2.1. Database of compressible turbulent channel flows

The database in this study contains five high-fidelity DNS cases of compressible turbulent channel flows driven by a volume force to maintain the constant mass flow rate. The physical model is displayed in figure 1. It should be mentioned that the averaged temperature is monotonically increasing from the wall to the channel centre, which is different from the boundary layer.

Figure 1. Diagram of the compressible turbulent channel flow model.

The governing equations are shown as

(2.1) \begin{align}&\qquad\qquad\quad \frac {{\partial \rho }}{{\partial t}} + \frac {{\partial \rho {u_j}}}{{\partial {x_j}}} = 0, \end{align}

(2.2) \begin{align}& \frac {{\partial \rho {u_i}}}{{\partial t}} + \frac {{\partial \rho {u_i}{u_j}}}{{\partial {x_j}}} = - \frac {{\partial p}}{{\partial {x_i}}} + \frac {{\partial {\tau _{\textit{ij}}}}}{{\partial {x_j}}} + {f_1}{\delta _{i1}}, \end{align}

(2.3) \begin{align}& \frac {{\partial \rho E}}{{\partial t}} + \frac {{\partial \rho H{u_j}}}{{\partial {x_j}}} = \frac {{\partial {\tau _{\textit{ij}}}{u_i}}}{{\partial {x_j}}} - \frac {{\partial {q_j}}}{{\partial {x_j}}} + {f_1}{u_1}, \end{align}

where $x_i(i = 1, 2, 3) = (x, y, z), u_i(i = 1, 2, 3) = (u, v, w)$ and

(2.4a,b) \begin{align} {\tau _{\textit{ij}}} = \mu \left ( {\frac {{\partial {u_i}}}{{\partial {x_j}}} + \frac {{\partial {u_j}}}{{\partial {x_i}}} - \frac {2}{3}\frac {{\partial {u_k}}}{{\partial {x_k}}}{\delta _{\textit{ij}}}} \right ),\quad {q_j} = - \kappa \frac {{\partial T}}{{\partial {x_j}}}. \end{align}

Here, $t$ denotes time and $u$ , $v$ and $w$ are the velocity components in the streamwise ( $x$ ), wall-normal ( $y$ ) and spanwise ( $z$ ) directions, respectively. Here $\rho ,T,p$ represent density, temperature and pressure which satisfy the ideal gas law $p=\rho RT$ , where $R$ is the gas constant. Here $E = (1/2){u_i}{u_i} + {C_v}T$ is the total energy and the total enthalpy is defined as $H = E + p/\rho$ . Here $\tau _{\textit{ij}}$ denotes the viscous stress, where $\mu$ is the dynamic viscosity obeying Sutherland’s law, and $q_j$ is the heat transfer term. Here $f_1$ is the body force. Additionally, the heat conductivity coefficient $\kappa = (C_p\mu ) / Pr$ . Here $C_v$ and $C_p$ stand for the constant-volume specific heat capacity and the constant-pressure specific heat capacity. The ratio of specific heat $\gamma = C_p/C_v = 1.4$ and the Prandtl number $Pr = 0.72$ .

All these simulations are conducted in a computational domain of $4\pi h \times 2h \times 2\pi h$ , where $h$ is the half-height of the channel. The convection terms are discretized by a seventh-order hybrid scheme between weighted essentially non-oscillatory and a linear upwind-biased scheme, and an eighth-order central scheme is adopted for the discretization of viscous terms. Time advancement is carried out through the third-order Runge–Kutta scheme. Periodic boundary conditions are imposed in the streamwise and spanwise directions. The no-slip and isothermal conditions are adopted for upper and lower walls. The fidelity of these data has been validated in our previous work (Cheng & Fu Reference Cheng and Fu2022). Other parameters have been listed in table 1.

Table 1. Parameter settings of the DNS database. Here, $Ma_b$ denotes the bulk Mach number, and ${\textit{Re}}_b, {\textit{Re}}_{\tau }, {\textit{Re}}_{\tau }^*$ denote the bulk Reynolds number, friction Reynolds number and semilocal friction Reynolds number, respectively. Additionally, $\Delta x^+$ and $\Delta z^+$ denote the streamwise and spanwise grid resolutions in viscous units, respectively, $\Delta y^+_{min}$ and $\Delta y^+_{max}$ denote the finest and coarsest resolution in the wall-normal direction.

In this paper, $\bar \varphi$ denotes the Reynolds average of a generic variable $ \varphi$ and the corresponding fluctuation is $\varphi '$ , while the Favre average is $\tilde \varphi = \overline {\rho \varphi }/\bar {\rho }$ and the corresponding fluctuation is $\varphi ''$ . The friction velocity ${u_\tau } = \sqrt {{\tau _w}/{{\bar \rho }_w}}$ (where $\tau _w$ is the wall shear stress) and the viscous length scale ${\delta _{\nu }} = {\bar \mu _w}/ ( {{{\bar \rho }_w}{u_\tau }} )$ . The variables normalized by wall units are denoted by a ‘ $+$ ’ superscript and the subscript ‘ $b$ ’ and ‘ $w$ ’ represent the bulk and wall values, respectively. For compressible flow, semilocal wall units are suitable to investigate Reynolds number effects (Griffin, Fu & Moin Reference Griffin, Fu and Moin2021; Huang, Duan & Choudhari Reference Huang, Duan and Choudhari2022) that are denoted by superscript ‘ $*$ ’, i.e. ${u_\tau ^* } = \sqrt {{\tau _w}/{{\bar \rho }}}$ and ${\delta _{\nu }^* } = {\bar \mu }/ ( {{{\bar \rho }}{u_\tau ^* }} )$ . Thus, the semilocal friction Reynolds number ${\textit{Re}}_{\tau }^*= h/{\delta _{\nu }^* }(h)={\textit{Re}}_{\tau }\sqrt {\bar {\rho }_c/\bar {\rho }_w}(\bar {\mu }_c/\bar {\mu }_w)$ , where subscript ‘ $c$ ’ denotes the value of the variable at the channel centre.

2.2. Identification of coherent structures

The typical coherent structures, which include intense fluctuation structures for both $u'$ and $T'$ , the Qs structures of tangential Reynolds stress $\widetilde {\ u^{\prime\prime}v^{\prime\prime}}$ and the corresponding heat flux $\widetilde {T^{\prime\prime}v^{\prime\prime}}$ , are considered in this study as they are highly related to each other. It should be mentioned that for Qs structures, Favre fluctuations (defined based on the Favre average) are adopted to maintain consistency with the formulation in the transport equations. This choice has nearly no impact on the results, particularly when the Mach number is not high, as Cogo et al. (Reference Cogo, Salvadore, Picano and Bernardini2022) reported. Fluctuation magnitudes are chosen as the criteria to identify these two structures. Moreover, the vortex structures for velocity field are also identified since they play an important role in the self-sustaining mechanism of wall turbulence (Toh & Itano Reference Toh and Itano2005; Del Á lamo et al. Reference del ÁLAMO, JIMÉNEZ, ZANDONADE and MOSER2006). To identify vortex structures, the $\varDelta$ criterion is adopted since it does not depend on the divergence-free condition. Here $\varDelta$ is the discriminant of the velocity gradient tensor $\boldsymbol{\nabla }\boldsymbol{u}$ that can be calculated through three Galilean invariants of $\boldsymbol{\nabla }\boldsymbol{u}$ , written as

(2.5a–c) \begin{align} P = - {\textrm {tr}}\left ( {\boldsymbol{\nabla }\boldsymbol{u}} \right ),\;\;\;\;Q = - \frac {1}{2}\left [ {{\textrm {tr}}\big( {\boldsymbol{\nabla }{\boldsymbol{u}^2}} \big) - {\textrm {tr}}{{\left ( {\boldsymbol{\nabla }\boldsymbol{u}} \right )}^2}} \right ],\;\;\;\;R = - \det \left ( {\boldsymbol{\nabla }\boldsymbol{u}} \right )\!. \end{align}

Then, $\varDelta$ can be obtained through the following formula:

(2.6) \begin{equation} \varDelta = {\left ( {\frac {{Q - {P^2}/3}}{3}} \right )^3} + {\left ( {\frac {{R + 2{P^3}/27 - PQ/3}}{2}} \right )^2}. \end{equation}

Chong, Perry & Cantwell (Reference Chong, Perry and Cantwell1990) pointed out that in a vortex core, $\boldsymbol{\nabla }\boldsymbol{u}$ is dominated by its rotational part that requires that $\varDelta$ is positive.

The thresholds for these criteria are ambiguous and if wall turbulence is considered, its inhomogeneity in the wall-normal direction makes the problem more challenging. However, this study does not dig into it and different thresholds have been used at different $(x,z)$ -planes according to their distance to the wall (Del Á lamo et al. Reference del ÁLAMO, JIMÉNEZ, ZANDONADE and MOSER2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Hwang & Sung Reference Hwang and Sung2018a ). Criteria for identifying all these structures are summarized in table 2. Percolation analysis is used to determine the thresholds $\alpha$ and the details can be found in the Appendix. It is found that all $\alpha$ can be set to 1.75 for all coherent structures in all cases except the vortex structures, which is the same value adopted by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) in incompressible channel flow for tangential Reynolds stress. It was mentioned by both Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Hwang & Sung (Reference Hwang and Sung2018a ) that the results are qualitatively similar when $\alpha$ does not deviate much from 1.5. Meanwhile, the $\alpha _{\Delta }$ for vortex structures is set as 0.1 according to the percolation analysis.

Table 2. Criteria for identifying different structures and $\alpha$ in each criterion is decided by percolation analysis. Here $\boldsymbol{x} = (x,z)$ and the subscript ‘rms’ denotes the root-mean-square of the variable.

Since the domain is discretized by the Cartesian mesh, the mesh points, which meet the structure criterion, form a point set and the connected points are grouped into a subset which represents a single structure. However, structures obtained through this way will contain some meaningless fragments that only consist of several points due to numerical issues. Thus, in this study, only structures with volume larger than the cube of 30 wall units will be counted and analysed, as del Á lamo et al. (Reference del Álamo and Jiménez2006) suggested. The famous Townsend’s attached-eddy model (Townsend Reference Townsend1976) provides an important perspective to inspect the features of coherent structures. Thus, the structures with their lowest parts below the first mesh node away from the wall are grouped into wall-attached structures. Otherwise, they will be regarded as wall-detached structures.

3.1. Intense fluctuation structures

The intense fluctuation structures are identified by these two criteria:

(3.1a,b) \begin{align} \left | {u'(\boldsymbol{x})} \right | \gt 1.75{u^{\prime}_{\textit{rms}}(y)},\;\;\;\;\left | {T'(\boldsymbol{x})} \right | \gt 1.75{T^{\prime}_{\textit{rms}}(y)}. \end{align}

The numbers of structures used to accumulate statistics are more than $3\times 10^4$ for low Reynolds number ( ${\textit{Re}}_b\lt 10^4$ ) cases and $10^5$ for high Reynolds number ( ${\textit{Re}}_b\geqslant 10^4$ ) cases. According to the sign of the integral of fluctuations in one specific structure, structures can be divided into two categories, i.e. positive (+) and negative (–) dominant structures, which are not the same as the pure positive/negative structures in Hwang & Sung (Reference Hwang and Sung2018b ) and Yuan et al. (Reference Yuan, Tong, Li, Chen and Dong2022). However, the differences between them are slight. The comparison between these two methods has been conducted and the results (not shown in this paper) have only slight difference. Here, Case Ma30Re15K will be focused on first. Table 3 displays that the attached structures account for almost 70 % of the total structure volume statistically, though their number is much smaller than the detached ones. Similar results are also found by Hwang & Sung (Reference Hwang and Sung2018a ). The fractions of the positive and negative structures are also included in table 3. For attached structures, the same phenomenon that fewer negative structures account for larger volume can also be noticed. It should be mentioned that the intense $T'^-$ structures get almost all the volume of attached $T'$ structures, which means that the tall and large attached structures almost all belong to intense $T'^-$ structures, while most of the attached intense $T'^+$ structures are near wall with small volume. This phenomenon differs from that observed in compressible turbulent boundary layers with heated and slightly cooled walls (Yuan et al. Reference Yuan, Tong, Li, Chen and Dong2022). In turbulent channel flow, the sole mechanism for transferring heat generated by viscous dissipation to the exterior is through the wall. Consequently, the temperature gradient near the wall is extremely steep. Given that the temperature increases monotonically from the wall towards the channel centre, the thermal condition can be analogized to that of an extremely cooled wall in a boundary layer. This unique characteristics of intense $T'$ structures will be investigated in § 4 carefully. Moreover, intense $T'$ structures tend to be attached to the wall more than the intense $u'$ structures.

Table 3. The number and volume fractions of the intense fluctuation structures in Case Ma30Re15K. Two numbers in the brackets represent the fractions of positive structures and negative structures, respectively.

3.1.1. Attached structures

The self-similarity of the attached structures, which has been verified by vast research in incompressible fields, will also be inspected in compressible channel flows here. Figure 2 depicts the extracted samples of attached structures from the instantaneous field of Case Ma30Re15K. Both the intense $u'$ and $T'$ structures extend from the wall to the central region of the channel, with lengths reaching approximately $3h$ . It is noteworthy that the intense $u'$ structures possess greater height and width in comparison with the $T'$ structures.

Figure 2. Samples of attached structures of (a) intense $u'$ and (b) intense $T'$ .

The structure sizes are further investigated. Firstly, the joint probability density functions (p.d.f.s) of $l_x$ and $l_y$ are estimated as shown in figure 3. It is important to note that due to the relatively small number of large-scale structures in each instantaneous field, it is challenging to accumulate a sufficient number of large-scale structures comparable to the smaller ones. Consequently, the estimation uncertainty for the large-scale portion of the p.d.f.s may be greater than that for the smaller scales. However, the trend towards larger scales remains discernible. Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Hwang & Sung (Reference Hwang and Sung2018a ) provided the fitted equations $l_x^+ \approx 3l_y^+$ and $l_x^+ = 17.98(l_y^+)^{0.74}$ , respectively, for attached structures in the logarithmic region. These two lines are also plotted in figure 3. Although the ${\textit{Re}}_{\tau } = 1243$ in this case is not sufficiently high to encompass a very broad range of scales for LSMs and VLSMs, a linear relationship can still be observed when $l_y^+ \gt 100$ . This observation aligns with the previously established relationship for both intense $u'$ and $T'$ structures. However, for this compressible case, the relation $l_x^+ \approx 7.5l_y^+$ is obtained, which exhibits a steeper slope compared (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012) with for Qs structures in incompressible flow. These results provide evidence for elongation of large-scale structures in compressible flow, which is also observed by Bross et al. (Reference Bross, Scharnowski and Kähler2021). Moreover, it can also be found that in the near-wall region ( $l_y^+\lt 50$ ), $l_x$ primarily exhibits a scale of $O(100)$ wall units, and these relations fail due to the significant influence of near-wall streaks. Subsequently, the width of the structures $l_z$ is also taken into consideration, which is supposed to meet $l_z \approx l_y$ (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Hwang & Sung Reference Hwang and Sung2018a ). The joint p.d.f.s of $l_z$ and $l_y$ with the equality relation are illustrated in figure 4. As the height of the structure increases, the width and height of the attached $u'$ structures become nearly identical. However, it is observed that the attached $T'$ structures are slightly stretched in the spanwise direction, with $l_z^+ \approx 1.1l_y^+$ . Furthermore, the peak value of each p.d.f. is located around the position $(l_z^+,l_y^+)=(50,15)$ , which aligns with the width of the near-wall streaks. It should be mentioned that since the thermal wall condition of the compressible channel flow can be analogized to the extremely cooled wall of the boundary layer, near-wall coherent structures tend to elongate (Coleman et al. Reference Coleman, Kim and Moser1995; Duan et al. Reference Duan, Beekman and MartÍN2010). However, large-scale structures are almost unaffected by compressibility (Ringuette et al. Reference Ringuette, Wu and Martín2008). Consequently, the lower part of the p.d.f.s shifts somewhat to the right compared with that in other compressible turbulent boundary layers with a not so cooled wall.

Figure 3. Joint p.d.f.s of the structure length and height for attached (a) intense $u'$ structures and (b) intense $T'$ structures. The yellow dashed line denotes $l_x^+ = 17.98(l_y^+)^{0.74}$ (Hwang & Sung Reference Hwang and Sung2018a ), the blue dashed line stands for $l_x^+ \approx 3l_y^+$ (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012) and the green one, $l_x^+ \approx 7.5l_y^+$ , is obtained in this paper. The contour levels are logarithmically distributed.

Figure 4. Joint p.d.f.s of the structure width and height for attached (a) intense $u'$ structures and (b) intense $T'$ structures. The yellow dashed line denotes $l_z^+ \approx l_y^+$ , while the blue dashed line denotes $l_z^+ \approx 1.1l_y^+$ . The contour levels are logarithmically distributed.

It can be seen that the attached intense $T'$ structures exhibit the same self-similarity as the attached intense $u'$ structures, based on the structure size. This observation provides a solid foundation for establishing the relation between $u^{\prime}_{\textit{rms}}$ - and $T^{\prime}_{\textit{rms}}$ -like SRA, since the attached structures account for nearly 70 % of the total structure volume.

3.1.2. Detached structures

The detached structures constitute a significant proportion of numbers, exceeding 70 %, yet they account for less than one-third of the total structural volume. The length, width and height of detached structures are almost at the same scale (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012). It turns out that the results for the intense $T'$ structures remain the same, which also has $l_x^+ \approx l_y^+ \approx l_z^+$ in this case, as illustrated in figure 5(b). However, the length of the detached intense $u'$ structures is slightly elongated in the streamwise direction, reaching approximately $1.5l_y^+$ . It can be concluded that the streamwise stretching of intense $u'$ fluctuation structures consistently occurs in compressible channel flows. Moreover, the width of these structures remains close to their height, regardless of whether they are attached or detached. Although the detached intense $T'$ structures exhibit slightly different features compared with the intense $u'$ structures, the relatively small volume of the detached $T'$ structures renders this discrepancy acceptable when establishing an analogy between $u^{\prime}_{\textit{rms}}$ and $T^{\prime}_{\textit{rms}}$ .

Figure 5. Joint p.d.f.s of the structure height and another dimension for detached (a) intense $u'$ structures and (b) intense $T'$ structures. The red line denotes $l_*^+ \approx l_y^+$ and the blue line denotes $l_*^+ \approx 1.5l_y^+$ , where subscript ‘ $*$ ’ can be $x$ or $z$ . The contour levels are logarithmically distributed.

3.2. The Qs structures

Quadrant events, particularly Q $^-$ s (Q2s and Q4s), play a crucial role in the transfer and redistribution of momentum and energy within wall-bounded turbulent flows. Herein, the Qs structures are identified based on the following criteria:

(3.2a,b) \begin{align} \left | {u^{\prime\prime}(\boldsymbol{x})v^{\prime\prime}(\boldsymbol{x})} \right | \gt 1.75{u^{\prime\prime}_{\textit{rms}}(y)v^{\prime\prime}_{\textit{rms}}(y)},\;\;\;\;\left | {T^{\prime\prime}(\boldsymbol{x})v^{\prime\prime}(\boldsymbol{x})} \right | \gt 1.75{T^{\prime\prime}_{\textit{rms}}(y)v^{\prime\prime}_{\textit{rms}}(y)}. \end{align}

For Case Ma15Re3K, over $6\times 10^4$ structures are utilized for statistical accumulation, while more than $10^5$ are used for other cases. Table 4 presents the number and volume fractions of Q1s–Q4s structures for both $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ . The attached structures are counted separately. Unlike the intense fluctuation structures, the total volume used to calculate the volume fractions for Qs structures is based on the entire channel volume. Results show that the volume fractions for Qs structures of $u^{\prime\prime}v^{\prime\prime}$ are almost consistent with the results reported by Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) in incompressible flows with ${\textit{Re}}_{\tau }=950$ . The Q $^-$ s structures surpass the Q $^+$ s structures (Q1s and Q3s) in both number and volume. In this study, we focus exclusively on the Q $^-$ s structures. It is evident that the volumes of the attached Q2s structures are more than three times larger than the attached Q4s structures, indicating that the predominant phenomenon in this case involves ejections, which transfer momentum and energy from the near-wall region to the outer region. Furthermore, the volume fraction of Q4s structures for $T^{\prime\prime}v^{\prime\prime}$ is extremely low, whereas the Q2s structures are significantly larger. This observation aligns with the findings in § 3.1, as less intense $T'^+$ structures directly contribute to fewer Q1s and Q4s structures, which possess positive temperature fluctuation components. This represents a notable distinction between streamwise velocity and temperature, a topic that will be further explored in § 4.

Table 4. The number and volume fractions of the Qs structures in Case Ma30Re15K. Here $N_1$ – $N_4$ denote the number fractions for Q1s–Q4s structures, respectively. Here $V_1$ – $V_4$ denote the volume fractions with respect to the whole channel volume for Q1s–Q4s structures, respectively.

Figure 6. Joint p.d.f.s of the structure length and height for attached Qs structures of (a) $u^{\prime\prime}v^{\prime\prime}$ and (b) $T^{\prime\prime}v^{\prime\prime}$ . The yellow dashed line denotes $l_x^+ = 17.98(l_y^+)^{0.74}$ (Hwang & Sung Reference Hwang and Sung2018a ), the blue dashed line stands for $l_x^+ \approx 3l_y^+$ (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012) and the green one, $l_x^+ \approx 25(l_y^+)^{0.7}$ , is obtained in this paper. The contour levels are logarithmically distributed.

3.2.1. Attached structures

The size distributions of attached Qs structures are first examined, and the joint p.d.f.s of the structure height and its other two dimensions are presented in figures 6 and 7. The Qs structures for $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ exhibit nearly identical sizes statistically, as indicated by the p.d.f.s, except that the original equality between structure width and height transitions into a slightly nonlinear relationship for $T^{\prime\prime}v^{\prime\prime}$ structures which is expressed as $l_z^+ \approx 1.6(l_y^+)^{0.95}$ , as shown in figure 7(b). In comparison with the intense fluctuation structures discussed in § 3.1, Qs structures have a greater number of flat streaks with heights of $O$ (1) wall units, since a turning point exists as $l_y^+$ approaching unity in figure 6. Furthermore, the fitted equation obtained in this study for the length and height of Qs structures when $l_y^+\gt 100$ is given by $l_x^+ \approx 25(l_y^+)^{0.7}$ , which aligns more closely with the findings reported in Hwang & Sung (Reference Hwang and Sung2018a ). They attributed the deviation from linearity to the inclination and meandering of the structures, as well as the relatively low Reynolds number ( ${\textit{Re}}_{\tau } \approx 1000$ ). However, this phenomenon will not be further explored in the present study, as the focus is primarily on the relationship between velocity and temperature coherent structures. The high degree of similarity in size between the attached Qs structures of $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ suggests the existence of similar momentum and energy transport results in compressible turbulent flows.

Figure 7. Joint p.d.f.s of the structure width and height for attached Qs structures of (a) $u^{\prime\prime}v^{\prime\prime}$ and (b) $T^{\prime\prime}v^{\prime\prime}$ . The yellow dashed line denotes $l_z^+ \approx l_y^+$ , while the blue dashed line denotes $l_z^+ \approx 1.6(l_y^+)^{0.95}$ . The contour levels are logarithmically distributed.

3.2.2. Detached structures

The size of the detached Qs structures is also studied by estimating the joint p.d.f.s across different dimensions. The results are illustrated in figure 8. The elongation in streamwise direction persists for detached Qs structures in this compressible flow case, with the dimensional relationship given as $l_x^+ \approx 1.5l_y^+ \approx 1.5l_z^+$ . Unlike the intense fluctuation structures, no significant discrepancy is observed between figures 8(a) and 8(b). To date, it has been found that both the attached and detached Qs structures exhibit negligible difference between $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ . Therefore, it is reasonable to anticipate that the turbulent Prandtl number ${\textit{Pr}}_t$ , which characterizes the relationship between the averaged Reynolds shear stress $\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ and averaged heat flux $\widetilde {T^{\prime\prime}v^{\prime\prime}}$ , will have a value close to unity, with deviations only in the extremely near-wall region.

Figure 8. Joint p.d.f.s of the structure height and another dimension for detached Qs structures of (a) $u^{\prime\prime}v^{\prime\prime}$ and (b) $T^{\prime\prime}v^{\prime\prime}$ . The red line denotes $l_*^+ \approx l_y^+$ and the blue line denotes $l_*^+ \approx 1.5l_y^+$ , where subscript ‘ $*$ ’ can be $x$ or $z$ . The contour levels are logarithmically distributed.

3.3. Reynolds and Mach numbers effects

All five cases are analysed to investigate the Reynolds and Mach number effects on both velocity and temperature structures. The semilocal friction Reynolds number ${\textit{Re}}_{\tau }^*$ is selected as the indicator. The analysis proceeds in an orderly manner, first examining intense fluctuation structures followed by Qs structures. It should be mentioned that the subsequent analysis in this chapter remains relatively preliminary and more cases with higher Mach and Reynolds numbers should be inspected to make the conclusions more solid in the future works.

3.3.1. Intense fluctuation structures

Table 5 lists the volume fractions of different structures with respect to the whole channel. It can be found that, except for the intense $u'^-$ structures whose volume fractions decrease monotonically with increasing Reynolds number in both Mach 3 and Mach 1.5 cases, the Reynolds number effects on other structures are non-monotonic. Furthermore, the influence of the Reynolds number varies depending on the Mach numbers, particularly for intense $T'$ structures. Consequently, the Reynolds number effects cannot be discussed separately without considering the Mach number. Additionally, a higher Mach number generally leads to larger volume fractions for almost all $u'$ structures and a greater proportion of attached components in negative structures. However, the combined influence of Reynolds and Mach numbers on the volume of $T'$ structure is highly complex and requires further investigation. Notably, the volume fractions for $T'^+$ structures remain small regardless of the Reynolds and Mach numbers, which is a remarkable difference in comparison with the $u'$ structures. This observation suggests that the temperature fluctuations exhibit negative skewness.

Table 5. The volume fractions of different intense fluctuation structures with respect to the whole channel in all five cases. Here ${\textit{Re}}_{\tau }^*$ is the semilocal friction Reynolds number. Subscript ‘att’ represents the attached structures.

Structure sizes are evaluated based on aspect ratios. The p.d.f.s of the length-to-height and width-to-height ratio are presented in figures 9 and 10, respectively. Given the distinct characteristics of attached and detached structures, estimations are performed separately for these two types. The elongation of the attached structures in the streamwise direction, as § 3.1 mentioned, can also be observed in figure 9. In contrast, detached structures exhibit isotropic features. The variation of the $l_x/l_y$ for detached structures exhibits no clear trend as the Reynolds and Mach numbers change. The peak values for all lines are located around $l_x/l_y=1$ for $u'$ structures and $l_x/l_y=0.9$ for $T'$ structures. However, it is also observed that the distributions of $l_x/l_y$ for detached $u'$ structures in the two cases with lowest ${\textit{Re}}_{\tau }^*=148$ deviate from the other three cases near the peak, as shown in figure 9(a,b). Therefore, the length-to-height ratio of detached intense fluctuation structures can achieve a fully developed state when the Reynolds number reaches a specific threshold. Regarding attached structures, it is worth noting that most attached structures with $l_x/l_y\gt 20$ have heights exceeding 100 wall units, which are regarded as large structures here. On the contrary, small structures with heights less than 100 wall units dominate the range $l_x/l_y\lt 3$ . The dashed lines in figure 9 exhibit a relatively flat plateau when $l_x/l_y$ falls within the range of $3-20$ . This plateau is flatter for $u'$ structures. Furthermore, focusing on the large structures, higher Mach numbers or smaller Reynolds numbers result in more highly elongated structures. However, the volume of these highly elongated structures cannot be excessively large, as their height must remain small enough to achieve the large elongation rate.

Figure 9. The p.d.f.s of the length-to-height ratio for intense fluctuation structures of (a) $u'$ and (b) $T'$ . Here denotes detached structures and denotes attached structures.

Figure 10. The p.d.f.s of the width-to-height ratio for intense fluctuation structures of (a) $u'$ and (b) $T'$ . Here denotes detached structures and denotes attached structures.

The width remains relatively small since the dashed lines in figure 10 have the similar plateau when $l_z/l_y$ is between 1 and 4. Unlike the elongation in the streamwise direction, the stretch in the spanwise direction is only sensitive to the Mach number. As the Mach number increases, the number of extremely stretched structures also increases. Comparing the attached $u'$ structures with attached $T'$ structures, it can be noticed that the $l_z/l_y$ of $T'$ structures tends to be slightly larger according to the location of the peak. However, for both detached $u'$ and $T'$ structures, the collapse behaviour is consistent across all cases when $l_z/l_y\gt 0.6$ despite the variations in Reynolds and Mach numbers.

3.3.2. The Qs structures

The Reynolds and Mach number effects on Q $^-$ s structures are investigated, since these structures play an important role in turbulence momentum and energy transport. The volume fractions of different types of Q $^-$ s structures are summarized in table 6. For Qs structures of both $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ , the Reynolds number effects exhibit a non-monotonic behaviour, but the latter ones are more sensitive. Furthermore, it is observed that the response of $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ Q $^-$ s structures to the variations in Mach number differs. Specifically, Cases Ma15Re9K and Ma30Re15K exhibit nearly identical volume fractions for Q $^-$ s structures $u^{\prime\prime}v^{\prime\prime}$ regardless of type, whereas distinct Mach numbers lead to markedly different volume fractions for Q2s structures of $T^{\prime\prime}v^{\prime\prime}$ . Therefore, the energy transfer is strongly correlated with the Mach number. Additionally, similar to intense fluctuation structures, the Q4s structures of $T^{\prime\prime}v^{\prime\prime}$ with $T'\gt 0$ account for only a small proportion of the total channel volume compared with those of $u^{\prime\prime}v^{\prime\prime}$ .

Table 6. The volume fractions of different Qs structures with respect to the whole channel in all the five cases. Here ${\textit{Re}}_{\tau }^*$ is the semilocal friction Reynolds number. The subscript ‘u’ denotes structures of $u^{\prime\prime}v^{\prime\prime}$ , ‘T’ denotes structures of $T^{\prime\prime}v^{\prime\prime}$ and ‘att’ represents the attached structures.

Figure 11. The p.d.f.s of the ratio of length to height for Qs structures of (a) $u^{\prime\prime}v^{\prime\prime}$ and (b) $T^{\prime\prime}v^{\prime\prime}$ . Here denotes detached structures and denotes attached structures.

Structure sizes of all the Qs structures are also examined, and distinct behaviours are identified in comparison with the intense fluctuation structures. Figure 11 illustrates the p.d.f.s of length-to-height ratio. A higher Mach number can still induce stronger elongation in the streamwise direction, with the effects being significantly more pronounced here than in § 3.3.1. Furthermore, the plateau observed in the dashed line in figure 9 is now replaced by a peak in figure 11. No noticeable difference is found between Qs structures of $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ .

Figure 12. The p.d.f.s of the ratio of width to height for Qs structures of (a) $u^{\prime\prime}v^{\prime\prime}$ and (b) $T^{\prime\prime}v^{\prime\prime}$ . Here denotes detached structures and denotes attached structures.

Although the width-to-height ratio of the detached Qs structures is insensitive to both Reynolds and Mach numbers, as evidenced by the good collapse of lines in figure 12, the Mach number effects on attached Qs structures are extremely notable. A large number of attached Qs structures with $l_z/l_y$ around 10 are generated in the cases at Mach 3. Upon closer examination, these structures predominantly reside in the near-wall region with $y^+_{{max}}\lt 30$ . Furthermore, among these wide structures in Case Ma30Re15K, Q3s structures account for 50 % of the volume, whereas the proportion of Q2s is 34 %.

4.1. Overlapping structures

Since all structures are extracted separately, their instantaneous distributions and shapes can be systematically compared one by one between the streamwise velocity and temperature to identify their similarities and differences. Turbulence is an extremely complicated system, and the instantaneous behaviours between two different variables are unlikely to exhibit high similarity without some form of average operation, such as root-mean-square. However, this approach can provide rich details. In this paper, the structures (remarked as $s_i$ ) of a specific variable are extracted and grouped into a structure set $S$ . Each $s_i$ consists of a mesh point set $P_i$ which is decided by the method introduced in § 2.2. A significant degree of overlap between the structures of streamwise velocity and temperature indicates that these paired structures occupy nearly identical spatial position with similar shapes. To obtain a quantitative result, the overlap ratio for two structures is defined as

(4.4) \begin{equation} {D_{i,j}} = 2\frac {{|{{\{ P_i\} }_{S_1}} \cap {{\{ {P_j}\} }_{S_2}}|}}{{|{{\{ {P_i}\} }_{S_1}}| + |{{\{ {P_j}\} }_{S_2}}|}}. \end{equation}

Here, $S_1$ and $S_2$ are structure sets corresponding to two distinct variables and $|\boldsymbol{\cdot }|$ denotes the cardinality of the set, specifically referring to the number of mesh points here. If $D_{i,j}\gt D_{cr}$ , the $i$ th structure in $S_1$ and the $j$ th structure in $S_2$ are identified as an overlapping pair. A higher value of $D_{i,j}$ indicates a larger common part between the two structures. To ensure that no single structure is assigned to more than one overlapping pair, $D_{cr}$ is set to 50 %. Structures within the same overlapping pairs are located at nearly identical spatial positions. With this criterion, the spatial distributions of the intense fluctuation structures and Qs structures for streamwise velocity and temperature are compared with each other in details.

4.1.1. Intense fluctuation structures

Table 7 lists the detailed information on the overlapping pairs, including their numbers and volume fractions with respect to all intense fluctuation structures in regions with $y^+\lt 50$ and $y^+\gt 50$ , separately. It is remarkable that both the number and volume fractions of the overlapping pairs are significantly high when $y^+\lt 50$ . For the instantaneous field, it is surprising that more than half of the intense $u'$ and $T'$ structures are positioned nearly identically, which gives the reason for the high correlation in the near-wall region in figure 13. This observation is consistent with the analysis from the governing equations that both the streamwise velocity and temperature exhibit behaviours akin to passive scalar. As the height increases, their instantaneous positions diverge completely. Additionally, at higher Reynolds numbers, the intense $u'$ and $T'$ structures exhibit reduced overlap. Therefore, identifying the characteristics (e.g. shape) of the overlapping pairs is essential for uncovering the underlying mechanisms.

Table 7. The number ( $N$ ) and volume ( $V$ ) fractions of the overlapping pairs between intense $u'$ and $T'$ structures. Here ${\textit{Re}}_{\tau }^*$ is the semilocal friction Reynolds number. Subscript ‘o’ denotes the overlapping pairs, ‘u+T’ denotes the summation of the variables of intense $u'$ and $T'$ structures and ‘1’, ‘2’ denote the regions with $y^+\lt 50$ and $y^+\gt 50$ , respectively. Specifically, $V_{{o}}$ is the volume of the common parts in overlapping pairs.

Figure 14. Samples of overlapping pairs in regions with (a) $y^+\lt 50$ and (b) $y^+\gt 50$ for Case Ma30Re15K. Red and blue structures denote intense $u'$ and $T'$ structures, respectively.

Figure 14 presents the typical overlapping pairs at two height regions for Case Ma30Re15K. Both the near-wall streaks and large-scale structures exhibit high overlapping rates. A further analysis is conducted by inspecting the volume-weighted averages of the intense $u'$ structures within the overlapping pairs (which are quite similar for intense $T'$ structures) through the following formula:

(4.5) \begin{equation} [l] = \left (\sum \limits _i {{V_i}{l_i}} \right ) \Big/ \sum \limits _i {{V_i}}, \end{equation}

where $l$ is a specific variable, $V$ denotes the volume of velocity structure, subscript ‘ $i$ ’ denotes the $i$ th overlapping pair, and ‘[ ]’ denotes the average operation. Table 8 summarizes the results. It shows that the near-wall overlapping pairs mainly consist of streamwise streaks, whereas the overlapping pairs of large-scale structures play a more significant role in sustaining the correlation in the outer region. Since the existence of these streaks constitutes a fundamental process in the self-sustaining cycle of turbulence (Toh & Itano Reference Toh and Itano2005; Del Á lamo et al. Reference del ÁLAMO, JIMÉNEZ, ZANDONADE and MOSER2006), the high overlapping rate between the instantaneous streaks of $u'$ and $T'$ also suggests that the temperature passively follows the self-sustaining cycle of velocity. Furthermore, the observed difference may result from the discrepancy of the large-scale structures in the outer layer through the inner–outer interaction, which has been widely reported by many researchers. Particularly in the high Reynolds number cases, where the number and effects of large-scale structures become considerable, the overlapping rate is extremely low, indicating that the kinetic processes of the LSMs and VLSMs for velocity and temperature have different mechanisms. These mechanisms will be further investigated in §§ 4.2 and 4.3.

Table 8. The volume-weighted average of the intense $u'$ structures in the overlapping pairs. Here ‘ $[\,]$ ’ denotes the average operation weighted by the volume of the circumscribing box of a specific structure and ‘1’, ‘2’ denote the regions with $y^+\lt 50$ and $y^+\gt 50$ , respectively.

Therefore, in the near-wall region, the similar diffusive feature, which is dominant in their governing equations (4.1) and (4.2), reflects in the similar kinematics behaviours between streamwise velocity and temperature, such as the overlapping streaks structures. The discrepancy in the skewness of their fluctuations may ascribe to the interactions with outer layer and different wall-normal distributions of the source terms. However, the highly nonlinear advection term in (4.1) begins to take effects in the logarithmic and outer regions, reducing the correlation between streamwise velocity and temperature. The remaining overlapping pairs likely originate from the near-wall region, which is associated with momentum and energy transfer in the channel and will be further discussed in § 4.1.2.

4.1.2. The Qs structures

The overlapping pairs of Qs structures of streamwise velocity and temperature are inspected, as they provide an important perspective for investigating momentum and energy transfer in turbulent flow. Table 9 presents the details about the overlapping pairs, including their number and volume fractions with respect to all Qs structures in the regions with $y^+\lt 50$ and $y^+\gt 50$ separately. The results exhibit strong resemblance to intense fluctuation structures in the near-wall region, as shown in table 7, while the volume fractions are higher in the outer layer. A negative correlation between the overlapping rate and the Reynolds number can also be found here. Furthermore, it also indicates that the instantaneous distributions of the large-scale Qs structures of $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ , which play a more critical role in cases with higher Reynolds number, are totally different.

Table 9. The number ( $N$ ) and volume ( $V$ ) fractions of the overlapping pairs between Qs structures of $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ . Here ${\textit{Re}}_{\tau }^*$ is the semilocal friction Reynolds number. Subscript ‘o’ denotes the overlapping pairs, ‘uv+Tv’ denotes the summation of the variables of Qs structures of $u^{\prime\prime}v^{\prime\prime}$ and $T^{\prime\prime}v^{\prime\prime}$ and ‘1’, ‘2’ denote the regions with $y^+\lt 50$ and $y^+\gt 50$ , respectively. Specially, $V_{{o}}$ is the volume of the common parts in overlapping pairs.

The correlation between streamwise velocity and temperature fluctuations is examined within the overlapping pairs of Q2s structures to investigate the transport of these fluctuations from the near-wall region to the outer layer. Figure 15 demonstrates high correlations, particularly for high Reynolds number cases, across nearly the entire domain except near the channel centre. The Reynolds and Mach numbers effects are also notable. Higher Mach numbers lead to a broader range of tight correlations, whereas higher Reynolds number enhance the correlation in the outer layer. Therefore, the overlapping ejection-event structures serve as one of the sources contributing to the remaining correlation of $u'$ and $T'$ in the outer layer. However, the effects of the highly nonlinear advection term in (2.2) diminishes the correlation progressively. Further discussion on these influences, arising from the different forms of governing equations in the outer layer, will be presented in subsequent parts from the perspective of self-similarity of the coherent structures.

Figure 15. Correlations of velocity fluctuations and temperature fluctuations in overlapping pairs of Q2s structures normalized by their root-mean-square.

4.2. Conditional average

The size of intense fluctuation structures for the $u'$ and $T'$ has been investigated in § 3. Here, the shapes of the attached $u'$ and $T'$ structures are analysed in conjunction with the streamwise vortex identified by $\varDelta$ criterion. Given that the vortex clusters exhibit self-similarity (del Á lamo et al. Reference del Álamo and Jiménez2006), a conditional average is performed within the wall-attached vortex clusters with different scales. The velocity and temperature fluctuation fields surrounding the vortex are rescaled based on the height of the structure’s centre. The formula for the streamwise velocity is presented below, and the other variables can be treated in the same way,

(4.6) \begin{equation} \langle {u'}\rangle \left ( \boldsymbol{R} \right ) = \sum \limits _i {y_{c,i}^3\left [ {{u}\left ( {\boldsymbol{R}{y_{c,i}} + {\boldsymbol{x}_{c,i}}} \right ) - \bar u\left ( {{r_y} + {y_{c,i}}} \right )} \right ]} /\sum \limits _i {y_{c,i}^3}, \end{equation}

where $\boldsymbol{x}_{c,i}$ is the position of the circumscribing box centre of the $i$ th structure, $\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{x}_c$ is the position vector respect to the box centre, with its three components $r_x,r_y,r_z$ , $\boldsymbol{R}=\boldsymbol{r}/y_c$ is the rescaled vector and $\left \langle \, \right \rangle$ denotes the conditional average. As shown in (4.6), $y_{c,i}^3$ is assigned as the weight for the average operation. Since the length, width and height of the vortex cluster are nearly proportional to the height of the structure’s centre, this weighting factor is effectively equivalent to the volume of the circumscribing box enclosing the structure.

Figure 16. (a) Three-dimensional plot of the average velocity and temperature fluctuation fields conditioned to the attached vortex clusters for Case Ma30Re15K. The blue and red structures in the middle are the isosurfaces of negative fluctuations of $u'$ and $T'$ , respectively. The yellow and green structures at two sides are the isosurfaces of positive fluctuations of $u'$ and $T'$ , respectively. The slice at $R_x=1.5$ shows the contour of the wall-normal velocity. (b) The streamwise distributions of the averaged centre height of the positive structures at two sides. Here denotes velocity structures and denotes temperature structures.

Figure 16(a) illustrate the velocity and temperature fluctuation fields after conditional average in Case Ma30Re15K. Results from other cases are highly similar and therefore not presented here. The negative fluctuation structures in the middle are accompanied by two positive fluctuation structures, as depicted by the isosurfaces for both streamwise velocity and temperature. Similar findings have also been reported by del Á lamo et al. (Reference del Álamo and Jiménez2006) in incompressible flow. Additionally, a slice at $R_x=1.5$ is provided, revealing that the negative fluctuation structures align with the positive $v$ regions in the spanwise direction, coincidentally meeting the definition of Q2 events. In this region, the streamwise velocity (blue structures) and temperature (red structures) fluctuation structures exhibit strong similarity, indicating that the streamwise vortex transports both velocity and temperature from lower height regions to higher regions via Q2 events. Conversely, in the regions on either side, the positive fluctuation structures align with the negative $v$ regions, forming Q4 events. Here, the characteristics of the positive fluctuation structures of streamwise velocity (yellow structures) and temperature (green structures) differ significantly. Figure 16(b) presents the streamwise distributions of the averaged centre height of the positive structures on both sides. The velocity structures rise rapidly, whereas the temperature structures remain close to the bottom across all cases. Dong et al. (Reference Dong, Tong, Yu, Chen, Yuan and Wang2022b ) computed the conditional averages of the temperature field based on positive and negative shear stress pairs or heat flux pairs in compressible boundary layer and found that the positive $T'$ structures have smaller inclination angles. This observation suggests that the positive temperature structures and Q4 structures of $T^{\prime\prime}v^{\prime\prime}$ are located at the bottom of the vortex packets, and the vortex can only transport velocity structures back to the lower region, not temperature structures.

As previously discussed, the streamwise velocity and temperature exhibit a strong correlation in the near-wall region, characterized by a high overlapping rate of the instantaneous structures (in § 4.1). Additionally, the p.d.f.s of structure size for both attached and detached structures of streamwise velocity and temperature are remarkably similar, including large-scale features (in § 3). The generation mechanism of the LSMs of velocity remains an open question. Neither the hairpin vortex packet mechanism (Kim & Adrian Reference Kim and Adrian1999) nor the theory of linear modes with the largest transient growth (del Á lamo & Jiménez Reference del Álamo and Jiménez2006) can explain the generation of the LSMs of temperature due to the scalar property of temperature and its governing equation (4.2), which conflict with these explanations. Consequently, this paper proposes that the formation process of LSMs of the velocity carries temperature to form these structures together at the same time. However, due to the different mechanisms in the outer layer, temperature on longer follows the transfer of velocity structures. This also elucidates the absence of large-volume sweep events for temperature around vortex clusters, which contrasts with the behaviour of streamwise velocity.

4.3. Energy transfer

The energy transfer between flow structures at different scales is investigated to explore the relationship between the streamwise velocity and temperature. Scale decomposition is performed in the real space using a Gaussian filter,

(4.7) \begin{equation} \hat u_i^{\sigma }(\boldsymbol{x})=C(\sigma )\int _V u'_i\left (\boldsymbol{x}'\right )\exp \left (-\frac {2}{\sigma ^2}(\boldsymbol{x}-\boldsymbol{x}')^2\right )\mathrm{d}\boldsymbol{x}', \end{equation}

where $\sigma$ is the filter scale and $C(\sigma )$ is the normalization coefficient that ensures the integral of the kernel equals one. Here $V$ is the integral domain, defined as a box centred around $\boldsymbol{x}$ with the dimensions of $2\sigma$ in each direction. The filter has also been utilized by Lozano-Durán et al. (Reference Lozano-Durán, Holzner and Jiménez2016) and Motoori & Goto (Reference Motoori and Goto2021). They further extended the filtering operation by reflecting the filter with the sign of $\hat u_2$ inverted when facing the wall to guarantee the boundary condition, which is also adopted in this study. By applying this low-pass filter, the fields corresponding to the specific scale $\sigma$ can be obtained by

(4.8) \begin{equation} u_i^{\sigma }(\boldsymbol{x})=\hat u_i^{\sigma }(\boldsymbol{x})-\hat u_i^{2\sigma }(\boldsymbol{x}). \end{equation}

Figure 17 illustrates the velocity fluctuation fields for Case Ma30Re15K under various scales. The original field without filtering is processed following (4.7) and (4.8) with $\sigma =0.1, 0.2$ and 0.4, which are normalized by the channel half-height $h$ . Three fields with different scales, which can also be directly identified from the streamwise streaks, are obtained. The results demonstrate the successful realization of scale decomposition. Although this decomposition is not strictly orthogonal like Fourier decomposition, the cross-correlation between different scales is sufficiently low, as noted by Motoori & Goto (Reference Motoori and Goto2021), to support subsequent analysis.

Figure 17. Contour of velocity fluctuation fields at $y^+ \approx 15$ for Case Ma30Re15K: (a) the original fields without filtering; (b) filter scale $\sigma =0.1$ ; (c) filter scale $\sigma =0.2$ ; (d) filter scale $\sigma =0.4$ . The scale sizes have been normalized by the channel half-height $h$ .

The transport equation of the turbulent kinetic energy $k=u_i''u_i''/2$ is presented as follows:

(4.9) \begin{equation} \begin{aligned} \frac {{\partial \bar \rho \tilde k}}{{\partial t}} + {{\tilde u}_j}\frac {{\partial \bar \rho \tilde k}}{{\partial {x_j}}} = &- \bar \rho \widetilde {{{u_i''}}{{u_j''}}}\frac {{\partial {{\tilde u}_i}}}{{\partial {x_j}}} - \bar \rho \tilde k\frac {{\partial {{\tilde u}_j}}}{{\partial {x_j}}} - \overline {{{u_j''}}} \frac {{\partial \bar p}}{{\partial {x_j}}}+ \overline {p'\frac {{\partial {{u_j''}}}}{{\partial {x_j}}}} \\ & - \frac {\partial }{{\partial {x_j}}}\left ( {\overline {p'{{u_i''}}} \;{\delta _{\textit{ij}}} + \bar \rho \widetilde {{{u_i''}}{{u_i''}}{{u_j''}}}/2}-{{{\bar \tau }_{\textit{ij}}}\overline {{{u_i''}}} - \overline {{{\tau _{\textit{ij}} '}}{{u_i''}}} } \right )\\ & - \left ( {{{\bar \tau }_{\textit{ij}}}\overline {\frac {{\partial {{u_j''}}}}{{\partial {x_j}}}} + \overline {{{\tau _{\textit{ij}} '}}\frac {{\partial {{u_j''}}}}{{\partial {x_j}}}} } \right )\!. \end{aligned} \end{equation}

On the right-hand side of (4.9), The first row contains the production term $- \bar \rho \widetilde {{{u_i''}}{{u_j''}}}({{\partial {{\tilde u}_i}}}/{{\partial {x_j}}})$ and terms related with compressibility. The second and third rows correspond to the diffusion terms and the dissipation terms, respectively. For compressible channel flow, only the wall-normal derivatives in production terms are considered.

To investigate energy production at a specific scale and energy transfer between different scales, fluctuation fields $u_i^{\sigma _1}$ and $u_i^{\sigma _2}$ corresponding to distinct scales $\sigma _1$ and $\sigma _2$ are extracted from the original flow field, as an example. If more than two scales are considered, the subsequent derivation can be similarly applied. In the similar manner as (4.9) is derived, The transport equation of $k^{\sigma _1}=u_i^{\sigma _1}u_i^{\sigma _1}/2$ can also be derived with the approximate orthogonality $\overline {\rho u_i^{\sigma _1}u_j^{\sigma _2}}=0$ . The terms concerned in this study are written as

(4.10) \begin{align}&\qquad\qquad\qquad {\textit{Pr}}_u(\sigma _1,y) = - \bar \rho \widetilde {u^{\sigma _1}v^{\sigma _1}}\frac {{\partial \tilde u}}{{\partial y}}, \end{align}

(4.11) \begin{align}& Tr_u(\sigma _2 \rightarrow \sigma _1;y) = - \overline {\rho u_i^{ {{\sigma _1}} }u_j^{ {{\sigma _1}}}\frac {{\partial u_i^{ {{\sigma _2}} }}}{{\partial {x_j}}}} - \left ( { - \overline {\rho u_i^{ {{\sigma _2}} }u_j^{ {{\sigma _2}} }\frac {{\partial u_i^{{{\sigma _1}}}}}{{\partial {x_j}}}}} \right )\!. \end{align}

The terms above present the energy production of the scale $\sigma _1$ and the net energy transfer from the scale $\sigma _2$ to the scale $\sigma _1$ , respectively. The first part on the right-hand side of (4.11) indicates the energy extracted by scale $\sigma _1$ from scale $\sigma _2$ , whereas the direction of the energy transfer for its second part reverses. The other terms are not relevant to the current discussions. The results are also similar to those reported in Motoori & Goto (Reference Motoori and Goto2021) in incompressible flow.

The turbulent scalar energy $\phi =T^{\prime\prime}T^{\prime\prime}/2$ is also defined to investigate the interscale energy transfer of the temperature structures. Its transport equation is also derived here:

(4.12) \begin{equation} \begin{aligned} \frac {{\partial \bar \rho \tilde \phi }}{{\partial t}} + {{\tilde u}_j}\frac {{\partial \bar \rho \tilde \phi }}{{\partial {x_j}}} = &- \bar \rho \widetilde {T^{\prime\prime}{{u_j''}}}\frac {{\partial \tilde T}}{{\partial {x_j}}} - \bar \rho \tilde \phi \left ( {1 + \frac {{2{R_g}}}{{{c_v}}}} \right )\frac {{\partial {{\tilde u}_j}}}{{\partial {x_j}}} \\ &+ \frac {1}{{{c_v}}}\overline {T^{\prime\prime}} {{\bar \tau }_{\textit{ij}}}\frac {{\partial {{\tilde u}_i}}}{{\partial {x_j}}} + \frac {1}{{{c_v}}}\overline {{{\tau _{\textit{ij}} '}}T^{\prime\prime}} \frac {{\partial {{\tilde u}_i}}}{{\partial {x_j}}} \\ &- \frac {1}{{{c_v}}}\bar p\overline {T^{\prime\prime}\frac {{\partial {{u_j''}}}}{{\partial {x_j}}}} - \left ( {\frac {1}{2} + \frac {{{R_g}}}{{{c_v}}}} \right )\frac {{\partial \bar \rho \widetilde {T^{\prime\prime}T^{\prime\prime}{{u_j''}}}}}{{\partial {x_j}}} \\ &- \frac {1}{{{c_v}}}\frac {\partial }{{\partial {x_j}}}\left ( {{{\bar q}_j}\overline {T^{\prime\prime}} + \overline {{{q_j'}}T^{\prime\prime}} } \right ) + \frac {1}{{{c_v}}}\left ( {{{\bar q}_j}\overline {\frac {{\partial T^{\prime\prime}}}{{\partial {x_j}}}} + \overline {{{q_j'}}\frac {{\partial T^{\prime\prime}}}{{\partial {x_j}}}} } \right ) \\ &- \frac {{2{R_g}}}{{{c_v}}} \overline {\rho {{u_j''}}T^{\prime\prime}\frac {{\partial T^{\prime\prime}}}{{\partial {x_j}}}} + \frac {1}{{{c_v}}}\left ( {{{\bar \tau }_{\textit{ij}}}\overline {T^{\prime\prime}\frac {{\partial {{u_i''}}}}{{\partial {x_j}}}} + \overline {T^{\prime\prime}{{\tau _{\textit{ij}} '}}\frac {{\partial {{u_i''}}}}{{\partial {x_j}}}} } \right )\!. \end{aligned} \end{equation}

Compared with the turbulent kinetic energy, the turbulent scalar energy has terms related to both velocity and temperature derivatives in its production terms. Similar to the derivation of $k^{\sigma _1}$ , the energy production and transfer terms for scaler energy $\phi ^{\sigma _1}$ of scale $\sigma _1$ can be written as

(4.13) \begin{align}&\qquad\quad {\textit{Pr}}_T(\sigma _1,y) = - \bar \rho \widetilde {T^{\sigma _1}v^{\sigma _1}}\frac {{\partial \tilde T}}{{\partial y}}+ \frac {1}{{{c_v}}}\overline {{ \tau }_{12}^{\sigma _1} T^{\sigma _1}} \frac {{\partial {\tilde u}}}{\partial {y}}, \end{align}

(4.14) \begin{align}& Tr_T(\sigma _2 \rightarrow \sigma _1;y) = - \overline {\rho T^{ {{\sigma _1}} }u_j^{ {{\sigma _1}}}\frac {{\partial T^{ {{\sigma _2}} }}}{{\partial {x_j}}}} - \left ( { - \overline {\rho T^{ {{\sigma _2}} }u_j^{ {{\sigma _2}} }\frac {{\partial T^{{{\sigma _1}}}}}{{\partial {x_j}}}}} \,\right )\!. \end{align}

The wall-normal distributions of the production terms of four different cases have been plotted in figure 18. The peak values of all scales in each case are all located within the range of $y^+=15$ –20 and the structures at this position have been illustrated in figure 17. Mach number effects are evident, as the production terms of the scalar energy are significantly higher when Ma = 3.0. For the low Reynolds number cases depicted in figure 18(a,c), the production terms of the large-scale velocity structures ( $\sigma =0.4$ ) at the logarithmic region begin to emerge, despite a monotonic decrease beyond the peak position in the near-wall region. However, when compared with the production terms of turbulent kinetic energy, it becomes apparent that the production terms for the large-scale temperature structures are considerably lower and exhibit a rapid decline without undergoing any variation stage. This difference becomes significantly more pronounced for high Reynolds number cases in figure 18(b,d). A nearly flat region is observed across almost the entire logarithmic region for the production terms of the large-scale velocity structures, a finding consistent with results from the incompressible flows (Motoori & Goto Reference Motoori and Goto2021). In contrast, no similar behaviour is evident for scalar temperature energy. It has been mentioned in § 4.2 that the large-scale negative temperature fluctuation structures are concurrently generated during the formation of velocity structures. However, their behaviours in the outer layer differ markedly. Figure 18 elucidates why temperature structures lack a production mechanism in the outer layer, further indicating that these large-scale temperature structures do not possess the self-sustaining process that is supposed to be featured by the velocity structures (Flores & Jimenez Reference Flores and Jimenez2006).

Figure 18. Wall-normal distributions of the production terms of (a) Case Ma15Re3K, (b) Case Ma15Re20K, (c) Case Ma30Re5K and (d) Case Ma30Re15K. Blue ones denote velocity terms and red ones denote temperature terms. From the lighter one to the darker one denote the scale $\sigma =0.1,0.2$ and 0.4, respectively.

The interscale energy transfer $Tr(\sigma _1\rightarrow \sigma _2;y)$ in four cases has been calculated by (4.11) and (4.14) and the results are presented in figure 19. Since the self-energy transfer is zero when $\sigma _1=\sigma _2$ , these lines are omitted from the plots. The blue and red lines denote the energy transfer for velocity and temperature, respectively. From lighter to darker shades of the lines, $\sigma _2=0.1, 0.2$ and 0.4. Line types are utilized to differentiate between various values of $\sigma _1$ for a given $\sigma _2$ . For both velocity and temperature, the interscale energy transfer consistently occurs from the large-scale structures to the small-scale structures, as presented in figure 19. Therefore, large-scale temperature fluctuation structures with large $\sigma$ , which lack sufficient energy gain from the production terms, experience only energy loss during the interscale energy transfer and dissipate rapidly in the outer layer. As a result, the correlation between the streamwise velocity and temperature is notably low in the outer layer with minimal overlap in their instantaneous distributions. In this study, only structures with an intermediate size ( $\sigma =0.2$ ) exhibit both energy gain and loss during the transfer process, which will be the primary focus in the subsequent analysis. The position of peak value for the transfer terms in each case, which indicates the height at which interscale interactions are most significant, differs between velocity and temperature structures. For velocity structures with $\sigma =0.2$ , as the Reynolds number increases, they tend to gain energy from larger structures at heights 1.5–2 times their own scale size. The findings for high Reynolds number results are consistent with those reported by Motoori & Goto (Reference Motoori and Goto2021) in incompressible flow. In contrast, temperature structures do not exhibit such behaviours, and the energy transfer primarily occurs at heights nearly equal to their own scale size. This phenomenon suggests that although the hierarchy of the temperature structures result from the velocity structures, their own mechanism is different from the self-sustaining process of the velocity structures. It should be mentioned that the scale decomposition used here is designed to compare the energy production and interscale energy transfer mechanisms between velocity and temperature. These findings do not correspond directly with those of the structures extracted by the clustering method in § 3. However, it provides a valuable perspective for understanding the decrease in correlation within the outer layer, where large-scale structures dominate. However, with all these phenomena and mechanisms uncovered in this paper, the overall picture of the correlation between velocity and temperature fluctuations in compressible turbulent channel flows from the perspective of coherent structures can be provided, which will be discussed in § 5.

Figure 19. Wall-normal distributions of interscale energy transfer of (a,b) Case Ma15Re3K, (c,d) Case Ma15Re20K, (e,f) Case Ma30Re5K and (g,h) Case Ma30Re15K. The blue lines on the left-hand side belong to velocity and the red lines on the right-hand side belong to temperature. The legend shows the values of $\sigma _1 \rightarrow \sigma _2$ that give the direction of the energy transfer.