ in “M8T08”, and this local secondary peak gradually disappears as the wall temperature decreases. It is also found that as the wall temperature decreases, the intensity $s_{rms}^{\prime}\gamma M^{2}$ decreases in the region between $y^{*}>10$ and $y/\delta<0.7$, while $s_{rms}^{\prime}\gamma M^{2}$ is significantly enhanced in the near-wall region ($y^{*}<10$). An interesting phenomenon is found that a strong local secondary peak is observed at $y^{*}\approx 5$ in “M8T015”. It is noted that the local secondary peaks also appear in the near-wall region in “M6Tw025” and “M14Tw018” cases in Zhang et al. (2018). However, to the best of our knowledge, previous studies have not provided a physical explanation for the strong local secondary peak of $s_{rms}^{\prime}\gamma M^{2}$ in the near-wall region when the wall is strongly cooled. In Section 4, it is found that the above phenomenon can be attributed to the appearance of a special entropic structure (“the streaky entropic structures” or SES) near the wall.

It is also found that the intensities $\rho_{rms}^{\prime}/\bar{\rho}$ and $T_{rms}^{\prime}/\bar{T}$ have similar behaviours with $s_{rms}^{\prime}\gamma M^{2}$ among most regions of the boundary layer, except for the much larger values of $\rho_{rms}^{\prime}/\bar{\rho}$ in the vicinity of the wall ($y^{*}<5$). The similarity among the intensities $\rho_{rms}^{\prime}/\bar{\rho}$, $T_{rms}^{\prime}/\bar{T}$ and $s_{rms}^{\prime}\gamma M^{2}$ can be explained based on the Kovasznay decomposition (Kovasznay, 1953; Chassaing et al., 2002; Gauthier, 2017; Wang et al., 2019). It is noted that the Kovasznay decomposition can decompose the thermodynamic variables into the acoustic modes and the entropic modes (Kovasznay, 1953; Chassaing et al., 2002; Gauthier, 2017; Wang et al., 2019).

In compressible turbulent flow, the acoustic modes of the thermodynamic variables can be defined as (Chassaing et al., 2002; Gauthier, 2017; Wang et al., 2019)

$$p^{\prime}_{I}=p-\bar{p},$$

(9)

$${\rho}^{\prime}_{I}=\frac{\bar{\rho}p^{\prime}_{I}}{\gamma\bar{p}},$$

(10)

$${T}^{\prime}_{I}=\frac{\left(\gamma-1\right)\bar{T}p^{\prime}_{I}}{\gamma\bar{p}};$$

(11)

and the entropic modes can be given by (Chassaing et al., 2002; Gauthier, 2017; Wang et al., 2019)

$$p^{\prime}_{E}=0,$$

(12)

$${\rho}^{\prime}_{E}=\rho-\bar{\rho}-{\rho}^{\prime}_{I},$$

(13)

$${T}^{\prime}_{E}=T-\bar{T}-{T}^{\prime}_{I}.$$

(14)

Therefore, the acoustic mode of the pressure $p^{\prime}_{I}$ is consistent with the fluctuating pressure $p^{\prime}$, and the fluctuating density and temperature can be divided into the acoustic modes and the entropic modes respectively: ${\rho}^{\prime}={\rho}^{\prime}_{I}+{\rho}^{\prime}_{E}$; ${T}^{\prime}={T}^{\prime}_{I}+{T}^{\prime}_{E}$. It is noted that the correlation coefficients between variables ${\varphi}^{\prime}$ and ${\psi}^{\prime}$ can be defined as

$$R\left({\varphi}^{\prime},{\psi}^{\prime}\right)=\frac{\overline{{\varphi}^{\prime}{\psi}^{\prime}}}{\sqrt{\overline{{\varphi}^{\prime 2}}}\sqrt{\overline{{\psi}^{\prime 2}}}}.$$

(15)

The correlation coefficient $R\left({\varphi}^{\prime},{\psi}^{\prime}\right)=1$ indicates that the variables ${\varphi}^{\prime}$ and ${\psi}^{\prime}$ are positively linearly correlated with each other; while $R\left({\varphi}^{\prime},{\psi}^{\prime}\right)=-1$ suggests that the variables ${\varphi}^{\prime}$ and ${\psi}^{\prime}$ are negatively linearly correlated with each other.

The correlation coefficients $R\left(\rho_{I}^{\prime},T_{I}^{\prime}\right)$, $R\left(\rho_{I}^{\prime},p^{\prime}\right)$, $R\left(\rho_{E}^{\prime},T_{E}^{\prime}\right)$ and $R\left(\rho_{E}^{\prime},s^{\prime}\right)$ along wall-normal direction are shown in figure 7. It is found that $R\left(\rho_{I}^{\prime},T_{I}^{\prime}\right)=1$ and $R\left(\rho_{I}^{\prime},p^{\prime}\right)=1$ along wall-normal direction, confirming that $\rho_{I}^{\prime}$, $T_{I}^{\prime}$ and $p^{\prime}$ are positively linearly correlated with each other. Furthermore, $R\left(\rho_{E}^{\prime},T_{E}^{\prime}\right)\approx-1$ and $R\left(\rho_{E}^{\prime},s^{\prime}\right)\approx-1$, suggesting that $\rho_{E}^{\prime}$ is almost negatively linearly correlated with $T_{E}^{\prime}$ and $s^{\prime}$ respectively. Accordingly, the acoustic modes of density and temperature $\rho_{I}^{\prime}$ and $T_{I}^{\prime}$ are linearly correlated with the fluctuating pressure $p^{\prime}$, and the entropic modes of density and temperature $\rho_{E}^{\prime}$ and $T_{E}^{\prime}$ are almost linearly correlated with the fluctuating entropy $s^{\prime}$.

The normalised turbulent intensities of the acoustic and entropic modes of density and temperature along wall-normal direction are shown in figure 8. It is found that the profiles of the intensities of the acoustic modes of density and temperature (figure 8 (a) (c)) are similar to that of the fluctuating pressure (figure 5 (a)), and the intensities of the entropic modes of density and temperature (figure 8 (b) (d)) also have similar behaviours with that of the fluctuating entropy (figure 6 (a)).

The relative contributions $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ and $T_{E,rms}^{\prime}/\left(T_{E,rms}^{\prime}+T_{I,rms}^{\prime}\right)$ along wall-normal direction are shown in figure 9. It is noted that if the relative contributions $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ and $T_{E,rms}^{\prime}/\left(T_{E,rms}^{\prime}+T_{I,rms}^{\prime}\right)$ are lower than 0.5, the acoustic modes of density and temperature are predominant over their entropic modes; on the contrary, if the relative contributions are larger than 0.5, the entropic modes are dominant. It is found that the relative contributions $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ and $T_{E,rms}^{\prime}/\left(T_{E,rms}^{\prime}+T_{I,rms}^{\prime}\right)$ increase as the wall-normal location increases. Furthermore, the relative contribution $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ is totally smaller than $T_{E,rms}^{\prime}/\left(T_{E,rms}^{\prime}+T_{I,rms}^{\prime}\right)$ among the boundary layer, suggesting that the entropic mode has a much larger contribution in fluctuating temperature than that in fluctuating density. In the near-wall region, the fluctuating density is mainly dominated by its acoustic mode, and the acoustic mode of temperature is slightly predominant over the entropic mode of temperature. However, in the far-wall region, the fluctuating density and temperature are mainly dominated by their entropic modes, which further lead to the similarity between the profiles of $s_{rms}^{\prime}\gamma M^{2}$, $\rho_{rms}^{\prime}/\bar{\rho}$ and $T_{rms}^{\prime}/\bar{T}$ shown in figure 6.

It is also found that the wall temperature has a significant influence on the relative contributions. In the near-wall region, as the wall temperature decreases, the acoustic modes and the entropic modes of density and temperature are all enhanced, but the amounts of the growth of the entropic modes are slightly larger than those of the acoustic modes, which result in the increase of the relative contributions $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ and $T_{E,rms}^{\prime}/\left(T_{E,rms}^{\prime}+T_{I,rms}^{\prime}\right)$ with colder wall temperature. In the far-wall region, when the wall temperature becomes colder, the acoustic modes are enhanced while the entropic modes are reduced, which lead to the decrease of the relative contributions of the entropic modes in strongly cooled wall case. Therefore, as the wall temperature decreases, the increase of $\rho_{rms}^{\prime}/\bar{\rho}$ and $T_{rms}^{\prime}/\bar{T}$ near the wall is mainly due to the contributions of both the acoustic and entropic modes, while the decrease of $\rho_{rms}^{\prime}/\bar{\rho}$ and $T_{rms}^{\prime}/\bar{T}$ far from the wall can be ascribed to the major contributions of the entropic modes.

It has been noted previously in figure 6 (b) that $\rho_{rms}^{\prime}/\bar{\rho}$ has much larger values than those of $s_{rms}^{\prime}\gamma M^{2}$ and $T_{rms}^{\prime}/\bar{T}$ in the vicinity of the wall ($y^{*}<5$), especially in “M8T04” and “M8T015”. This can be ascribed to the dominant contribution of the acoustic mode to the fluctuating density near the wall (figure 9 (a)), and the fact that the acoustic mode of density attains the primary peak at the wall in “M8T04” and “M8T015” (figure 8 (a)). Moreover, it is also found in figure 6 (e) that the primary peak value of $\rho_{rms}^{\prime}/\bar{\rho}$ at $y/\delta\approx 0.6$ in “M8T015” is slightly larger than that in “M8T04”, which is opposite to the behaviours of $s_{rms}^{\prime}\gamma M^{2}$ and $T_{rms}^{\prime}/\bar{T}$. This can be attributed to the facts that the intensity of the acoustic mode of density in “M8T015” is much larger than that in “M8T04” (figure 8 (a)), and the acoustic mode of density in “M8T015” has a much larger contribution to fluctuating density than that in “M8T04” near the edge of the boundary layer (figure 9 (a)).

In order to quantitatively characterize the characteristic length scales of the energetic structures of the streamwise velocity and thermodynamic variables, the premultiplied streamwise and spanwise spectra are further investigated. It is noted that $k_{x}$ and $k_{z}$ are the wavenumber in the streamwise and spanwise directions respectively. ${\lambda}_{x}$ and ${\lambda}_{z}$ are the corresponding wavelength in the streamwise and spanwise directions respectively. Furthermore, ${\lambda}_{x}^{+}={\lambda}_{x}/\delta_{\nu}$ and ${\lambda}_{x}^{*}={\lambda}_{x}/\delta_{\nu}^{*}$.

The normalised premultiplied streamwise and spanwise spectra of the fluctuating streamwise velocity $k_{x}E_{\rho u^{\prime\prime}u^{\prime\prime}}/\left(\bar{\rho}{u_{\tau}^{*}}^{2}\right)$ and $k_{z}E_{\rho u^{\prime\prime}u^{\prime\prime}}/\left(\bar{\rho}{u_{\tau}^{*}}^{2}\right)$ are shown in figure 10. It is found in figure 10 (a) and (c) that the premultiplied streamwise and spanwise spectra of the fluctuating streamwise velocity achieve their primary peaks at $y^{*}\approx 18$, which are consistent with the primary peak location of $u_{rms}^{\prime\prime}/u_{\tau}^{*}$ (figure 4 (a)). The peak of $u_{rms}^{\prime\prime}/u_{\tau}^{*}$ corresponds to the cycle of the near-wall streak generation (Jim$\acute{e}$nez & Pinelli, 1999; Jim$\acute{e}$nez, 2013; Hutchins & Marusic, 2007b; Monty et al., 2009; Pirozzoli & Bernardini, 2013; Huang et al., 2022). Furthermore, $k_{z}E_{\rho u^{\prime\prime}u^{\prime\prime}}/\left(\bar{\rho}{u_{\tau}^{*}}^{2}\right)$ has a weak outer peak at nearly $y/\delta\approx 0.2$ (figure 10 (d)), indicating the long streaky motion in the outer region (i.e. LSMs or VLSMs) (Hutchins & Marusic, 2007a; Monty et al., 2009; Pirozzoli & Bernardini, 2011, 2013; Hwang, 2016; Huang et al., 2022). However, the outer peak of $k_{x}E_{\rho u^{\prime\prime}u^{\prime\prime}}/\left(\bar{\rho}{u_{\tau}^{*}}^{2}\right)$ is not evident (figure 10 (b)), mainly due to the relatively low Reynolds number of the DNS database (Hwang, 2016).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating streamwise velocity $k_{x}E_{\rho u^{\prime\prime}u^{\prime\prime}}/\left(\bar{\rho}{u_{\tau}^{*}}^{2}\right)$ and $k_{z}E_{\rho u^{\prime\prime}u^{\prime\prime}}/\left(\bar{\rho}{u_{\tau}^{*}}^{2}\right)$ at $y^{*}=18$ and $y/\delta=0.2$ are shown in figure 11 and figure 12 respectively. It is found in figure 11 (a) and (c) that the peak locations of the premultiplied streamwise and spanwise spectra of the fluctuating streamwise velocity in wall unit $\left({\lambda}_{x}^{+},{\lambda}_{z}^{+}\right)$ increase significantly as the wall temperature decreases, especially in strongly cooled wall case “M8T015”. These peak locations represent the characteristic streamwise length and spanwise spacing of the near-wall streaks. However, the semi-local scaling $\left({\lambda}_{x}^{*},{\lambda}_{z}^{*}\right)$ can significantly reduce the disparity between the peak locations of spectra (figure 11 (b) (d)), yielding the characteristic streamwise length ${\lambda}_{x}^{*}\approx 10^{3}$ and spanwise spacing ${\lambda}_{z}^{*}\approx 150$ in different wall temperature cases. Similar values of the characteristic streamwise length and spanwise spacing of the near-wall streaks have also been found in incompressible boundary layers (Hutchins & Marusic, 2007b; Monty et al., 2009) and compressible boundary layers (Pirozzoli & Bernardini, 2013; Huang et al., 2022). Furthermore, it is shown in figure 12 that the streamwise and spanwise spectra achieve their peak values at ${\lambda}_{x}/\delta\approx 2$ and ${\lambda}_{z}/\delta\approx 1$ respectively at $y/\delta=0.2$, which represent the characteristic streamwise length and spanwise spacing of the long streaky motions in the outer region. Similar values of the characteristic spanwise spacing at $y/\delta=0.2$ was also obtained in previous study of hypersonic turbulent boundary layers (Cogo et al., 2022).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating pressure $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ are shown in figure 13. The $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ in the nearly adiabatic wall case “M8T08” achieve their primary peaks at $y^{*}\approx 60$, while the pressure spectra of “M8T04” and “M8T015” achieve their primary peaks at the wall. These observations are consistent with the primary peak locations of $p_{rms}^{\prime}/\bar{p}$ (figure 5 (a)).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating pressure $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ at $y^{*}=60$, $y^{*}=2$ and $y^{*}=5$ are shown in figure 14, figure 15 and figure 17 respectively. It is shown in figure 14 that as the wall temperature decreases, the peak values of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ are drastically enhanced, and the peak locations also increase under the wall unit $({\lambda}_{x}^{+},{\lambda}_{z}^{+})$ at $y^{*}=60$. The semi-local scaling $({\lambda}_{x}^{*},{\lambda}_{z}^{*})$ significantly reduces the disparity between the peak locations, and $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ achieve their peaks at ${\lambda}_{x}^{*}\approx 400$ and ${\lambda}_{z}^{*}\approx 250$ respectively at $y^{*}=60$.

However, in the near-wall region ($y^{*}=2$), the behaviours of the pressure spectra are quite different. It is found in figure 15 that peak locations of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ in the nearly adiabatic wall case “M8T08” at $y^{*}=2$ are significantly different from those in cooled wall cases “M8T04” and “M8T015”. This difference can be explained according to the instantaneous fields of the normalised fluctuating pressure $p^{\prime}/\bar{p}$ at $y^{*}=2$ as shown in figure 16. It is shown in figure 16 (a) and (b) that the special structures marked by red dashed boxes are observed, and these structures are named as “the travelling-wave-like alternating positive and negative structures” (TAPNS). These structures are well organized as wavelike alternating positive and negative patterns along the streamwsie direction, and have also been found in compressible channel flows (Yu et al., 2019; Tang et al., 2020) and turbulent boundary layers (Xu et al., 2021b; Zhang et al., 2022). Moreover, the TAPNS disappear in nearly adiabatic wall case “M8T08”, and are strongly enhanced as wall temperature decreases. Particularly, in the strongly cooled wall case “M8T015” (figure 16 (a)), the TAPNS are prevalent in the whole field. The extreme positive and negative values of $p^{\prime}/\bar{p}$ are mainly located among the TAPNS, which further lead to the significant enhancement of $p_{rms}^{\prime}/\bar{p}$ near the wall (figure 5 (a)). The peak locations of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ and $k_{z}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ in the nearly adiabatic wall case “M8T08” are ${\lambda}_{x}^{*}\approx 300$ and ${\lambda}_{z}^{*}\approx 200$ respectively at $y^{*}=2$, which have similar aspect ratio ${\lambda}_{x}^{*}/{\lambda}_{z}^{*}$ to the characteristic scales at $y^{*}=60$. Nevertheless, the pressure spectra achieve their peaks at ${\lambda}_{x}^{*}\approx 120$ and ${\lambda}_{z}^{*}\approx 200$ in “M8T04”, and ${\lambda}_{x}^{*}\approx 210$ and ${\lambda}_{z}^{*}\approx 250$ in “M8T015”. As the wall temperature decreases from “M8T04” to “M8T015”, the characteristic streamwise length scale ${\lambda}_{x}^{*}$ is significantly increased, while the characteristic spanwise spacing scale ${\lambda}_{z}^{*}$ is slightly enhanced, which is similar to the behaviour of the instantaneous wavelike alternating positive and negative structures shown in figure 16 (a) and (b). Therefore, it can be inferred that the characteristic streamwise length scale ${\lambda}_{x}^{*}$ and spanwise spacing scale ${\lambda}_{z}^{*}$ of the cooled wall cases “M8T04” and “M8T015” represent the scales of the TAPNS. The TAPNS is short and fat (i.e. ${\lambda}_{x}^{*}<{\lambda}_{z}^{*})$. As the wall temperature decreases, the aspect ratio ${\lambda}_{x}^{*}/{\lambda}_{z}^{*}$ of TAPNS increases from almost 0.6 in “M8T04” to 0.84 in “M8T015”, which indicates that the strongly cooled wall prefers to increase the streamwise length scale compared with the spanwise spacing scale of TAPNS. Moreover, the intensity of the TAPNS is strongly enhanced as the wall temperature decreases. It is noted that these wavelike alternating positive and negative patterns have also been reported in the fluctuating dilatation ${\theta}^{\prime\prime}\equiv\partial{u}^{\prime\prime}_{k}/\partial x_{k}$ in figure 6 of Xu et al. (2021b). Therefore, it can be deduced that as the wall temperature decreases, the peak values of the pressure spectra and the turbulent intensities of the pressure are significantly enhanced in the near-wall region, which further lead to the enhanced compressibility near the wall (Duan et al., 2010; Zhang et al., 2018; Xu et al., 2021a, b; Zhang et al., 2022). Moreover, it is shown above in figure 5 (a) that the maximum values of $p^{\prime}/\bar{p}$ appear at the wall in “M8T04” and “M8T015”. This phenomenon is attributed to the appearance of TAPNS near the wall in the cooled wall cases.

As the wall-normal location moves further away from the wall (at $y^{*}=5$), it is shown in figure 17 that most of the peak locations of the streamwise and spanwise spectra are similar to those at $y^{*}=2$, except for the peak location of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ in “M8T04”. It is found in figure 17 (b) that the peak of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ in “M8T04” becomes much wider than that at $y^{*}=2$, and the characteristic streamwise length scale of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ in “M8T08” (${\lambda}_{x}^{*}\approx 300$) is also found in the streamwise spectra in “M8T04”. Here for better description, the pressure structures with the characteristic streamwise length scale in “M8T08” are named as “the base acoustic structures”. The above observation indicates that the base acoustic structures also exist in the cooled wall cases “M8T04” and “M8T015”. If the wall is significantly cooled and the wall-normal location $y^{*}$ is close to the wall, the strength of the TAPNS is much larger than that of the base pressure structures, and only the characteristic streamwise length scale ${\lambda}_{x}^{*}$ of TAPNS is dominant in $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ (i.e. “M8T04” and “M8T015” in figure 15 (b) and “M8T015” in figure 17 (b)). As the wall temperature increases and $y^{*}$ moves away from the wall, the intensity of the TAPNS becomes weaker. Therefore, the TAPNS and the base acoustic structures have similar intensities in these situations, which result in the wide peak in “M8T04” in figure 17 (b). When the wall temperature is nearly adiabatic (i.e. “M8T08”), the TAPNS disappear and only the base acoustic structures are dominant in the near-wall region.

The normalised premultiplied streamwise and spanwise spectra of the fluctuating entropy $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ and $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ are depicted in figure 18. The $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ and $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ achieve their primary peaks near the edge of the boundary layer, which are consistent with the primary peak location of $s_{rms}^{\prime}\gamma M^{2}$ (figure 6 (d)). The secondary peaks of entropy spectra appear at $y^{*}\approx 50$ in “M8T08” and “M8T04”, which are coincident with the secondary peak location of $s_{rms}^{\prime}\gamma M^{2}$ (figure 6 (a)). However, for the strongly cooled wall case “M8T015”, the secondary peaks of entropy spectra appear at nearly $y^{*}=5$. The normalised premultiplied streamwise and spanwise spectra of the fluctuating entropy $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ and $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ at $y/\delta=0.7$, $y^{*}=50$, $y^{*}=2$ and $y^{*}=5$ are plotted in figure 19, figure 20, figure 21 and figure 23 respectively. It is shown in figure 19 that $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ and $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ attain their peaks at ${\lambda}_{x}/\delta\approx 1.3$ and ${\lambda}_{z}/\delta\approx 1$ respectively. Furthermore, the peak values of $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ and $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ decrease as the wall temperature decreases. It can be seen in figure 20 that the semi-local scaling can reduce the disparity between the peak locations of spectra. The $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ and $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ achieve their peaks at ${\lambda}_{x}^{*}\approx 700$ and ${\lambda}_{z}^{*}\approx 250$ respectively at $y^{*}=50$. Furthermore, the peak values of the normalised premultiplied streamwise and spanwise spectra of the fluctuating entropy decrease as the wall temperature decreases, which are consistent with the observation that the values of $s_{rms}^{\prime}\gamma M^{2}$ decrease as the wall temperature decreases at $y^{*}=50$ (figure 6 (a)).

It is shown in figure 6 (a) that a local secondary peak of $s_{rms}^{\prime}\gamma M^{2}$ appears near the wall in “M8T015”. The underlying mechanism of this phenomenon is revealed as follows. It is found in figure 21 that the peak values of the entropy spectra in “M8T04” and “M8T015” are significantly larger than those in “M8T08”, and the peak values increase as the wall temperature decreases. Furthermore, the $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ in “M8T015” and “M8T04” attain their peaks at ${\lambda}_{x}/\delta\approx 2.3$ and 1.3 respectively, and the peak locations of the $k_{z}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ in “M8T015” and “M8T04” are ${\lambda}_{z}/\delta\approx 0.25$ and 0.2 respectively. The underlying structures of the characteristic scales of the entropy spectra are revealed by the instantaneous fields of the normalised fluctuating entropy $s^{\prime}\gamma M^{2}$ at $y^{*}=2$ shown in figure 22. It can be seen in figure 22 (a) and (b) that the long and thin fluctuating entropy structures only appear when the wall is cooled (“M8T015” and “M8T04”). These special entropic structures exhibit the streaky patterns with alternating stripes of the high and low entropy, which is similar to the streaks of $u^{\prime}$ (shown in figure 5 (a) in Xu et al. (2021a)) with $R\left(s^{\prime},u^{\prime}\right)=0.93$ at $y^{*}=2$ in “M8T015”. However, the entropy structures in the nearly adiabatic wall case “M8T08” are pretty weak and fragmented. Here for the sake of description, the entropy structures in “M8T08” are named as “the base entropic structures”, while the generated streaky patterns in “M8T04” and “M8T015” are called “the streaky entropic structures” (SES). It is shown in figure 18 that the characteristic length scale is significantly larger than the characteristic spanwise spacing scale (i.e. ${\lambda}_{x}/\delta\gg{\lambda}_{z}/\delta$) in “M8T015” and “M8T04”, which is consistent with the long and thin nature of SES. Furthermore, the ${\lambda}_{x}/\delta$ and ${\lambda}_{z}/\delta$ in “M8T015” are larger than those in “M8T04”, which is coincident with the observation that the SES become longer in the streamwise direction and fatter in the spanwise direction as the wall temperature decreases (figure 22 (a) and (b)). Accordingly, it can be inferred that the characteristic streamwise length scale ${\lambda}_{x}/\delta$ and spanwise spacing scale ${\lambda}_{z}/\delta$ of the entropy spectra in “M8T015” and “M8T04” represent the scales of the SES. Furthermore, the aspect ratio ${\lambda}_{x}/{\lambda}_{z}$ of SES increases from 6.5 in “M8T04” to 9.2 in “M8T015”, suggesting that the strongly cooled wall prefers to increase the streamwise length scale compared with the spanwise spacing scale of SES. Moreover, the intensity of the SES is significantly enhanced as the wall temperature decreases, which further leads to the significant enhancement of $s_{rms}^{\prime}\gamma M^{2}$ near the wall in the cooled wall cases (figure 6 (a)). Specifically, the local secondary peak of $s_{rms}^{\prime}\gamma M^{2}$ near the wall in “M8T015” is attributed to the strong intensity of the SES.

As the wall-normal location increases to $y^{*}=5$, it is seen in figure 23 that the entropy spectra in “M8T08” and “M8T04” have pretty small peak values, and attain their peaks at ${\lambda}_{x}^{*}\approx 450$ and ${\lambda}_{z}^{*}\approx 140$ respectively at $y^{*}=5$, which have a similar characteristic aspect ratio ${\lambda}_{x}^{*}/{\lambda}_{z}^{*}$ with the characteristic scales at $y^{*}=50$. Furthermore, the peak values of the entropy spectra in “M8T08” and “M8T04” decrease as the wall temperature decreases, which are coincident with the behaviours at $y^{*}=50$ and $y/\delta=0.7$. However, the peak values of the entropy spectra in strongly cooled wall case “M8T015” are significantly larger than those in other two cases. The entropy spectra in “M8T015” attain their peaks at ${\lambda}_{x}/\delta\approx 2.3$ and ${\lambda}_{z}/\delta\approx 0.25$ respectively, which are similar to the characteristic scales at $y^{*}=2$. The above observations indicate that the SES exist in “M8T015”, while disappear in “M8T04” at $y^{*}=5$. Therefore, it is implied that the SES can exist in a larger range of wall-normal distance $y^{*}$ as the wall temperature decreases.

The normalised premultiplied streamwise and spanwise spectra of the fluctuating density $k_{x}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ and $k_{z}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ are shown in figure 24. It is found that the spectra of the fluctuating density are similar to those of the fluctuating entropy at $y^{*}>20$, which is consistent with the observation in figure 9 (a) that the fluctuating density is dominated by its entropic mode at $y^{*}>20$. The $k_{x}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ and $k_{z}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ in three cases achieve their primary peaks at $y/\delta\approx 0.6$, and the secondary peak locations of the density spectra in “M8T08” and “M8T04” are $y^{*}\approx 40$, which are coincident with peak locations of $\rho_{rms}^{\prime}/\bar{\rho}$ (figure 6 (b) (e)). However, a complicated behaviour of the density spectra appears in “M8T015”, mainly due to the strong variation of the relative contribution $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ at $y^{*}<20$ with strongly cooled wall (figure 9 (a)).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating density $k_{x}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ and $k_{z}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ at $y/\delta=0.6$ and $y^{*}=40$ are depicted in figure 25 and figure 26 respectively. The $k_{x}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ and $k_{z}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ achieve their peaks at ${\lambda}_{x}/\delta\approx 1.3$ and ${\lambda}_{z}/\delta\approx 1$ respectively at $y/\delta=0.6$ (figure 25), and the peak locations of the density spectra are ${\lambda}_{x}^{*}\approx 700$ and ${\lambda}_{z}^{*}\approx 250$ respectively at $y^{*}=40$ (figure 26). It is noted that the peak locations of the density spectra are totally similar to those of the entropy spectra (figure 19 and 20), which are mainly due to the reason that the fluctuating density is dominated by its entropic mode at $y^{*}>20$ (figure 9 (a)).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating density $k_{x}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ and $k_{z}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ at $y^{*}=2$ and $y^{*}=5$ are depicted in figure 27 and figure 28 respectively.

At $y^{*}=2$ (figure 27), the peak locations of the density spectra are similar to those of the pressure spectra at $y^{*}=2$ in three cases, suggesting that the acoustic mode of density is dominant in the fluctuating density at $y^{*}=2$. However, the values of the density spectra at large values of ${\lambda}_{x}/\delta$ and ${\lambda}_{z}/\delta$ are much larger than those of the pressure spectra at $y^{*}=2$ in “M8T04” and “M8T015”, which indicate that the relative contribution of the entropic mode to the fluctuating density is enhanced at $y^{*}=2$ when the wall is cooled.

The enhancement of the relative contribution of the entropic mode to the fluctuating density in strongly cooled wall case becomes much more significant at $y^{*}=5$. It is found in figure 28 that the peak locations of the density spectra are also similar to those of the pressure spectra at $y^{*}=5$ in three cases, indicating that the acoustic mode of density still has a major contribution to the fluctuating density at $y^{*}=5$. However, a secondary peak of $k_{x}E_{\rho^{\prime}\rho^{\prime}}/\bar{\rho}^{2}$ appears at ${\lambda}_{x}/\delta\approx 2.3$ in “M8T015”, suggesting that the relative contribution of the entropic mode to the fluctuating density becomes significantly larger in the strongly cooled wall case “M8T015”. It is shown in figure 9 (a) that the relative contribution $\rho_{E,rms}^{\prime}/\left(\rho_{E,rms}^{\prime}+\rho_{I,rms}^{\prime}\right)$ in “M8T015” has a hump marked by the green dashed box. Here, the density spectra at $y^{*}=5$ in “M8T015” reveal that the hump is mainly due to the appearance of SES when the wall is strongly cooled. The SES significantly enhance the intensity of the entropic mode of density, which further enhance the relative contribution of the entropic mode to the fluctuating density.

The normalised premultiplied streamwise and spanwise spectra of the fluctuating temperature $k_{x}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ and $k_{z}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ are shown in figure 29. It is found that the temperature spectra are similar to the entropy spectra (figure 18), indicating the dominant contribution of the entropic mode of temperature as shown in figure 9 (b).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating temperature $k_{x}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ and $k_{z}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ at $y/\delta=0.8$ and $y^{*}=50$ are shown in figure 30 and figure 31 respectively. The temperature spectra attain their peaks at ${\lambda}_{x}/\delta\approx 1.3$ and ${\lambda}_{z}/\delta\approx 1$ at $y/\delta=0.8$, which are similar to the observation in Cogo et al. (2022). Furthermore, the peak locations of the temperature spectra are ${\lambda}_{x}^{*}\approx 700$ and ${\lambda}_{z}^{*}\approx 250$ respectively at $y^{*}=50$. It is noted that the peak locations of the temperature spectra far from the wall are similar to the behaviours of the entropy spectra, suggesting the dominance of the entropic mode in fluctuating temperature at $y^{*}>20$ (figure 9 (b)).

The normalised premultiplied streamwise and spanwise spectra of the fluctuating temperature $k_{x}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ and $k_{z}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ at $y^{*}=2$ and $y^{*}=5$ are shown in figure 32 and figure 33 respectively. The values of the temperature spectra at $y^{*}=2$ in “M8T08” are pretty small, which is consistent with the small values of $T_{rms}^{\prime}/\bar{T}$ at $y^{*}=2$ in nearly adiabatic wall case (figure 6 (c)). The primary peak locations of $k_{x}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ in “M8T04” and “M8T015” are ${\lambda}_{x}^{*}\approx 120$ and 210 at $y^{*}=2$ respectively, which are similar to the peak locations of $k_{x}E_{p^{\prime}p^{\prime}}/\bar{p}^{2}$ at $y^{*}=2$ (figure 15 (b)). Moreover, the $k_{x}E_{T^{\prime}T^{\prime}}/\bar{T}^{2}$ in “M8T04” and “M8T015” attain their secondary peaks at ${\lambda}_{x}/\delta\approx 1.3$ and 2.3 respectively, which are also coincident with the peak locations of $k_{x}E_{s^{\prime}s^{\prime}}\left(\gamma M^{2}\right)^{2}$ at $y^{*}=2$ (figure 21 (c)). These observations indicate that both the acoustic and entropic modes have significant contributions to the fluctuating temperature at $y^{*}=2$ when the wall is cooled.