3.1. Population density

The population density of the intense fluctuation structures is investigated to examine the existence of hierarchical length scales (Perry & Chong Reference Perry and Chong1982), which has been identified by Hwang & Sung (Reference Hwang and Sung2018) in incompressible boundary layers. The following formula provides the definition of population density:

(3.1) \begin{equation} n_s(l_{\!y}) = \frac {N_s(l_{\!y})}{mL_xL_z}, \end{equation}

in which $n_s$ is the population density, $N_s(l_{\!y})$ is the number of structures with the structure height of $l_{\!y}$ , m is the number of fields used to count the structure number, and $L_x, L_z$ are the streamwise and spanwise length of subdomains (see table 2).

Figure 3.

Population density of the intense (a) $u'$ and (b) $T'$ structures with respect to their height in Set T10S. The slopes of the dashed lines are also displayed. The part of the lines near their right ends are smoothed to elucidate the trends, since the number of the large-scale structures is small which causes the original statistical data denoted by points to oscillate.

Figure 4.

Population density of the intense (a) $u'$ and (b) $T'$ structures with respect to their height in Set T05S. The slopes of the dashed lines are also displayed. The part of the lines near their right ends are smoothed to elucidate the trends, since the number of the large-scale structures is small, which causes the original statistical data denoted by points to oscillate.

Figure 4. Long description

Two line graphs depict the population density of intense structures with respect to their height. Panel A: The line graph shows the population density of intense streamwise velocity fluctuation structures. The x-axis represents the height in wall units (l_y+) on a logarithmic scale ranging from 10^0 to 10^4. The y-axis represents the population density (n_s) on a logarithmic scale ranging from 10^-6 to 10^0. The graph includes three lines: Attached u’ (light blue), Attached u’’ (blue), and Detached u’ (gray). The slopes of the dashed lines are labeled as -2.3, -2.8, and -1.0. Panel B: The line graph shows the population density of intense temperature fluctuation structures. The x-axis represents the height in wall units (l_y+) on a logarithmic scale ranging from 10^0 to 10^4. The y-axis represents the population density (n_s) on a logarithmic scale ranging from 10^-6 to 10^0. The graph includes three lines: Attached T’’ (light red), Attached T’ (red), and Detached T’ (gray). The slopes of the dashed lines are labeled as -2.4, -1.0, and -1.1.

Figures 3 and 4 present the population densities of attached positive, attached negative and detached structures in Sets T10S and T05S. The dashed lines in these figures illustrate the decreasing trend in population density with increasing structure height, following an inverse power-law relationship $n_s \propto (l_{\!y}^+)^k$ , where $k$ denotes the slope of each dashed line, for $l_{\!y}^+ \gt 100$ beyond the buffer layer. The slope $k$ varies across different flows, although the theory of Perry & Chong (Reference Perry and Chong1982) indicates $k = -1$ . Cheng et al. (Reference Cheng, Li, Lozano-Durán and Liu2020) reported values of $-1.3$ for intense $u'^+$ structures and $-1.9$ for intense $u'^-$ in incompressible channel flows, whereas Hwang & Sung (Reference Hwang and Sung2018) observed the $-1$ scaling law for attached $u'$ structures without distinguishing their sign, within a limited range of $l_{\!y}^+$ in incompressible boundary layer flows, which is consistent with the current findings in figures 3(a) and 4(a) for attached $u'^+$ structures. Moreover, similar to Cheng et al. (Reference Cheng, Li, Lozano-Durán and Liu2020), attached $u'^-$ structures exhibit a higher decay rate than attached $u'^+$ structures in the logarithmic layer. In addition, the cooling effect of the wall appears to have no influence on the slope $k$ . Regarding intense temperature structures, previous research is scarce and Yuan et al. (Reference Yuan, Tong, Li, Chen and Dong2022) only reports variation trends primarily confined to the buffer layer, with $l_{\!y}^+ \lt 200$ . In the current study, the slope $k$ is nearly identical for attached $T'^+$ and $T'^-$ structures, and wall cooling reduces the value of $k$ . For Set T05S, attached $T'$ structures nearly follow the $-1$ scaling law shown in figure 4(b). Furthermore, detached structures also exhibit an inverse power-law behaviour, but with a significantly higher decay rate compared with attached structures.

The deviation from the $-1$ scaling law in the population density of attached structures may be attributed to three primary factors. First, the Reynolds number in the current study is insufficiently high, meaning that scale separation is not fully developed. Second, real flow structures interact with one another and cannot be treated as perfectly isolated entities, as assumed by Perry & Chong (Reference Perry and Chong1982). Third, Townsend’s attached-eddy hypothesis may not be perfectly valid in real flows for all types of structures. It highlights a critical area for refining existing theoretical assumptions and demonstrates the potential for future improvements in turbulence modelling.

3.2. Shape characteristics

The shape characteristics of the intense fluctuation structures are investigated in terms of the length-to-height and width-to-height aspect ratios, which are also regarded as indicators of self-similarity (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Hwang & Sung Reference Hwang and Sung2018; Li et al. Reference Li, Fu and Xu2025). Importantly, the objective of this analysis is not to categorise the structures into idealised shapes, such as hairpin vortices, nor to reconstruct their exact geometric boundaries; such detailed topological characterisations fall outside the scope of the current study. Same as in § 3.1, attached positive, attached negative and detached structures are analysed separately, which also provides a perspective for unified consideration.

The structures from Set T10S with adiabatic wall condition are analysed first. It should be noted that the adiabatic wall condition is implemented by setting the wall temperature to the recovery temperature with a recovery coefficient $r=0.9$ . Therefore, this adiabatic condition is not strictly satisfied and results show that a tiny negative wall-normal gradient of temperature occurs close to the wall. This configuration can be regarded as a slightly heating wall when considering small-scale structures near the wall. However, for overall analysis, these slight heating effects can be neglected. Figure 5 displays the joint probability density functions (j.p.d.f.s) of the structure length ( $l_x$ ) and height ( $l_{\!y}$ ). The variation trend, represented by the dashed line, is also indicated in the figures along with its corresponding equation. In the incompressible regime, Lozano-Durán et al. (Reference Lozano-Durán, Flores and Jiménez2012) and Hwang & Sung (Reference Hwang and Sung2018) provided the fitted equations $l_x^+ \approx 3l_{\!y}^+$ and $l_x^+ = 17.98(l_{\!y}^+)^{0.74}$ for attached structures in channel flows and boundary layer flows, respectively. For compressible channel flows, Li et al. (Reference Li, Fu and Xu2025) reported the linear relation $l_x^+ \approx 7.5l_{\!y}^+$ without distinguishing the sign of the structures. In the current study of compressible boundary layer flows with adiabatic wall condition, deviation from linearity is observed, and the equations $l_x^+ = 18(l_{\!y}^+)^{0.71}$ and $l_x^+ = 25(l_{\!y}^+)^{0.71}$ are obtained for attached $u'^+$ and $u'^-$ structures, respectively, which are closer to the results of Hwang & Sung (Reference Hwang and Sung2018). Furthermore, compressibility noticeably alters the scaling relations relative to incompressible flows (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012; Hwang & Sung Reference Hwang and Sung2018), leading to a streamwise elongation of the attached $u'$ structures, which is especially prominent in the $u'^-$ structures.

Figure 5.

Joint p.d.f.s of the structure length and height for (a,b,c) intense $u'$ structures and (d,e, f) intense $T'$ structures in Set T10S. Panels (a) and (d) correspond to attached positive structures, panels (b) and (e) to attached negative structures, and panels (c) and (f) to detached structures. The equation of the dashed line, which represents the variation trend, is provided as well. The contour levels are logarithmically distributed.

When examining the temperature structures, the most striking observation is the emergence of a secondary peak near $y^+=5$ in the attached $T'^+$ structures, as illustrated in figure 5(d). As previously noted, the near-wall temperature distribution in the wall-normal direction plays a critical role. Away from the wall, the wall-normal gradients of streamwise velocity and temperature have opposite signs, indicating a stronger correlation between the attached $u'$ structures and the corresponding attached $T'$ structures of opposite sign, as suggested by the strong Reynolds analogy (SRA) (Morkovin Reference Morkovin1962),

(3.2) \begin{equation} \frac {T'_{\textit{rms}}/\bar T}{{\left ( {\gamma - 1} \right ){{\textit{Ma}}^2}u'_{\textit{rms}}/\bar u}} = \frac {1}{{a\left ( {1 - \partial {{\bar T}_t}/\partial \bar T} \right )}}, \end{equation}

where $\bar T_t$ is the averaged total temperature and the subscript ‘rms’ denotes the root-mean-square of the variable. The coefficient $a$ is set to different positive values in previous research (Huang et al. Reference Huang, Coleman and Bradshaw1995; Brun et al. Reference Brun, Petrovan Boiarciuc, Haberkorn and Comte2008). Cheng & Fu (Reference Cheng and Fu2025) further proposed that only the energy-containing motions within the flow adheres to the SRA. As mentioned previously, the attached structures convey the largest portions of Reynold shear stress, thus, this SRA correlation can also be evident from the similarity between the upper portion of figure 5(d) and that of figure 5(b). This double peak distribution of the j.p.d.f.s has also been observed by Yuan et al. (Reference Yuan, Tong, Li, Chen and Dong2022) in compressible boundary layer flows with mild wall cooling effects ( $T_r/T_w=0.79$ ), where a temperature peak occurs quite near the wall. Therefore, when the wall-normal gradient of temperature is nearly zero, the relationship between $u'$ and $T'$ undergoes significant changes, as the SRA encounters a singularity. The influence of this difference on the $u'{-}T'$ relationship and its underlying mechanism, which was investigated in compressible channel flows by Li et al. (Reference Li, Fu and Xu2025), will be the focus of future work. A more direct comparison is provided in § 3.3. Furthermore, the detached structures exhibit a scaling relation of $l_x^+ \propto (l_{\!y}^+)^{1.5}$ in the logarithmic layer for both $u'$ and $T'$ structures. It should be mentioned that the tall but very short structures (left local peak in figure 5 c, f) are caused by the sudden jump of the instantaneous TNTI and will not be investigated in this study.

Figure 6.

Joint p.d.f.s of the structure width and height for (a,b,c) intense $u'$ structures and (d,e, f) intense $T'$ structures in T10S. Panels (a) and (d) correspond to attached positive structures, panels (b) and (e) to attached negative structures, and panels (c) and (f) to detached structures. The equation of the dashed line, which represents the variation trend, is provided as well. The contour levels are logarithmically distributed.

Figure 6. Long description

Panel A: A scatter plot showing the relationship between structure width (l_z^+) and height (l_y^+) for attached positive structures. The axes are logarithmic. The plot shows a positive correlation with a dashed blue line representing the trend. Panel B: A scatter plot showing the relationship between structure width (l_z^+) and height (l_y^+) for attached negative structures. The axes are logarithmic. The plot shows a positive correlation with a dashed blue line representing the trend. Panel C: A scatter plot showing the relationship between structure width (l_z^+) and height (l_y^+) for detached structures. The axes are logarithmic. The plot shows a positive correlation with a dashed blue line representing the trend. Panel D: A scatter plot showing the relationship between structure width (l_z^+) and height (l_y^+) for attached positive structures. The axes are logarithmic. The plot shows a positive correlation with a dashed purple line representing the trend. Panel E: A scatter plot showing the relationship between structure width (l_z^+) and height (l_y^+) for attached negative structures. The axes are logarithmic. The plot shows a positive correlation with a dashed purple line representing the trend. Panel F: A scatter plot showing the relationship between structure width (l_z^+) and height (l_y^+) for detached structures. The axes are logarithmic. The plot shows a positive correlation with a dashed purple line representing the trend.

Figure 6 displays the j.p.d.f.s of the structure width ( $l_z$ ) and height ( $l_{\!y}$ ). For attached $u'$ structures, the width and height are at the same scale following $l_z^+=l_{\!y}^+$ in the logarithmic layer. Together with figure 9(a,b), the equality between the width scale and height scale of the attached velocity-related structures in the logarithmic layer is consistently observed in incompressible channel flows (Lozano-Durán et al. Reference Lozano-Durán, Flores and Jiménez2012), incompressible boundary layer flows (Hwang & Sung Reference Hwang and Sung2018), compressible channel flows (Li et al. Reference Li, Fu and Xu2025) and compressible boundary layer flows under varying wall cooling conditions in the current study, indicating a high degree of universality at this Mach number. For attached $T'$ structures, a slight deviation from the equality is observed. Furthermore, a second near-wall peak is also evident in the distribution of the j.p.d.f. in figure 6(d). Consequently, the very small near-wall wall-normal temperature gradient gives rise to a number of plate-like structures with $l_x^+ \approx l_z^+ \approx 100, l_{\!y}^+ \approx 5$ , as figure 7 shows. Their appearences reveal the velocity–temperature relationship near the wall as mentioned before through SRA.

Figure 7.

Samples of the near-wall plate-like attached $T'^+$ structures in Set T10S. The horizontal positions of the structures have been shifted for better visualisation.

In addition, the detached structures exhibit nearly uniform length, width and height for both $u'$ and $T'$ .

Figure 8.

Joint p.d.f.s of the structure length and height for (a,b,c) intense $u'$ structures and (d,e, f) intense $T'$ structures in Set T05S. Panels (a) and (d) correspond to attached positive structures, panels (b) and (e) to attached negative structures, and panels (c) and (f) to detached structures. The equation of the dashed line, which represents the variation trend, is provided as well. The contour levels are logarithmically distributed.

Structures in Set T05S with moderate wall cooling effects have also been analysed to further investigate the effects of wall temperature. In this case, the region with a small wall-normal temperature gradient shifts upward and a temperature peak occurs at $y^+ \approx 30$ , representing the key difference from the adiabatic case. Figure 8 shows the j.p.d.f.s of the structure length ( $l_x$ ) and height ( $l_{\!y}$ ). The wall cooling effects have a more notable influence on the attached $u'^-$ structures, and the deviation from the linear relationship between $l_x$ and $l_{\!y}$ increases (figure 8 b). Moreover, the intense $T'$ structures are more sensitive to the wall temperature condition. Compared with the adiabatic case, here, the second peak in the distribution of the p.d.f.s shifts upward along with the rise of the small gradient region of temperature mentioned previously and it merges partially with the original upper peak, as shown in figure 8(d). Taking together the results of compressible channel flows (Li et al. Reference Li, Fu and Xu2025) and compressible boundary layer flows with $T_r/T_w \approx 0.5$ , $0.79$ (Yuan et al. Reference Yuan, Tong, Li, Chen and Dong2022), $1.0$ and $1.58$ (Yuan et al. Reference Yuan, Tong, Li, Chen and Dong2022), which correspond to ultimate, moderate, mild, no wall cooling effects and wall heating effects, the overall variation trend of the shape characteristics of attached $T'^+$ structures with changing wall temperature becomes clear. At first, when the wall is heated, the wall-normal gradient of temperature is negative, with moderate heat transfer at the wall and only one peak occurring in the distribution of j.p.d.f.s of length and height. When close to the adiabatic condition, the existence of a very small wall-normal temperature gradient region leads to a second near-wall peak. As the wall becomes cooler, this second peak shifts upward until it finally merges with the original peak. Therefore, the double-peak distribution of j.p.d.f.s only occurs when $T_r/T_w$ does not deviate significantly from unity. In addition, the wall cooling effects strengthen the elongation of the near-wall attached $T'^-$ and detached $T'$ structures in the streamwise direction, leading to a distinct variation trend in the near-wall region, as shown in figure 8(e, f). Based on the SRA, near the wall, since both the wall-normal gradients of temperature and velocity are positive, $T'$ and $u'$ are positively correlated, which can also be observed by comparing the near-wall regions of figure 8(b,e). As the wall cooling effect continuously strengthens, this elongation effect extends throughout the entire domain; see Li et al. (Reference Li, Fu and Xu2025).

When considering the aspect ratio of width to height, as figure 9 displays, the linear relationship still remains for intense $u'$ structures and is not affected by wall temperature. However, figure 9(d,e) shows a similar phenomenon to the aspect ratio of length to height for attached $T'$ structures here, and the variation trend with changing wall temperature is also identical.

Figure 9.

Joint p.d.f.s of the structure width and height for (a,b,c) intense $u'$ structures and (d,e, f) intense $T'$ structures in Set T05S. Panels (a) and (d) correspond to attached positive structures, panels (b) and (e) to attached negative structures, and panels (c) and (f) to detached structures. The equation of the dashed line, which represents the variation trend, is provided as well. The contour levels are logarithmically distributed.

Figure 9. Long description

Panel A: A scatter plot showing the relationship between the structure width (l_z^+) and height (l_y^+) for attached positive structures. The horizontal axis represents the structure width (l_z^+) with a range from 10^1 to 10^3. The vertical axis represents the structure height (l_y^+) with a range from 10^1 to 10^3. The plot includes a dashed line representing the variation trend with the equation l_z^+ = l_y^+. The contour levels are logarithmically distributed. Panel B: A scatter plot showing the relationship between the structure width (l_z^+) and height (l_y^+) for attached negative structures. The horizontal axis represents the structure width (l_z^+) with a range from 10^1 to 10^3. The vertical axis represents the structure height (l_y^+) with a range from 10^1 to 10^3. The plot includes a dashed line representing the variation trend with the equation l_z^+ = l_y^+. The contour levels are logarithmically distributed. Panel C: A scatter plot showing the relationship between the structure width (l_z^+) and height (l_y^+) for detached structures. The horizontal axis represents the structure width (l_z^+) with a range from 10^1 to 10^3. The vertical axis represents the structure height (l_y^+) with a range from 10^1 to 10^3. The plot includes a dashed line representing the variation trend with the equation l_z^+ = 0.6(l_y^+)^1.1. The contour levels are logarithmically distributed. Panel D: A scatter plot showing the relationship between the structure width (l_z^+) and height (l_y^+) for attached positive structures. The horizontal axis represents the structure width (l_z^+) with a range from 10^1 to 10^3. The vertical axis represents the structure height (l_y^+) with a range from 10^1 to 10^3. The plot includes a dashed line representing the variation trend with the equation l_z^+ = 1.3(l_y^+). The contour levels are logarithmically distributed. Panel E: A scatter plot showing the relationship between the structure width (l_z^+) and height (l_y^+) for attached negative structures. The horizontal axis represents the structure width (l_z^+) with a range from 10^1 to 10^3. The vertical axis represents the structure height (l_y^+) with a range from 10^1 to 10^3. The plot includes a dashed line representing the variation trend with the equation l_z^+ = 5(l_y^+)^0.8. The contour levels are logarithmically distributed. Panel F: A scatter plot showing the relationship between the structure width (l_z^+) and height (l_y^+) for detached structures. The horizontal axis represents the structure width (l_z^+) with a range from 10^1 to 10^3. The vertical axis represents the structure height (l_y^+) with a range from 10^1 to 10^3. The plot includes a dashed line representing the variation trend with the equation l_z^+ = 0.7(l_y^+)^1.1. The contour levels are logarithmically distributed.

3.3. Conditional average fields

To further investigate the differences between intense $u'$ and $T'$ structures, the conditionally averaged fields of velocity and temperature fluctuations around attached $u'^-$ structures are conducted as follows:

(3.3) \begin{equation} \left \langle {u'} \right \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, $\boldsymbol{r}(r_x,r_{\!y},r_z)=\boldsymbol{x}-\boldsymbol{x}_c$ is the position vector with respect to the box centre, $\boldsymbol{R}=\boldsymbol{r}/y_c$ is the rescaled vector and $\left \langle \, \right \rangle$ denotes the conditional average. As shown in (3.3), the fields are normalised by $y_{c,i}^3$ . Since the length, width and height of the vortex cluster are nearly proportional to the height of the structure’s centre as the scaling laws suggest, this normalisation effectively accounts for the varying sizes of the structures.

Figure 10.

Three-dimensional conditionally averaged velocity and temperature fluctuation fields associated with attached $u'^-$ structures for (a) Case M2T10 and (b) Case M2T05. Blue and red isosurfaces denote the $u'$ and $T'$ structures, respectively. The central region features negative $u'$ and positive $T'$ , flanked by lateral structures representing positive $u'$ and negative $T'$ .

Figure 10 presents the conditionally averaged isosurfaces of the velocity and temperature fluctuations for cases M2T10 and M2T05. The conditionally averaged fields centred on the attached $u'^-$ structures exhibit partial similarities to those centred on vortical structures in channel flows (Li et al. Reference Li, Fu and Xu2025), although substantial differences remain. For the adiabatic-wall case, as illustrated in figure 10(a), the central structure of negative velocity fluctuation, $u'^-$ , is accompanied by a positive temperature fluctuation, $T'^+$ , whereas the flanking positive velocity fluctuations, $u'^+$ , are associated with negative temperature fluctuations, $T'^-$ . The isosurfaces of the velocity and temperature fluctuations display similar topological features; both exhibit a distinct upward inclination in the streamwise direction, implying comparable evolutionary mechanisms. However, as Li et al. (Reference Li, Fu and Xu2025) reported, the temperature isosurfaces on the flanks of the conditionally averaged field in channel turbulence are almost completely attached to the wall, with no evident lift-up. Moreover, figure 10(b) reveals that under the cold-wall condition, the isosurfaces of the temperature fluctuations exhibit a global elevation at their lower bounds relative to the velocity fluctuations. This phenomenon arises from the presence of a near-wall temperature peak in the cold-wall case, which causes the wall-normal temperature gradient to switch from positive to negative. Consequently, the correlation between the velocity and temperature fluctuations transitions from positive to negative. As a result, the lower boundary of the positive temperature fluctuations is restricted to the region above the temperature peak, a constraint not imposed on the velocity fluctuations, thereby causing the overall lift-up of the temperature isosurfaces. Furthermore, despite this elevation at the bottom, the velocity and temperature structures still retain certain similarities in their upper regions. Therefore, in contrast to the high degree of similarity observed over the adiabatic wall, the coupling between the velocity and temperature fields is considerably more complex under cold-wall conditions.

4.1. Advection velocity

The group velocity of the coherent structures is investigated, represented by the velocity at the structure centre (as shown in figure 1) in this study. The calculation of the advection velocity does not require capturing the entire lifetime of large-scale structures; instead, a temporally consecutive flow field is necessary. Therefore, Sets T10S and T05S with high temporal resolution are analysed.

Figure 11.

Conditional p.d.f.s $p(U_{adv}|y_c)$ of the advection velocity for (a) intense $u'^+$ structures, (b) intense $u'^-$ structures, (c) intense $T'^+$ structures and (d) intense $T'^-$ structures at the structure centre in Set T10S. The contour levels are logarithmically spaced. The red line in each panel represents the location of the conditional p.d.f. maximum at each height, while the black line denotes the mean streamwise velocity $\bar u$ at the streamwise position corresponding to $\textit{Re}_{\tau }=985$ in Case M2T10.

Figure 11 displays the conditional p.d.f.s $p(U_{adv}|y_c)$ of the advection velocity of the intense fluctuation structures. Compared with the j.p.d.f.s adopted in the previous section, $p(U_{adv}|y_c)$ provides much clearer information at each height, as the number of near-wall structures dominates, making it difficult to observe the characteristics of the advection velocity of structures at higher wall-normal distances in the figure. The location of the $p(U_{adv}|y_c)$ maximum at each height is also plotted in comparison with the mean streamwise velocity $\bar u$ at the streamwise centre of the subdomains with $\textit{Re}_{\tau }=985$ . Since the definition of attached and detached structures for branches depends on their entire lifetime, which cannot be captured from two consecutive fields, the following analyses will not distinguish between them. It can be observed in figure 11 that the advection velocity of intense $u'^+$ and $T'^-$ structures is greater than the mean streamwise velocity, whereas that of intense $u'^-$ and $T'^+$ structures is lower. However, the deviation is not significant above the buffer layer. Therefore, the mean streamwise velocity can be used to approximate the advection velocity of the structures, with the near-wall region corrected by a constant value as introduced in § 2.3. These near-wall advection velocities for intense $u'$ and $T'$ structures are approximately $9.5u_\tau$ and $11.5u_\tau$ , respectively, where $u_\tau$ is the friction velocity.

Figure 12.

Conditional p.d.f.s $p(U_{adv}|y_c)$ of the advection velocity for (a) intense $u'^+$ structures, (b) intense $u'^-$ structures, (c) intense $T'^+$ structures and (d) intense $T'^-$ structures at the structure centre in Set T05S. The contour levels are logarithmically spaced. The red line in each panel represents the location of the conditional p.d.f. maximum at each height, while the black line denotes the mean streamwise velocity $\bar u$ at the streamwise position corresponding to $\textit{Re}_{\tau }=985$ in Case M2T05.

Figure 12. Long description

Four line graphs depict the conditional probability density functions of advection velocity for different structures at the structure center in Set T05S. Panel A: The line graph shows the conditional p.d.f. of the advection velocity for intense structures. The x-axis is labeled y_c^+ and ranges from 10^0 to 10^3. The y-axis is labeled U_adv and ranges from 0 to 1.25. The contour levels are logarithmically spaced. The red line represents the location of the conditional p.d.f. maximum at each height, while the black line denotes the mean streamwise velocity. Panel B: The line graph shows the conditional p.d.f. of the advection velocity for intense structures. The axes and contour levels are the same as in Panel A. The red and black lines are also present. Panel C: The line graph shows the conditional p.d.f. of the advection velocity for intense structures. The axes and contour levels are the same as in Panel A. The red and black lines are also present. Panel D: The line graph shows the conditional p.d.f. of the advection velocity for intense structures. The axes and contour levels are the same as in Panel A. The red and black lines are also present.

The wall cooling effects on intense $T'$ structures can be observed in figure 12. Due to the near-wall temperature peak, the correlation between $u'$ and $T'$ reverses when passing through it. Therefore, the advection velocity undergoes intense variation around $y^+ \approx 20$ . Taking figure 12(c) as an example, the relatively high near-wall advection velocity of intense $T'^+$ structures decreases to below the mean velocity $\bar u$ , similar to what is observed for intense $u'^+$ structures. The difference is that above the temperature peak, the advection velocity of intense $T'^+$ structures is higher than that of intense $u'^+$ structures. In addition, these near-wall advection velocities for intense $u'$ and $T'$ structures are approximately $11u_\tau$ and $13.5u_\tau$ , respectively, slightly higher than those in the adiabatic case.

As the current study is restricted to a single Mach number, the influence of compressibility on the advection velocity of the structures cannot be directly isolated. However, Zhou et al. (Reference Zhou, Wang, Zhang, Huang and Xu2024) noted that the near-wall advection velocity of dilatational structures is approximately $16u_\tau$ , considerably higher than that of the intense $u'$ and $T'$ structures documented in the present work. It is therefore anticipated that as the Mach number increases, the enhanced contribution of dilatational structures will induce measurable deviations near the wall. Furthermore, because the outer region is populated by LSMs and VLSMs which are inherently insensitive to Mach number effects (Ringuette et al. Reference Ringuette, Wu and Martín2008), its advection characteristics are expected to remain unaffected.

4.2. Lifetime

The lifetime $L_t$ is defined as the duration of an evolution branch of a specific structure. Different from Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014), this study also includes branches that originate from splitting or terminate through merging, as splitting and merging are also regarded as mechanisms of birth and death, respectively, apart from the natural generation or dissipation mechanisms within the flow field. To investigate the lifetime, all four sets introduced in table 2 are used. The lifetimes of structures with centre height lower than $0.2\delta _{\textit{in}}$ are statistically calculated from Sets T10S and T05S, as these two sets have higher time resolution, enabling complete identification of near-wall structure evolution. However, for structures with centre height greater than $0.2\delta _{\textit{in}}$ , a long streamwise domain is required to ensure that the advection of the structures during their lifetimes is fully captured. Therefore, Sets T10L and T05L are used. The previous discussion presents an additional challenge in the study of boundary layer flows compared with channel flows. Combining these two results, the statistical results of the lifetimes of all structures are obtained, as shown in figures 13 and 14. Similar to Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014), percentiles of the data are used to provide a clearer representation.

Figure 13.

Lifetime $L_t$ of the (a) detached $u'$ structures, (b) attached $u'$ structures, (c) detached $T'$ structures and (d) attached $T'$ structures in Case M2T10. The lifetimes of structures with $y_c/\delta _{\textit{in}}\lt 0.2$ are statistically analysed from Set T10S, while those with $y_c/\delta _{\textit{in}}\gt 0.2$ are analysed from Set T10L. Panels (a) and (c) display the 95th (top) and 90th (bottom) percentiles of lifetimes as a function of centre height, represented by circles, while panels (b) and (d) show the 90th percentile. The slope of the variation trend is also shown.

Figure 13 displays the lifetime of structures in Case M2T10. In general, the lifetime increases as detached structures move away from the wall or as the height of attached structures increases, with the exception of a small lifetime peak observed around $y/\delta _{\textit{in}} \approx 0.1$ for detached structures. For all these types of structures, an approximately linear relationship between the length scale and the lifetime is observed, a finding also reported by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014). However, because the streamwise domain is long in Set T10L, it is inconvenient to use viscous units to non-dimensionalise the lifetime and the length scale, as $\textit{Re}_{\tau }$ and $\delta$ vary significantly. Comparing the intense $u'$ and $T'$ structures, it is observed that the detached $u'$ and attached $T'$ structures exhibit steeper variation slopes. It should be noted that the lifetime of large-scale structures is extremely difficult to estimate accurately using statistical methods, as the sample size is severely limited by the spatial extent of the current computational domain and the total time range. Moreover, increasing the domain size would require simulations at higher $\textit{Re}_{\tau }$ , while extending the time range demands significantly greater storage capacity – both of which are strongly constrained by the capacity of computation and storage resource. Therefore, for these structures, only a statistical trend is presented in this study.

Figure 14.

Lifetime $L_t$ of the (a) detached $u'$ structures, (b) attached $u'$ structures, (c) detached $T'$ structures and (d) attached $T'$ structures in Case M2T05. The lifetimes of structures with $y_c/\delta _{\textit{in}}\lt 0.2$ are statistically analysed from Set T05S, while those with $y_c/\delta _{\textit{in}}\gt 0.2$ are analysed from Set T05L. Panels (a) and (c) display the 95th (top) and 90th (bottom) percentiles of lifetimes as a function of centre height, represented by circles, while panels (b) and (d) show the 90th percentile. The slope of the variation trend shown as a dashed line is also shown.

Figure 14. Long description

Panel A: A scatter plot showing the lifetimes of detached structures as a function of center height. The vertical axis is labeled ‘L_t / (δ_in / u_∞)’ and the horizontal axis is labeled ‘y_c / δ_in’. The plot includes two sets of data points represented by circles, with the top set indicating the 95th percentile and the bottom set indicating the 90th percentile. Two lines with slopes of 1.3 and 0.78 are also shown. Panel B: A scatter plot showing the lifetimes of attached structures as a function of a different variable. The vertical axis is labeled ‘L_t / (δ_in / u_∞)’ and the horizontal axis is labeled ‘l_y / δ_in’. The plot includes data points represented by circles and a dashed line with a slope of 0.81. Panel C: A scatter plot showing the lifetimes of detached structures as a function of center height. The vertical axis is labeled ‘L_t / (δ_in / u_∞)’ and the horizontal axis is labeled ‘y_c / δ_in’. The plot includes two sets of data points represented by circles, with the top set indicating the 95th percentile and the bottom set indicating the 90th percentile. Two lines with slopes of 0.94 and 0.52 are also shown. Panel D: A scatter plot showing the lifetimes of attached structures as a function of a different variable. The vertical axis is labeled ‘L_t / (δ_in / u_∞)’ and the horizontal axis is labeled ‘l_y / δ_in’. The plot includes data points represented by circles and a dashed line with a slope of 2.3.

Figure 15.

Sample structures at the time of (a) birth and (b) death in Set T10S. Blue and orange structures represent intense $u'$ and $T'$ structures, respectively. The positions of the structures have been shifted for better visualisation.

The wall cooling effect significantly extends the region with the linear relation into the near-wall region and increases the slope for temperature structures, while the intense $u'$ structures remain almost unchanged as figure 14 displays. This phenomenon reveals the different sustaining mechanisms underlying the coherent structures of temperature and streamwise velocity, which were reported by Li et al. (Reference Li, Fu and Xu2025) in channel flows and will be further investigated in future work.

The investigation of the lifetime of structures in the current study provides an important time scale for the compressible turbulent boundary layer flows, which can be used to understand the dynamics of more complex flows, such as particle-laden flows. Jie et al. (Reference Jie, Cui, Xu and Zhao2022) argued that particle organisation is more naturally parametrised by a Stokes number based on the lifetime of the sweep/ejection structures responsible for wall-normal transport than by a time scale based solely on viscous or Kolmogorov units. The present observation that the lifetime of coherent structures increases approximately linearly with their wall-normal distance has a direct implication for the time scale governing inertial-particle response in turbulent boundary layers.

4.3. Birth and death

The birth and death are two critical stages in the life cycle of a coherent structure. This study investigates the shapes and positions of the structures at these two stages. Figure 15 illustrates selected structures in Set T10S at the moments of their formation and dissipation. It is evident that the streamwise extent does not consistently dominate the overall dimensions of the structures. It should be noted that structures can form naturally from the flow and can also arise from the splitting of a larger structure; both processes are regarded as birth. Accordingly, dissipation within the flow and merging with another structure are both considered death events. The differences between these two birth mechanisms and the corresponding death mechanisms will also be examined. Here, Set T10S is used as an example because the complete life cycle is not required and a higher time resolution is preferred. The p.d.f.s of the length-to-height and width-to-height aspect ratios, as well as the centre of structures at the time of structure birth and death are shown in figure 16. Clear unimodal distributions of the length-to-height and width-to-height aspect ratios are observed for both intense $u'$ and $T'$ structures, while the p.d.f. of the structure centre exhibits a bimodal distribution with two peaks located in the near-wall region. It can be observed that these characteristics exhibit notable differences at the times of birth and death. To quantitatively compare the differences, the values of the aspect ratios and centre positions at the peak of the p.d.f. are summarised in table 5, together with the results of Set T05S. In the short sets, the variation of $\textit{Re}_{\tau }$ is relatively small and the viscous units at the midpoint of the subdomain are used to non-dimensionalise the wall-normal distance of the structure centre.

Figure 16.

P.d.f.s of the (a,d) length-to-height and (b,e) width-to-height aspect ratios, as well as (c, f) the centre of structures at the time of structure birth and death in Set T10S. Panels (a), (b) and (c) correspond to intense $u'$ structures, while panels (d), (e) and (f) correspond to intense $T'$ structures. Solid lines and dashed lines represent birth time and death time, respectively.

Figure 16. Long description

The image contains six line graphs. Panel A shows the probability density function (P.d.f.) of the length-to-height aspect ratio (l_x/l_y) for intense structures, with solid and dashed lines representing birth and death times, respectively. The x-axis is labeled l_x/l_y and ranges from 10^-2 to 10^1, while the y-axis is labeled P.d.f. and ranges from 0 to 0.8. Panel B displays the P.d.f. of the width-to-height aspect ratio (l_z/l_y) for intense structures, with the same axis labels and ranges as Panel A. Panel C illustrates the P.d.f. of the center of structures (y_c^+) at the time of structure birth and death for intense structures, with the x-axis labeled y_c^+ ranging from 10^0 to 10^3 and the y-axis labeled P.d.f. ranging from 0 to 20. Panel D shows the P.d.f. of the length-to-height aspect ratio (l_x/l_y) for intense structures, similar to Panel A but with different data. Panel E displays the P.d.f. of the width-to-height aspect ratio (l_z/l_y) for intense structures, similar to Panel B but with different data. Panel F illustrates the P.d.f. of the center of structures (y_c^+) at the time of structure birth and death for intense structures, similar to Panel C but with different data. Each panel uses solid lines for birth time and dashed lines for death time.

Table 5.

Location of the maximum value in the p.d.f. of the length-to-height and width-to-height aspect ratios, as well as the centre of structures at the time of their birth and death. The p.d.f. of the structure centre exhibits two distinct peaks, labelled as ’inner’ and ’outer’. The results of Sets T10S and T05S are presented separately, with values distinguished by a comma.

Table 5. Long description

A table with 8 rows and 5 columns comparing the location of the maximum value in the probability density function of the length-to-height and width-to-height aspect ratios, as well as the center of structures at the time of their birth and death. The columns are labeled as Type, lx/ly, lz/ly, yc+ inner, and yc+ outer. The table presents values for different types of structure events, including Birth (Flow), Birth (Splitting), Death (Flow), Death (Merging), and their counterparts for Sets T10S and T05S. Each row provides specific values for the aspect ratios and center positions at the peak of the probability density function. The values are distinguished by a comma for Sets T10S and T05S. Row 1: u Birth (Flow), 0.70, 0.64, 0.55, 0.53, 5.31, 4.95, 19.2, 21.3. Row 2: u Birth (Splitting), 0.93, 0.86, 0.76, 0.80, 6.44, 6.79, 24.0, 27.9. Row 3: u Death (Flow), 0.70, 0.66, 0.58, 0.59, 4.57, 4.51, 24.7, 26.4. Row 4: u Death (Merging), 0.85, 0.89, 0.74, 0.78, 6.26, 6.49, 25.8, 29.1. Row 5: T Birth (Flow), 0.45, 0.48, 0.79, 0.85, 6.93, 3.60, 28.7, 11.9. Row 6: T Birth (Splitting), 0.72, 0.87, 0.90, 0.92, 8.20, 4.52, 33.2, 14.6. Row 7: T Death (Flow), 0.58, 0.67, 0.77, 0.75, 6.64, 3.89, 33.0, 14.9. Row 8: T Death (Merging), 0.63, 0.82, 0.84, 0.81, 8.28, 4.84, 38.4, 14.4.

First, the structure shape is examined. It can be observed in table 5 that most structures have the largest extent in height compared with the length and width at birth and death, as both aspect ratios for all cases and all structure types are less than unity. However, for intense $u'$ structures, the length exceeds the width, whereas for temperature structures, the opposite is observed. The precise underlying mechanisms remain incompletely understood. To fully elucidate this phenomenon, further investigations integrating the energy mechanisms detailed by Li et al. (Reference Li, Fu and Xu2025) with transient-growth dynamics may be required. Considering the differences in birth and death mechanisms, both splitting and merging tend to generate structures with larger length-to-height and width-to-height aspect ratios compared with those from natural generation and dissipation in the flow, respectively. As discussed in § 3.2, the length-to-height ratio tends to be larger than 1 and the structure width is almost equal to the structure height. Since the ratio of the structures at birth or death is very small at any given time, these statistical results do not provide complementary information. Instead, they provide new insights into the structure shapes at these two special times. In addition, the cooling wall appears to have notable influences on the length-to-height ratio of the $T'$ structures at death, which is not the focus of this study.

As for the wall-normal position of the structures, both splitting and merging tend to result in higher structure positions compared with those from natural generation and dissipation in the flow, which is reasonable since both the splitting and merging processes involve larger structures, and as Townsend’s attached-eddy hypothesis (Townsend Reference Townsend1976) suggests, larger structures are located farther away from the wall. Moreover, in the adiabatic case, intense $u'$ structures tend to be closer to the wall at birth and death, whereas when the wall is cooling, the birth and death locations of intense $T'$ structures move towards the wall and become closer to the wall than those of intense $u'$ structures in Case M2T05. Results show that intense $T'$ structures are more sensitive to the thermal boundary conditions on the wall.

5.1. Position

The maximum and minimum wall-normal distances of the points contained within the structures, along with the structure centre (shown in figure 1), are analysed to provide a more complete description of the evolution of the structure position. The maximum height of the structure centre during its entire lifetime is selected as the reference length scale. Here, the streamwise and spanwise positions are not taken into consideration.

Figure 17.

Joint p.d.f.s of the life stage and the position characteristics of the attached structures in Set T10L, including the (a,d) central, (b,e) maximum and (c, f) minimum wall-normal distances. Panels (a), (b) and (c) correspond to intense $u'$ structures, and panels (d), (e) and (f) correspond to intense $T'$ structures.

Figure 17 displays the j.p.d.f.s of the life stage and the position characteristics of the attached structures in Set T10L. The results are quite similar between intense $u'$ and $T'$ structures. As can be observed in figure 17(a,d), most structures reach their maximum centre height at death time, while some attain it at birth time. This is reasonable, as the structures may reach their highest position more than once during their lifetime. A rising trend can also be identified from $y_c/y_{c,\textit{max}}=0.4$ at birth time to the highest position at death time. Therefore, the evolution of the structure centre can generally be divided into two categories: one is approximately symmetric, descending from the highest position, but eventually rising again to the peak at death; the other originates at a lower location and rises continuously throughout its lifetime. However, the maximum wall-normal distance (figure 17 b,e) is likely to slightly decrease in the early stage for many structures and remains nearly constant around $y_{\textit{max}}/y_{c,\textit{max}}=1.18$ , while the lowest part of the structures (figure 17 c, f) tends to remain attached to the wall for most of their lifetime. Based on the above-mentioned analyses, the most common complete position evolution can be described as follows: the wall-normal distances of the top and bottom of the structure remain nearly constant and the structure stays attached to the wall for most of its lifetime, while the structure centre undergoes a downward and then a rising stage. It should be noted that since large-scale structures account for a small proportion among all structures, this common evolution pattern may not apply to them, and further investigation is required after solving the challenge of storing a significantly larger number of long subdomains of the flow field than those used in this study. From the process, the evolution of the structure shape can also reveal that the variation of the structure centre is not caused by extension of the top or bottom of the structure in the wall-normal direction, but is instead related to elongation in the other two directions, which will be discussed in § 5.2.

Figure 18.

Joint p.d.f.s of the life stage and the position characteristics of the attached structures in Set T05L, including the (a,d) central, (b,e) maximum and (c, f) minimum wall-normal distances. Panels (a), (b) and (c) correspond to intense $u'$ structures, and panels (d), (e) and (f) correspond to intense $T'$ structures.

When the wall becomes cooling, it can be observed from figure 18 that the characteristics of the position evolution are almost the same, except for the minimum wall-normal distance of intense $T'$ structures. Due to the existence of a peak in the wall-normal distribution of temperature, the bottom of the structure tends to be located at $y_{\textit{min}}/y_{c,\textit{max}}=0.045$ , rather than on the wall, as shown in figure 18(f). According to the definition of attached structures of temporal evolution in this study, as introduced in § 2.3, a branch is considered attached if its corresponding structure’s bottom reaches the wall at least once during its lifetime. Therefore, unlike the attached $u'$ structures whose bottoms tend to remain attached to the wall, the bottoms of the attached $T'$ structures tend to detach from the wall after attachment and remain near the location of the near-wall temperature peak. Similar results have been observed in figure 10(b) in § 3.3. The effect of the mean temperature gradient dominates the overall spatial distribution of the intense $T'$ structures in the near-wall region.

5.2. Shape

To investigate the evolution of the overall spatial dimensions (i.e. length, height and width), their j.p.d.f.s with respect to the life stages are estimated, as shown in figures 19 and 20, with particular emphasis given to the general growth and stretching of the structures. The maximum length, height and width are used as the normalisation scales for the three directions, respectively. In Case M2T10, as figure 19(a,d) shows, both the intense $u'$ and $T'$ structures tend to have $l_x/l_{x,max} \approx 0.35$ , indicating that their length is elongated to nearly three times the original size. When the structures disappear, however, their length becomes much smaller, although the intense $T'$ structures remain relatively longer. The tendency of length variation is to increase during the first half of the life stage and then decrease rapidly. Considering the evolution of structure height, intense $u'$ structures exhibit an additional peak in the p.d.f. distribution at birth time, around $l_{\!y}/l_{y,\textit{max}} \approx 0.5$ , as shown in figure 19(b), whereas intense $T'$ structures tend to form at nearly their maximum height.

Figure 19.

Joint p.d.f.s of the life stage and the shape characteristics of the attached structures in Set T10L, including (a,d) length, (b,e) height and (c, f) width. Panels (a), (b) and (c) correspond to intense $u'$ structures, and panels (d), (e) and (f) correspond to intense $T'$ structures.

Figure 20.

Joint p.d.f.s of the life stage and the shape characteristics of the attached structures in Set T05L, including (a,d) length, (b,e) height and (c, f) width. Panels (a), (b) and (c) correspond to intense $u'$ structures, and panels (d), (e) and (f) correspond to intense $T'$ structures.

Since the difference between velocity and temperature is not the main focus in this study, the underlying mechanism will be investigated in the future work. As discussed in § 5.1, the bottom of the structure tends to remain attached to the wall, so the evolution of structure height results from a small decrease in the position of the top point of the structure. It can also be observed in figure 19(b,e) that the probability of structures with small height at death time remains moderate, indicating a distinct death shape. Furthermore, the evolution of structure width is relatively symmetric for intense $u'$ structures, as shown in figure 19(c), with $l_z/l_{z,\textit{max}} \approx 0.5$ at both birth and death times, whereas for intense $T'$ structures, larger width at birth and smaller width at death are more likely to appear. In addition, the p.d.f. of the life stage at which the maximum length, height and width is reached appears relatively uniform despite the birth and death time.

Figure 20 displays the shape evolution in Case M2T05 with wall cooling effects. Compared with the adiabatic case, the maximum locations of the j.p.d.f.s for length and width shift downward for both intense $u'$ and $T'$ structures. This is because the cooled wall elongates the structures in both the streamwise and spanwise directions (Li et al. Reference Li, Fu and Xu2025), resulting in increased maximum length and width. Moreover, an additional peak appears in the p.d.f.s of height at death around $l_{\!y}/l_{y,\textit{max}} \approx 0.15$ , as shown in figure 20(b,e). Therefore, in this case, at death time, the intense $T'$ structures become significantly smaller in all three directions, which is not the case for intense $u'$ structures.