2. Physical model and numerical method

The compressible turbulent boundary layers over a flat wall considered in the present study are governed by the Navier–Stokes equations for Newtonian perfect gases, written as

(2.1) \begin{align} \frac {\partial \rho }{\partial t} + \frac {\partial \rho u_i}{\partial x_i} = 0, \\[-28pt] \nonumber \end{align}

(2.2) \begin{align} \frac {\partial \rho u_i}{\partial t} + \frac {\partial \rho u_i u_j}{\partial x_{\kern-1pt j}} = -\frac {\partial p}{\partial x_i} + \frac {\partial \tau _{\textit{ij}}}{\partial x_{\kern-1pt j}}, \\[-28pt] \nonumber \end{align}

(2.3) \begin{align} \frac {\partial \rho E}{\partial t} + \frac {\partial \rho u_j E}{\partial x_{\kern-1pt j}} = -\frac {\partial p u_i}{\partial x_i} + \frac {\partial \tau _{\textit{ij}} u_i}{\partial x_{\kern-1pt j}} -\frac {\partial q_{\kern-1pt j}}{\partial x_{\kern-1pt j}}. \\[-12pt] \nonumber \end{align}

The velocity components in the $x_i$ direction ( $i=1,2,3$ , corresponding to the streamwise, wall-normal and spanwise directions, also represented by $x$ , $y$ and $z$ ) are denoted by $u_i$ (also $u$ , $v$ and $w$ , respectively). Here, $\rho$ , $p$ and $E$ are density, pressure and total energy, related to temperature $T$ by the state equation for perfect gases

(2.4) \begin{align} p = \rho \textit{RT},\quad E = \frac {1}{2} u_i u_i + C_V T, \end{align}

with $R$ the gas constant, and $C_V$ the constant volume specific heat. The viscous stress $\tau _{\textit{ij}}$ is determined by the constitutive equation for Newtonian fluid, and the molecular heat conduction $q_{\kern-1pt j}$ by Fourier’s law

(2.5) \begin{align} \tau _{\textit{ij}} = \mu \left ( \frac {\partial u_i}{\partial x_{\kern-1pt j}} + \frac {\partial u_j}{\partial x_i} \right ) -\frac {2}{3} \mu \delta _{\textit{ij}} \frac {\partial u_k}{\partial x_k},\quad q_{\kern-1pt j} = - \kappa \frac {\partial T}{\partial x_{\kern-1pt j}}, \end{align}

where $\mu$ is the fluid viscosity, calculated by Sutherland’s law, and $\kappa$ is the heat conductivity.

The boundary conditions are specified as follows. The flow inlet is given to be a synthetic turbulent boundary layer, composed of the mean velocity prescribed by the empirical formula (Musker Reference Musker1979), the mean temperature by the generalised Reynolds analogy (Zhang et al. Reference Zhang, Bi, Hussain and She2014) and turbulent fluctuations by the digital filtering method (Kempf, Wysocki & Pettit Reference Kempf, Wysocki and Pettit2012). The no-slip and no-penetration conditions for velocity, and the isothermal condition for temperature, are set at the wall. No-reflection conditions are enforced at the flow outlet and the upper boundary. Periodic conditions are adopted in the spanwise direction.

Direct numerical simulations are conducted by the finite difference method utilising the open-source code STREAmS developed by Bernardini et al. (Reference Bernardini, Modesti, Salvadore and Pirozzoli2021). The convective terms are discretised by the kinetic energy preserving scheme hybrid (Pirozzoli Reference Pirozzoli2010) with the fifth-order weighted essentially non-oscillation scheme (Shu & Osher Reference Shu and Osher1988) where the flow discontinuities are detected by the Ducros shock sensor (Ducros et al. Reference Ducros, Ferrand, Nicoud, Weber, Darracq, Gacherieu and Poinsot1999)

(2.6) \begin{align} s = \frac { - \boldsymbol{\nabla }\boldsymbol{\cdot }{\boldsymbol u}}{\sqrt {(\boldsymbol{\nabla }\boldsymbol{\cdot }{\boldsymbol u})^2 + |\boldsymbol{\nabla }\times {\boldsymbol u}| + (U_0/\delta _{\textit{in}})^2}}\\[-14pt]\nonumber \end{align}

by setting the criterion as $s\gt 0.3$ , with $U_0$ the free-stream velocity, and $\delta _{\textit{in}}$ the nominal boundary layer thickness at the turbulent inlet. The viscous terms are approximated by the sixth-order central scheme after being expanded into Laplacian form. The third-order low-storage Runge–Kutta scheme is applied for time advancement (Wray Reference Wray1990).

Hereinafter, the ensemble average of a generic flow quantity $\varphi$ is denoted by $\bar \varphi$ , and the corresponding fluctuations by $\varphi '$ . The Favre average, i.e. density-weighted average, is represented by $\tilde \varphi$ , and the corresponding fluctuations by $\varphi ''$ . The free-stream flow quantities are marked by the subscript $0$ . The superscript $+$ denotes the normalisation by the viscous scales, constituted by the wall shear stress $\tau _w$ , and mean fluid density and viscosity at the wall $\rho _w$ and $\mu _w$ , based on which the friction velocity $u_\tau$ , length scale $\delta _\nu$ and friction Reynolds number $ \textit{Re}_\tau$ can be defined as

(2.7) \begin{align} u_\tau = \sqrt {\frac {\tau _w}{\rho _w}},\quad \delta _\nu = \frac {\mu _w}{\rho _w u_\tau },\quad Re_\tau = \frac {\rho _w u_\tau \delta }{\mu _w}. \end{align}

In the cases with significant wall heat transfer, semi-local viscous scales are commonly adopted as well, the definition of which can be obtained by substituting the wall viscosity $\mu _w$ and density $\rho _w$ with local mean viscosity $\bar \mu$ and density $\bar \rho$ in defining viscous scales:

(2.8) \begin{align} u^*_\tau = \sqrt {\frac {\tau _w}{\bar \rho }},\quad \delta ^*_\nu = \frac {\bar \mu }{\bar \rho u^*_\tau },\quad Re^*_\tau = \frac {\bar \rho u^*_\tau \delta }{\bar \mu }. \end{align}

The flow quantities normalised by these semi-local viscous scales are marked by the superscript $*$ .

The flow parameters are reported in table 1. As we have stated previously, the free-stream Mach number $M_0 = U_0/c_0$ (where $c_0$ is the free-stream sound speed) is set as $6.0$ . The friction Reynolds number at the turbulent inlet is given as $ \textit{Re}_{\tau ,in} = 800$ , according to which the free-stream Reynolds numbers $ \textit{Re}_0$ are estimated. The estimated $ \textit{Re}_0$ values are listed in table 1. The wall temperatures in four cases are $0.75$ , $0.5$ , $0.35$ and $0.2$ times the recovery temperature $T_r = T_0 (1+r (\gamma -1)M^2_0/2)$ (where $r=0.89$ is the recovery factor). The streamwise, wall-normal and spanwise sizes of the computational domain are $L_x = 80 \delta _{\textit{in}}$ , $L_y = 6 \delta _{\textit{in}}$ and $L_z = 8 \delta _{\textit{in}}$ , discretised by $(N_x,N_y,N_z) = (4000, 800, 1200)$ grid points. In the streamwise direction, the grid points are uniformly distributed in the ranges $x=(0{-}50) \delta _{\textit{in}}$ and $x=(50{-}80) \delta _{\textit{in}}$ , respectively, with the streamwise grid interval in the former approximately four times larger than that in the latter to reduce computational costs in the development of the boundary layer to reach statistical equilibrium state, especially the generation of the large-scale motions and very-large-scale motions in the outer region. The small-scale motions in the near-wall region, on the other hand, are capable of recovering rapidly once the grids become finer due to their significantly shorter lifetime (Lozano-Durán & Jiménez Reference Lozano-Durán and Jiménez2014). A similar numerical strategy has been adopted in the previous study of Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2013). The grids are stretched by a hyperbolic sine function in the wall-normal direction, and uniformly distributed in the spanwise direction. The grid intervals under the viscous length scales at this location are reported in table 1. With the low-dissipative numerical schemes used in the present study (Pirozzoli Reference Pirozzoli2011), we deem the grid intervals to be sufficiently fine to capture the smallest-scale motions in turbulence. A preliminary examination of the turbulent statistics shows that the statistical equilibrium state can be reached at $x = 60 \delta _{\textit{in}}$ , so we report the results at $x= 70 \delta _{\textit{in}}$ herein unless otherwise stated. The simulations have been run for $150 \delta _{\textit{in}}/U_0$ before data samples are collected for another period $100 \delta _{\textit{in}}/U_0$ .

Table 1.

Flow parameters. Here, $ \textit{Re}_0 = \rho _0 U_0 \delta /\mu _0$ . The boundary layer thickness $\delta$ and the viscous scales are calculated at $x=70\delta _{\textit{in}}$ .

3. Instantaneous flow organisations

We commence our discussions by presenting the instantaneous flow fields. Figure 1 shows the instantaneous density distributions in the $x{-}y$ plane. The low-density regions are organised as forward-inclined back structures, with angle approximately $45^\circ$ (Cogo et al. Reference Cogo, Salvadore, Picano and Bernardini2022), forming bulges with fine fractal structures at the turbulent–non-turbulent interfaces (Zhang et al. Reference Zhang, Bi, Hussain and She2018b ; Su et al. Reference Su, Long, Wang and Li2024). In the free stream, density fluctuations radiate outwards, resembling the acoustic waves that can also be reflected by pressure fluctuations (Zhang, Duan & Choudhari Reference Zhang, Duan and Choudhari2017). With the lower wall temperature, the fluid density near the wall becomes higher. In the outer region, the fluid density remains approximately $(0.6{-}0.8)\rho _0$ for a comparatively high extent, and shows an evident interface between the turbulent and non-turbulent regions. Small-scale structures are becoming less obvious, probably due to the lower free-stream Reynolds number and the correspondingly higher dynamic viscosity that dissipates the small-scale motions more effectively. Such a low local Reynolds number effect can be reflected in flow statistics as well, as we will demonstrate in subsequent discussions.

Figure 1.

Instantaneous distribution of density $\rho /\rho _0$ in the $x{-}y$ plane for cases (a) C075, (b) C050, (c) C035, (d) C020.

Figure 2.

Instantaneous distribution of streamwise velocity fluctuations $u''/U_0$ at (a,c,e,g) $y^*=15$ and (b,d,f,h) $y=0.2\delta$ in cases (a,b) C075, (c,d) C050, (e,f) C035, (g,h) C020.

In figure 2, we present the instantaneous streamwise velocity distributions $u''/U_0$ at $y^* = 15$ in the buffer region (figures 2 a,c,e,g), where it is the most intense, and those at $y=0.2\delta$ in the outer region (figures 2 b,d,f,h) to highlight the characteristics of large-scale motions. In the buffer region at $y^*=15$ , the streamwise velocity fluctuations $u''/U_0$ are organised as streamwise-elongated streaks, commonly observed in wall-bounded turbulence of all types (Jiménez Reference Jiménez2018). Although different in length scales under global coordinates, the spanwise intervals are statistically the same under semi-local viscous scales with $\lambda ^*_z \approx 100$ , which we will show later in discussing the spanwise spectra. Moreover, there seem to manifest small-scale travelling-wave-like structures beneath the low-speed streaks in case C020 with the coldest wall, suggesting the significance of dilatational motions (Yu et al. Reference Zhang and Chernyshenko2024c ) under such flow conditions.

In the outer region, low-momentum regions are organised as streamwise-elongated structures as well, with the spanwise interval of approximately the boundary layer thickness $\delta$ , known as very-large-scale motions (Hutchins & Marusic Reference Hutchins and Marusic2007b ; Ringuette et al. Reference Ringuette, Wu and Martin2008; Bross et al. Reference Bross, Scharnowski and Kähler2021). They extend in the streamwise direction by approximately $(6{-}8) \delta$ . With the decreasing wall temperature, the velocity fluctuation intensities are slightly enhanced, but such large-scale streamwise-elongated low-momentum regions seem to be less manifest, indicating that the very-large-scale motions could be weaker under global scales.

4. One-point statistics of velocity and temperature

In this section, we consider the one-point statistics of velocity and temperature, including their average and fluctuation intensities. This will serve as the validation of the direct numerical simulation results in the present study, as well as the exploration of the wall temperature effects on the scaling in both the inner and outer regions.

Figure 3.

Wall-normal distribution of the integral transformed mean velocity by (a) van Driest transformation (4.1) and (b) total-stress-based transformation (4.2).

In figure 3, we display the wall-normal distribution of the mean velocity under two types of integration transformations. One is the classical van Driest transformation (van Driest Reference van Driest1951)

(4.1) \begin{align} u^+_D=\frac {1}{u_\tau } \int ^{\tilde u}_0 \sqrt {\frac {\bar \rho }{\bar \rho _w}}\, \textrm {d} u, \end{align}

plotted against the viscous coordinate $y^+$ , incorporating only the mean density variation. The other is the total-stress-based transformation proposed by Griffin et al. (Reference Griffin, Fu and Moin2021)

(4.2) \begin{align} u^+_{G} = \frac {1}{u_\tau } \int ^{y^*}_0 \frac {S^+_{\textit{eq}}}{1 + S^+_{\textit{eq}} - S^+_{\textit{TL}}}\, \textrm {d} y, \end{align}

plotted against the semi-local viscous coordinate $y^*$ , where the equivalent strain rates $S^+_{\textit{eq}}$ and $S^+_{\textit{TL}}$ are defined as

(4.3) \begin{align} S^+_{\textit{eq}} = \frac {\bar \mu _w}{\bar \mu } \frac {\partial \bar u^+_1}{\partial y^*},\quad S^+_{\textit{TL}} = \frac {\bar \mu }{\bar \mu _w} \frac {\partial \bar u^+_1}{\partial y^+}. \end{align}

As reported in figure 3(a), the van Driest transformed mean velocity in the viscous sublayer conforms with the linear law reasonably well for case C075, in which the wall heat transfer is not significant, while it deviates from it with decreasing wall temperature. In the log region in the vicinity of $y^+ \approx 100$ , the logarithmic increasing trend can also be observed, with the same slope as the classical log law $u^+_{D} = 2.44 \ln (y^+)+ 5.2$ , but higher intercepts with the lower wall temperature. These phenomena have also been reported in previous related studies (Duan et al. Reference Duan, Beekman and Martin2010; Liang & Li Reference Liang and Li2015; Zhang et al. Reference Zhang, Wan, Liu, Sun and Lu2018a ). We also show the results of Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) at $ \textit{Re}_\tau \approx 2000$ , $M_0=6.0$ and $T_w = 0.75 T_r$ for comparison. The mean velocity given in this paper with the same free-stream Mach number $M_0$ and wall temperature ratio $T_w/T_r$ agrees well with the reference data, proving the accuracy of the results herein. Under the total-stress-based transformation, the mean velocity distributions in these cases with different wall temperatures collapse with each other, obeying both the linear law in the viscous sublayer and the log law in the log region, supporting the good performance of the velocity transformation proposed by Griffin et al. (Reference Griffin, Fu and Moin2021).

Figure 4.

Wall-normal distribution of Reynolds normal stresses (a,b) $R^+_{11}$ , (c,d) $R^+_{22}$ and (e,f) $R^+_{33}$ plotted against (a,c,e) the semi-local viscous coordinate and (b,d,f) the global coordinate.

Figure 4 shows the Reynolds stress normalised by wall shear stress, defined as

(4.4) \begin{align} R^+_{\textit{ij}} = \frac {\overline {\rho u''_i u''_j}}{\tau _w}, \end{align}

along with the results reported by Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) at $M_0 = 5.86$ , $ \textit{Re}_\tau \approx 2000$ , $T_w/T_r = 0.76$ , those by Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) at $M_0 = 2$ , $ \textit{Re}_\tau \approx 1000$ , $T_w/T_r = 1.0$ , and those by Szajnecki et al. (Reference Szajnecki, Roy, Duan and Bisek2025) at $M_0 = 4.9$ , $ \textit{Re}_\tau \approx 1116$ , $T_w/T_r=0.6$ , for comparison. For the streamwise-normal component $R^+_{11}$ , the peak locations are the same ( $y^* \approx 15$ ) under semi-local scaling (figure 4 a), but the peak values are higher with the increasing friction Reynolds number $ \textit{Re}_\tau$ and the lower wall temperature. In the outer region (figure 4 b), the variation of $R^+_{11}$ follows approximately the log law. Such a logarithimic variation is proposed based on the attached-eddy hypothesis, as given by Perry & Li (Reference Perry and Li1990),

(4.5) \begin{align} R^+_{11} = B_1 - A_1 \log \left ( \frac {y}{\delta } \right ),\quad A_1=1.03,\ B_1 =2.39, \end{align}

for the streamwise-normal component. The vertical extension obeying this law becomes shorter with the lower wall temperature. Compared with the reference data, the results obtained herein are smaller in magnitude due to the lower wall temperature ratio and higher Mach numbers (compared with the results of Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011), and lower friction Reynolds number (compared with values of Cogo et al. Reference Cogo, Salvadore, Picano and Bernardini2022). This suggests that the attached eddies as well as the very-large-scale motions, if there are any at such moderate Reynolds numbers, are weaker at lower wall temperatures, and that their separations from the near-wall motions are shorter in the wall-normal direction. This will be further elaborated in the subsequent discussions.

The wall-normal component $R^+_{22}$ collapses below $y^* \approx 100$ under semi-local viscous coordinate (figure 4 c) and above $y=0.4 \delta$ under global coordinate (figure 4 d). It is also interesting to notice that $R^+_{22}$ increases slightly within the viscous sublayer ( $y^* \lesssim 5$ ) due to the more prominent dilatational motions near the wall (Yu, Xu & Pirozzoli Reference Yu, Xu and Pirozzoli2019; Yu & Xu Reference Yu and Xu2021). The spanwise component $R^+_{33}$ collapses with the reference data below $y^* \approx 50$ for cases C075 and C050, while the magnitudes are lower with the lower wall temperature, similar to the variation of the streamwise-normal component. In the outer region, $R^+_{33}$ manifests a decaying trend of variation within $y \approx (0.1{-}0.4) \delta$ following the log law, which is also given by Perry & Li (Reference Perry and Li1990),

(4.6) \begin{align} R^+_{33} = B_3 - A_3 \log \left ( \frac {y}{\delta } \right ),\quad A_3=0.475,\ B_3 =1.2, \end{align}

and similar to $R^+_{11}$ , the vertical extent of the region obeying the log law gets shorter with the lower temperature.

Figure 5.

Wall-normal distributions of mean temperature distribution (a) against $y/\delta$ and (b) against $\tilde u /U_0$ , compared with the GRA (symbols) (4.7).

Figure 6.

Wall-normal distributions of (a) temperature fluctuation intensity $T'^*$ , (b) ratio $H_s$ between the left- and right-hand sides of (4.8), (c) wall-normal turbulent heat flux $R^+_{vT}$ , (d) turbulent Prandtl number $ \textit{Pr}_T$ .

The wall-normal distribution of the mean temperature is displayed in figure 5(a). In case C075, the wall heat flux is not yet significant, explaining the small deviations of its mean velocity and Reynolds stresses from those of incompressible flows and compressible turbulence over adiabatic or quasi-adiabatic walls. As the wall temperature gets lower, the trend of non-monotonic variation becomes more evident, with the highest mean temperature gradually moving towards higher off-wall locations. We can also compare with the generalised Reynolds analogy (GRA) proposed by Zhang et al. (Reference Zhang, Bi, Hussain and She2014),

(4.7) \begin{align} \frac {\bar T}{T_0} = \frac {T_w}{T_0} + \frac {T_{rg}-T_w}{T_0} \frac {\tilde u}{U_0} + \frac {T_0 - T_{rg}}{T_0} \left ( \frac {\tilde u}{U_0} \right )^2, \end{align}

as shown in figure 5(b), with $T_{rg}$ being the generalised recovery temperature. The mean temperature and velocity are well associated by the GRA, further proving the validity of the mean temperature obtained herein.

Figure 6(a) reports the wall-normal distribution of the root mean square of temperature fluctuation, normalised by $\gamma T_w M^2_\tau$ , with friction Mach number $M_\tau = u_\tau /\sqrt {\gamma R T_w}$ and specific heat ratio $\gamma =1.4$ . Unlike the velocity fluctuations, the temperature fluctuation intensity depends highly on the wall temperature, showing no universal scaling laws either in the near wall or in the outer regions. However, it can also be correlated with the streamwise velocity fluctuation intensity $u''_{\textit{rms}}$ by the refined strong Reynolds analogy (Huang et al. Reference Huang, Coleman and Bradshaw1995)

(4.8) \begin{align} \frac {T'_{\textit{rms}}/\bar T}{(\gamma -1) M^2_l u''_{\textit{rms}}/\tilde u} = \frac {1}{|Pr_T\,(\partial \bar T_t/\partial \bar T -1)|}, \end{align}

where $M_l = \tilde u/\sqrt {\gamma R \bar T}$ is the local Mach number, and $ \textit{Pr}_T$ is the turbulent Prandtl number, defined as

(4.9) \begin{align} \textit{Pr}_T = \frac {\overline {\rho u''v''}}{\overline {\rho v''T'}} \frac {\partial \bar T/\partial y}{\partial \tilde u/\partial y}. \end{align}

We denote the ratio between the left- and right-hand sides of (4.8) as $H_s$ , and plot their profiles in each case in figure 6(b). Besides the near-wall region below $y=0.1\delta$ , this ratio remains almost unity across the boundary layer, further supporting the refined strong Reynolds analogy under the presently considered flow conditions.

The wall-normal turbulent heat flux $R_{vT} = \overline {\rho v'' T'}$ normalised by viscous scales $\rho _w u_\tau (\gamma T_w M^2_\tau )$ is shown in figure 6(c). The trend of variation of these profiles is similar, except for the more prominent negative value in the near-wall region in cold-wall case C025 due to the non-monotonic variation of the mean temperature. The peaks are reached at approximately the same off-wall location $y^* \approx 200{-}300$ under semi-local viscous coordinates, but the values decrease with the lowering wall temperature. However, we can infer from the distribution of the turbulent Prandtl number $ \textit{Pr}_T$ in figure 6(d), which falls in the range $0.9{-}1.0$ across the boundary layer except for the near-wall region, that the momentum and heat transport by turbulent fluctuations bear some resemblance to each other. This will be shown further in the next section by discussing their integral length scales.

5. Spectra properties and coherent structures

The results given in the previous sections demonstrate that wall temperature leaves significant impacts on turbulent motions, especially the velocity fluctuation intensities and coherent structures. The temperature fluctuations resemble the features of passive scalars, and the corresponding flow statistics obey the refined Reynolds analogy. In this section, we investigate the statistics in spectral space to explore the variation of turbulent fluctuation intensities at all scales, and their implications for turbulent motions, with the focus on velocity fluctuations due to their dominance in turbulent dynamics.

Figure 7.

Pre-multiplied spanwise spectra (a,b) $k_z E^+_{uu}(k_z)$ , (c,d) $k_z E^+_{vv}(k_z)$ (e,f) $k_z E^+_{ww}(k_z)$ under (a,c,e) semi-local viscous coordinates and (b,d,f) global coordinates.

Figure 8.

Pre-multiplied spanwise spectra of $\sqrt {\rho }\, u''$ at (a) $y^*=15$ and (b) $y/\delta =0.2$ , and $\sqrt {\rho } v''$ and $\sqrt {\rho } w''$ at (c,e) $y^*=50$ and (d,f) $y/\delta =0.2$ . The spectra in insets of (b,d,f) are normalised by global scales $\rho _0 U^2_0$ , and amplified by a factor $1000$ .

In figures 7(a,b), we display the pre-multiplied spanwise spectra of the density-weighted streamwise velocity $\sqrt {\rho }\, u''$ , plotted against the semi-local viscous coordinates and global coordinates, respectively. Under semi-local coordinates, the contour lines of the spectra are almost collapsed in the near-wall region at small scales, in particular near the peak at $y^* \approx 15$ and $\lambda ^*_z \approx 100$ that corresponds to the near-wall velocity streaks. A further inspection of the spectra at this height is given in figure 8(a), compared with the results given by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) for incompressible turbulent boundary layers at $ \textit{Re}_\tau = 1300$ . It can be found that the spanwise lengths corresponding to the peaks of these one-dimensional pre-multiplied spectra remain independent of the wall temperature, but the peak values increase as the wall temperature decreases, indicating the stronger intensities of the near-wall low-speed streaks. This is consistent with the findings at lower Reynolds numbers (Yu & Xu Reference Yu and Xu2021). Moreover, at length scales larger than $\lambda ^*_z \approx 500$ , which correspond to the large-scale and very-large-scale motions superposed upon the near-wall turbulence (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2013), the spectra intensities are lower at lower wall temperature. The generation of these large-scale motions near the wall can be attributed to two primary mechanisms: first, the nonlinear interactions among small-scale motions from the perspective of turbulent kinetic energy transport (Mizuno Reference Mizuno2016; Lee & Moser Reference Lee and Moser2019; Yu et al. Reference Yu, Zhao, Tang, Yuan and Xu2022c ), and second, the linear superposition of large-scale and very-large-scale motions originating from the outer region. With the statistics in the present study, we are capable of validating only the second mechanism, as will be demonstrated subsequently. The first mechanism will be left for future work.

In the outer region, there is inclined to emerge a second peak in each of these cases, suggesting the prevalence of the very-large-scale motions at different wall temperatures. However, the spectra intensities in the outer region are weaker in lower wall temperature cases. This can be more clearly observed in the one-dimensional spectra at $y=0.2\delta$ given in figure 8(b). We can identify that the characteristic length scales of these very-large-scale motions are $\lambda _z \approx 0.8\delta$ , and the intensities decrease with the lower wall temperature. This seems to contradict our previous elucidation using theoretical analysis and numerical experiments (Yu et al. Reference Yu, Dong, Guo, Tang, Yuan and Xu2024a ). However, if normalised by the global scales, the trend of variation of the spectra intensities is reversed, in accordance with the conclusions of our previous studies (Yu et al. Reference Yu, Dong, Guo, Tang, Yuan and Xu2024a ). This is also consistent with the observations in instantaneous flow visualisation in figure 2, which shows stronger large-scale velocity fluctuations at lower wall temperatures. The reduced prominence of very-large-scale motions can be attributed to their weaker tendency to merge into streamwise-elongated structures.

In figures 7(c–f), we present the pre-multiplied spanwise spectra of density-weighted velocity components $\sqrt {\rho } v''$ and $\sqrt {\rho } w''$ . These components represent cross-stream motions, such as streamwise vortices and large-scale circulations, which induce significant vertical momentum transport. When plotted against semi-local inner scales, the spectral contour lines nearly collapse at scales smaller than $\lambda ^*_z \approx 100$ , but show a slight decrease at larger scales, particularly for the peak values located in the vicinity of $\lambda ^*_z = 200$ and $y^* = 100$ . This suggests weaker near-wall quasi-streamwise vortices and reduced cross-stream momentum transport. The one-dimensional spectra at $y^* = 50$ , shown in figures 8(c,e), further support this observation, indicating that energetic cross-stream motions near the wall, probably organised in the shape of streamwise vortices, are weaker as the wall temperature decreases (Shadloo, Hadjadj & Hussain Reference Shadloo, Hadjadj and Hussain2015). Under global scaling, however, the spectra collapse at large scales in the outer region, while the smaller-scale motions appear less intense (figures 7(d,f) and 8(d,f)). Notably, when normalised by global scales, the cross-stream motions in the outer region also become stronger as the wall temperature decreases, similar to the behaviour of streamwise velocity fluctuations.

Figure 9.

Wall-normal distribution of filtered streamwise Reynolds normal stresses: (a) small-scale $R^+_{11,S}$ with $\lambda ^*_z \leqslant 450$ , (b) large-scale $R^+_{11,L}$ with $\lambda _z \geqslant 0.5 \delta$ , (c) integration of $R_{11,L}$ above $y=0.1\delta$ , $E_{1,L}$ normalised by $\bar \rho _w U_0 u_\tau$ , (d) $R_{11,L}$ normalised by $E_{1,L}$ . The dashed red line in (c) is the curve fitting by a power function $ \textit{Re}^{0.38}$ .

To further quantify the influences of the wall temperature on near-wall and outer motions, and their contributions to the turbulent kinetic energy, we integrate the spanwise spectra $E_{11}$ with scales smaller than $\lambda ^*_z = 450$ and those with scales greater than $\lambda _z = 0.5\delta$ (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011; Yu & Xu Reference Yu and Xu2022) to obtain the small-scale and large-scale Reynolds normal stresses $R^+_{11,S}$ and $R^+_{11,L}$ , respectively. The results are shown in figures 9(a,b). The small-scale turbulent intensities $R^+_{11,S}$ are weakly dependent on the wall temperature across the boundary layer under semi-local viscous scales, except for the slightly increasing peak values at lower wall temperatures. This is consistent with the findings in low-Reynolds-number flows (Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023). The large-scale turbulent intensities $R^+_{11,L}$ decrease with the decreasing wall temperature, as observed in the corresponding spanwise spectra. The energy of the large-scale motions $E_{1,L}$ can be obtained by integrating $R_{11,L}$ in the wall-normal direction from $y=0.1 \delta$ to the free stream. When normalised by $\bar \rho _w U_0 u_\tau$ , the energy of the very-large-scale motions follows approximately a power law at sufficiently high free-stream Reynolds numbers (figure 9 c). This suggests that the lower intensities of the very-large-scale motions with the decreasing wall temperature should be attributed to the lower free-stream Reynolds number, or equivalently, the higher dynamic viscosity in the outer region. If normalised by $E_{1,L}$ (figure 9 d), then the profiles of $R_{11,L}$ above $0.2 \delta$ almost collapse with each other, indicating the similar morphology of these very-large-scale motions.

Figure 10.

Wall-normal distributions of (a) $L_u/l_m$ , (b) $L_u/l^*_{12}$ , (c) $L_u/l^*_{12,G}$ , (d) $L_T/L_u$ .

Another perspective of characterising the length scales of energetic motions is the integral length scales, defined as

(5.1) \begin{align} L_\varphi (y) = \int ^{+\infty }_{-\infty } C_{\varphi \varphi }(\Delta z,y)\, \textrm {d}\Delta z, \quad \textrm {with}\ C_{\varphi \varphi }(\Delta z,y) = \frac {\overline {\varphi (x, y,z+\Delta z,t)\,\varphi (x,y,z,t)}} {\overline {\varphi ^2(x,y,z,t)}}. \end{align}

The classical theory of Prandtl (Reference Prandtl1925) and attached-eddy models (Townsend Reference Townsend1976) suggest that the characteristic length scale of the eddies, namely the mixing length, which can be defined as

(5.2) \begin{align} l_m \sim u^*_\tau \left ( \frac {\partial u_{D}}{\partial y} \right )^{-1} \end{align}

in compressible flows, is proportional to their wall distance, $l_m \propto \kappa y$ , which leads to the derivation of the log law of the mean velocity profile with the constant-stress layer hypothesis. Pirozzoli (Reference Pirozzoli2012) proposed an improved scaling of the characteristic length scales of the eddies

(5.3) \begin{align} l^*_{12} \sim (u^*_\tau \delta ) \left ( \frac {\partial u_{D}}{\partial y} \right )^{-1/2}, \end{align}

which shows good accuracy in characterising the size of eddies in both compressible and incompressible channels and turbulent boundary layers. In cases of strong wall heat transfer, Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) proposed to substitute the van Driest transformed mean velocity with that proposed by Griffin et al. (Reference Griffin, Fu and Moin2021),

(5.4) \begin{align} l^*_{12,G} \sim (u^*_\tau \delta ) \left ( \frac {\partial u_{G}}{\partial y} \right )^{-1/2}, \end{align}

in the hope of obtaining better consistency, considering its good performance in collapsing the mean velocity profiles with the classical log law.

The ratios between the integral length scales of velocity fluctuations $L_u$ and the three eddy length scales $l_m$ , $l^*_{12}$ and $l^*_{12,G}$ given above are reported in figures 10(a–c). The profiles of these ratios approximately collapse with each other. Amongst these results, we can find that the ratios $L_u/l^*_{12}$ and $L_u/l^*_{12,G}$ are close to unity in the outer region, with the former showing less wall temperature dependency while the latter manifesting less obvious variation with the wall-normal coordinate. This indicates that both of these constructed length scales, whether or not the viscosity is taken into account, yield similar estimations of the energy-containing eddy sizes.

In figure 10(d), we further show the ratio between the integral length scales of temperature and velocity fluctuations $L_T/L_u$ . These profiles are independent of the wall temperature in the outer region, being almost a constant value of $1/Pr_T$ if $ \textit{Pr}_T$ is taken to be $0.9$ , as is usually done in turbulent modelling. This is intriguing, for the strong Reynolds analogy suggests that based on the approximation $T'_{\textit{rms}} \sim l_T (\partial \bar T/\partial y)$ and $u''_{\textit{rms}} \sim l_u (\partial \tilde u/\partial y)$ , with $l_u$ and $l_T$ the characteristic scales of energy-containing eddies, (4.8) can be formulated as

(5.5) \begin{align} T'_{\textit{rms}} = \frac {1}{Pr_T} \frac {l_T}{l_u} \frac {\partial \bar T}{\partial \tilde u} u''_{\textit{rms}}, \end{align}

with $l_T = l_u$ . The results presented herein suggest that a more plausible explanation for the factor $1/Pr_T$ should be attributed to the slightly different eddy sizes of temperature and velocity fluctuations.

The discussions above only concern energy-containing motions. It is expected that the energy spectra display some features of high-Reynolds-number scaling laws in the outer region where the mean shear rate is comparatively weak. For a further inspection of such aspects, in figure 11 we plot the spanwise spectra under Kolmogorov scales. We also provide the results of incompressible turbulent boundary layers reported by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2014) for comparison. The spectra distributions at the wavenumbers greater than $k_z \eta \gt 0.05$ are consistent with those of incompressible flows in the outer region, independent of the off-wall location and wall temperature. For the spectra of both the density-weighted streamwise $\sqrt {\rho }\, u''$ and cross-stream velocities $\sqrt {\rho } v''$ and $\sqrt {\rho } w''$ , the $-5/3$ power law can be observed. This is also supported by the evident plateau of the pre-multiplied (by $(k_z \eta )^{5/3}$ ) spectra within the range approximately $k_z \eta = 0.01{-}0.1$ , as given in the insets of figure 11. For cases with lower wall temperatures, the range of wavenumbers obeying the $-5/3$ power law is shorter, which can be attributed to the weaker mean density and viscosity in the outer region. In the dissipative range, all these spectra are well-collapsed at the wavenumbers higher than $k_z \eta \approx 0.05$ , indicating the universality of the spectra away from the energy-containing scales (Lyu & Xu Reference Lyu and Xu2024). In general, we can conclude that the turbulent motions away from the wall at the scales within the inertia subrange and the dissipative range, along with the energy cascade processes that they entail, are no longer subject to the influences of the wall temperature, which has relatively profound impacts on the energy-containing motions, namely the large-scale and very-large-scale motions.

Figure 11.

Spanwise spectra (a) $E_{uu}$ , (b) $E_{vv}$ , (c) $E_{ww}$ at $y=0.2\delta$ (solid lines), $y=0.4\delta$ (dashed lines), $y=0.6\delta$ (dash-dotted lines), normalised by the Kolmogorov scale. Insets: the corresponding spectra pre-multiplied by $(k_z \eta )^{5/3}$ . Black lines indicate case C075, red lines indicate case C050, green lines indicate case C035, blue lines indicate case C020, and grey lines indicate $(k_z \eta )^{-5/3}$ . Black symbols indicate data from Sillero et al. (Reference Sillero, Jiménez and Moser2013).

6. Superposition and modulation effects

We have observed from the pre-multiplied spectra in the near-wall region in figure 8 that the imprints of the very-large-scale motions vary from case to case, depending on the wall temperature at the same free-stream Mach number $M_0$ and friction Reynolds number $ \textit{Re}_\tau$ .

In this section, we focus on the superposition and amplitude modulation effects of the very-large-scale motions in the outer region on near-wall turbulence.

Figure 12.

Spanwise linear coherence spectra $\varGamma _{\varphi \varphi }$ for (a) $\varphi = u''$ , (b) $\varphi = \sqrt {\rho }\, u''$ , (c) $\varphi =\rho u''$ , (d) $\varphi =T'$ . Black solid lines indicate case C075, red dashed lines indicate case C050, green dash-dotted lines indicate case C035, and blue solid lines indicate case C020.

Figure 12 shows the correlation between spanwise spectra coefficients $\hat \varphi (y,k_z)$ at $y=0.2\delta$ and those at the wall-normal locations below it, also known as the linear coherence spectra (Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015, Reference Baars, Hutchins and Marusic2016), defined as

(6.1) \begin{align} \varGamma _{\varphi \varphi } = \frac {{\rm Re} \left (\overline {\hat \varphi (y,k_z)\, \hat \varphi ^\star (y=0.2\delta ,k_z)} \right )} {\sqrt {\overline {|\hat \varphi (y,k_z)|^2}}\, \sqrt {\overline {|\hat \varphi (y=0.2\delta ,k_z)|^2}}}, \end{align}

for the purpose of evaluating superposition effects of the outer-scale motions on the near-wall region at different scales. Here, Re gives the real part of the complex function, and the superscript $\star$ represents the complex conjugate of the spectra coefficient. For the streamwise velocity, we consider three different flow quantities, depending on how fluid density $\rho$ is incorporated. The first quantity is $u''$ (figure 12 a), a direct inheritance from incompressible wall-bounded turbulence. The second flow quantity is $\sqrt {\rho }\, u''$ (figure 12 b), the square of which is representative of the turbulent kinetic energy. The third flow quantity is $\rho u''$ (figure 12 c), indicating the correlation in the momentum sense. Amongst these results, we can find that the linear coherence is strong merely at large scales, and decays to insignificant levels at scales smaller than $\lambda _z^* \approx 200$ . When weighted by density, the linear coherence spectra $\varGamma _{\varphi \varphi }$ are augmented at large scales but abated at small scales. If inspected at the scale $\lambda ^*_z \approx 1000$ , the spectra intensities are decreased for cases C075 and C050 with slightly higher wall temperatures, but increased for cases C035 and C020 with cold walls. Better collapse amongst these profiles in different cases is observed when using $\rho u''$ . This suggests that superposition effects should be interpreted in the sense of the momentum of the fluid, rather than that of kinetics or energy.

For temperature fluctuations $T'$ (figure 12 d), the wavenumber extent where the linear coherence spectra are high, namely the large-scale motions with significant superposition effects, is similar to that of the streamwise velocity. Due to the non-monotonic variation of mean temperature profile, the superposition effects in cases C035 and C020 are much lower. It is also intriguing to notice that the value of $\varGamma _{TT}$ becomes negative in case C020 at length scales $\lambda ^*_z \approx 100{-}500$ close to the wall, but remains positive at larger scales at various wall-normal locations. Such a phenomenon indicates the non-synergistic superposition effects for temperature fluctuations at different scales. This also explains the difficulties of predicting near-wall temperature fluctuations when such an effect is not considered (Yu & Xu Reference Yu and Xu2022; Yu et al. Reference Yu, Fu, Tang, Yuan and Xu2023), in particular the position in the adjacent to the mean temperature peak values.

Figure 13.

Magnitudes of the linear kernel functions of superposition coefficients $|H_{\varphi \varphi }(y,k_z)|$ for (a) $\varphi = u''$ , (b) $\varphi = \sqrt {\rho }\, u''$ , (c) $\varphi =\rho u''$ , (d) $\varphi =T'$ . Black solid lines indicate case C075, red dashed lines indicate case C050, green dash-dotted lines indicate case C035, and blue solid lines indicate case C020.

In predicting the near-wall turbulent motions based on inner–outer turbulent interactions, the superposition of the large-scale and very-large-scale motions on the near-wall turbulent motions can be achieved in the spectral space, allowing the incorporation of the variation of superposition effects with length scales (Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2016; Yin, Huang & Xu Reference Yin, Huang and Xu2018). It can be cast as for scales greater than (or wavenumbers smaller than) a certain threshold:

(6.2) \begin{align} \hat \varphi (y,k_z) = H_{\varphi \varphi }(y,k_z)\, \hat \varphi (y=0.2\delta ,k_z), \end{align}

where $H_{\varphi \varphi }$ is the linear kernel function, a complex function in the spectral space. It is desirable that the universality of the linear kernel function against the variation of the wall temperature can be achieved, for such a circumstance allows the development of universal prediction models. Using the direct numerical simulation datasets herein, the linear kernel function can be calculated as

(6.3) \begin{align} H_{\varphi \varphi }(y,k_z) = \frac {\overline {\hat \varphi (y,k_z)\, \hat \varphi ^\star (y=0.2\delta ,k_z)}} {\overline {\hat \varphi (y=0.2\delta ,k_z)\, \hat \varphi ^\star (y=0.2\delta ,k_z)}}. \end{align}

The results of the moduli of $H_{\varphi \varphi }(y,k_z)$ are reported in figure 13, in which the flow quantities $\varphi$ are identical to those given in figure 12. For velocity fluctuations $u''$ , the universality of the linear kernel function is better achieved when weighted by density (figure 13 c), which is reflected by the better collapsed profiles amongst these cases, except for case C020 with the lowest wall temperature. The temperature fluctuations $T'$ , unfortunately, do not manifest wall temperature independence. Based on the spectral linear coherence and the moduli of linear function, we can conclude that the large-scale temperature fluctuations in cold-wall cases are not capable of penetrating to the near-wall region, and leave significant imprints as in velocity fluctuations.

Figure 14.

Amplitude modulation correlation coefficient $A_{uu}(y_1,y_2)$ in cases (a) C075, (b) C050, (c) C035, (d) C020. Flooded contours indicate red for positive, and blue for negative, with interval $0.1$ . Grey dashed lines indicate $y^*_2 = y^*_1$ .

Another aspect of the influences of the large-scale and very-large-scale motions on the near-wall turbulence is the amplitude modulation effect. It depicts the effect that the small-scale motions are strengthened/weakened beneath positive/negative large-scale streamwise velocity regions (Mathis et al. Reference Mathis, Hutchins and Marusic2009; Marusic et al. Reference Marusic, Mathis and Hutchins2010), so it can be evaluated by the correlation coefficient

(6.4) \begin{align} A_{u\varphi } (y_1, y_2) = \frac {\overline {u''_L (y_1)\, E_{L,\varphi _s}(y_2)}} {\sqrt {\overline {u''^2_L(y_1)}}\sqrt {\overline {E^2_{L,\varphi _s}(y_2)}}}, \end{align}

with the subscripts $L$ and $s$ denoting the large-scale and small-scale portions of the flow quantity, respectively, and $E$ being the envelope of the fluctuations. Note that the large-scale velocity $u''_L$ is not weighted by density, for the modulation effects, which can be interpreted by the quasi-steady quasi-homogeneous hypothesis (Zhang & Chernyshenko Reference Zhang, Duan and Choudhari2016), can be regarded as the interaction between the large- and small-scale motions in the kinetic sense. Herein, we separate the large- and small-scale portions by spectra filtering, and the separation scale corresponding to the cut-off wavenumber is determined to be $0.5\delta$ . The envelope of the small-scale fluctuations is obtained by the Hilbert transformation. The procedures for calculating these variables can be found in previous literature (Agostini & Leschziner Reference Agostini and Leschziner2014, Reference Agostini and Leschziner2016) and will not be elaborated upon herein for brevity.

In figure 14, we present the contour map of $A_{uu}$ , the amplitude modulation correlation coefficients for $u''$ . As can be inferred from the positive-valued peak off the diagonal line, the most significant amplitude modulation occurs for the large-scale motions at $y^*\approx 100$ , on the small-scale motions at $y^* \approx 15$ and below in cases C075, C050 and C035. The modulation effects are weakened with the lower wall temperature, and almost disappear in case C020. This is consistent with the observation in the instantaneous distribution of $u''$ (figure 2) that the local intensification of the near-wall small-scale motions is less evident with the decreasing wall temperature.

Figure 15.

Amplitude modulation correlation coefficient $A_{uT}(y_1,y_2)$ in cases (a) C075, (b) C050, (c) C035, (d) C020. Flooded contours indicate red for positive, and blue for negative, with interval $0.1$ . Grey dashed lines indicate $y^*_2 = y^*_1$ .

The maps of $A_{uT}(y_1,y_2)$ are displayed in figure 15. Compared with $u''$ , the modulation effect on $T'$ is more prominent. Moreover, due to the non-monotonic variation of the mean temperature near the wall, there manifest two peaks off the diagonal, one at $y^* \approx 20$ , and the other in the viscous sublayer very close to the wall, modulated by the large-scale motions at $y^* \approx 100$ . Similar to velocity fluctuations, the modulation effects also become less significant as the wall temperature gets lower. However, there remains a small positive region below $y^*=3.0$ in case C020, indicating that the wall heat flux fluctuations are intensified below the large-scale and very-large-scale high-momentum regions. This is an important observation, for the more intense fluctuating wall heat flux due to the modulation effects should be carefully treated in the design of the thermal protection system of high-speed vehicles, which will be considered in our future work.