4.1. Conditionally averaged field based on extreme events

For identifying the coherent structures associated with specific extreme events, the volumetric conditional average is commonly used (Pan & Kwon Reference Pan and Kwon2018; Cardesa et al. Reference Cardesa, Monty, Soria and Chong2019; Guerrero et al. Reference Guerrero, Lambert and Chin2020; Tong et al. Reference Tong, Dong, Lai, Yuan and Li2022). The conditionally averaged quantity is denoted as $\langle \bullet \rangle (\boldsymbol {x},t)$. Referring to Pan & Kwon (Reference Pan and Kwon2018) and Tong et al. (Reference Tong, Dong, Lai, Yuan and Li2022), the size of the average box is selected as $-300<\Delta x^+<300$ and $0< y^+<300$ in streamwise and wall-normal directions, respectively, as well as the whole spanwise domain for the sake of performing Helmholtz decomposition in the following. The indicative extreme $\tau _w^\prime$ and $q_w^\prime$ are set at $\Delta x^+=0$, $\Delta z^+=0$ to locate the average box. The subscripts $\langle \bullet \rangle _{EP}$ and $\langle \bullet \rangle _{EN}$ represent the conditionally averaged quantities associated with EP and EN events, respectively.

Table 5 lists the parameters and results of the conditional average. The number of extreme events $n_E$ exceeds $10^6$, guaranteeing the convergence of the conditional average. The ratios of the conditionally averaged skin-friction coefficient to the overall skin-friction coefficient $\langle C_f\rangle _{EP}/C_f$ are approximately the same for both cases, while the ratios of the conditionally averaged heat-transfer coefficient to the overall heat-transfer coefficient $\langle C_h\rangle _{EP}/C_f$ for M8T05 is much larger than that for M2T05, also implying the more intense extreme events in the $q_w^\prime$ field for M8T05. Furthermore, associated with extremely negative events, $\langle C_f\rangle _{EN}$ and $\langle C_h\rangle _{EN}$ for M8T05 become negative values, implying flow states of backflow and reverse heat transfer in the sense of conditional average. According to the conditionally averaged coefficients, it can be expected that the transports of momentum and energy in extreme events are much more intense than those in mean flow. Due to the strong positive correlation of $\tau _w^\prime$ and $q_w^\prime$ for M2T05, the fluctuations that coexist in extreme $\tau _w^\prime$ and $q_w^\prime$ events occupy the majority, resulting in the percentages of $86\,\%$ and $76\,\%$ for EP and EN events, respectively. These percentages indicate that the same flow structures, i.e. near-wall streaks, play a dominating role in both extreme $\tau _w^\prime$ and $q_w^\prime$ events. For M8T05, the overlapping percentages decrease significantly for EP events and slightly for EN events, suggesting there are different dominating flow structures associated with extreme $\tau _w^\prime$ and $q_w^\prime$ events.

Table 5. Parameters and results of conditional average: $n_E$ is the sample number of extreme events; $C_f$ and $C_h$ are the skin-friction and heat-transfer coefficients, respectively; $\langle C_f\rangle$ and $\langle C_h\rangle$ indicate the conditionally averaged skin-friction and heat-transfer coefficients, respectively; $P$ is the percentage of the events which coexist both in extreme $\tau _w^\prime$ and $q_w^\prime$ events.

Conditionally averaged $\langle \tau _w^\prime \rangle$ and $\langle q_w^\prime \rangle$ fields in the vicinity of extreme events at $\Delta x^+=0$ and $\Delta z^+=0$ are plotted in figure 8, with the white lines circling the regions belonging to extreme WSS and WHF fluctuations. For M2T05, the $\langle \tau _w^\prime \rangle$ and $\langle q_w^\prime \rangle$ fields are similar to each other. Regarding EP events, the fluctuation patterns are of elongated shapes, and the extreme regions extend to a relatively large area. Regarding EN events, the areas of extreme regions are quite small, and high-value regions are observed at two sides. However, for M8T05, only in the $\langle \tau _w^\prime \rangle _{EP}$ field is an elongated structure present. All of the extreme regions circled with white lines show spotty shapes. The streamwise structures in the $\langle \tau _w^\prime \rangle _{EN}$ field seem to be split around the extreme region, with the centre structure extending in the spanwise direction. As far as the $\langle q_w^\prime \rangle$ fields are concerned, the fine-scale alternating positive and negative patterns are shown, reminding us of the dominant role of APNSs in the extreme events of $q_w^\prime$. For the three-dimensional coherent structures associated with extreme events, they will be discussed in detail with a combination of the conditional skin-friction and heat-transfer decompositions in §§ 4.3 and 4.4.

Figure 8. Conditionally averaged $\langle \tau _w^\prime \rangle$ and $\langle q_w^\prime \rangle$ fields around extreme events at $\Delta x^+=0$ and $\Delta z^+=0$ of (a–d) M2T05 and (e–h) M8T05. The left and right panels stand for EP and EN events, respectively. The white lines circle the regions belonging to extreme WSS and WHF fluctuations.

4.2. Conditional skin-friction and heat-transfer decompositions

The skin-friction and heat-transfer decompositions based on integral identities are able to provide valuable insight into transport mechanisms contributing to WSS and WHF (Wenzel et al. Reference Wenzel, Gibis and Kloker2022; Xu et al. Reference Xu, Wang and Chen2022). Based on integrating the conservation laws of different quantities, such as momentum (Fukagata et al. Reference Fukagata, Iwamoto and Kasagi2002), mean kinetic energy (Renard & Deck Reference Renard and Deck2016), mechanical energy (Drela Reference Drela2009), total enthalpy (Wenzel et al. Reference Wenzel, Gibis and Kloker2022), internal energy (Xu et al. Reference Xu, Wang and Chen2022) and so on, there are many kinds of different integral identities. However, all these identities are based on overall averaged equations and are proposed to describe the mean flow and corresponding transport mechanisms; some transient behaviours, such as in extreme events of $\tau _w^\prime$ and $q_w^\prime$ discussed here, cannot be taken into account in isolation yet. For depicting extreme events and demonstrating how coherent structures associated with extreme events contribute to $\tau _w^\prime$ and $q_w^\prime$, we have developed the conditional skin-friction and heat-transfer decompositions based on the conditionally averaged governing equations.

Referring to recently developed skin-friction and heat-transfer decompositions on mean flow (Wenzel et al. Reference Wenzel, Gibis and Kloker2022; Xu et al. Reference Xu, Wang and Chen2022), the FIK-like identity with twofold integration is adopted here, which owns a more intuitive physical interpretation of the underlying mechanisms of transferring momentum and energy than the RD and original FIK decompositions (Wenzel et al. Reference Wenzel, Gibis and Kloker2022). We start from the fully compressible Navier–Stokes equations and use the conditional average on extreme events, so that the corresponding physical mechanisms associated with extreme $\tau _w^\prime$ and $q_w^\prime$ can be particularly assessed. In combination with the continuity equation, the conditionally averaged streamwise-momentum equation can be written as

(4.1)\begin{equation} \left\langle\rho\right\rangle\frac{\partial \tilde{u}}{\partial t}+ \left\langle\rho\right\rangle\tilde{u}_j\frac{\partial \tilde{u} }{\partial x_j}+ \frac{\partial \left\langle\rho\right\rangle \widetilde{u^{\prime \prime} u_j^{\prime \prime}}}{\partial x_j}=-\frac{\partial \left\langle \,p\right\rangle}{\partial x}+\sqrt{\gamma}\frac{ M}{{\textit{Re}}} \frac{\partial \langle\tau_{1 j}\rangle}{\partial x_j}, \end{equation}

where the symbol $\widetilde {(\bullet )}$ denotes the conditional Favre average $\tilde {\phi }=\langle \rho \phi \rangle /\langle \rho \rangle$, the symbol $(\bullet )^{\prime \prime }$ denotes the fluctuation in the context of the conditional Favre average $\phi ^{\prime \prime }=\phi -\tilde {\phi }$. Based on the FIK-like identity, the conditionally averaged skin-friction coefficient can be decomposed as

(4.2)\begin{equation} \left\langle C_f\right\rangle=C_f^M+C_f^T+C_f^P+C_f^B+C_f^U, \end{equation}

specifically,

(4.3)$$\begin{gather} C_f^M=-\frac{1}{q_{\infty} y_b} \int_0^{y_b}\left(y_b-y\right) \langle\rho\rangle \tilde{u}_j \frac{\partial \tilde{u} }{\partial x_j} {{\rm d}y}, \end{gather}$$

(4.4)$$\begin{gather}C_f^T=-\frac{1}{q_{\infty} y_b} \int_0^{y_b}\left(y_b-y\right)\left(\frac{\partial\langle\rho\rangle \widetilde{ u^{\prime \prime} u^{\prime \prime}} }{\partial x}+\frac{\partial\langle\rho\rangle \widetilde{ u^{\prime \prime} w^{\prime \prime}} }{\partial z}\right)+\langle\rho\rangle \widetilde{ u^{\prime \prime} v^{\prime \prime}} {{\rm d}y}, \end{gather}$$

(4.5)$$\begin{gather}C_f^P=-\frac{1}{q_{\infty} y_b} \int_0^{y_b}\left(y_b-y\right) \frac{\partial\langle \,p\rangle}{\partial x} {{\rm d}y}, \end{gather}$$

(4.6)$$\begin{gather}C_f^B=\frac{1}{q_{\infty} y_b} \frac{\sqrt{\gamma} M}{\operatorname{{\textit{Re}}}} \int_0^{y_b}\left(y_b-y\right)\left(\frac{\partial\left\langle\tau_{x x}\right\rangle}{\partial x}+\frac{\partial\left\langle\tau_{x z}\right\rangle}{\partial z}\right)+\left\langle\tau_{x y}\right\rangle {{\rm d}y}, \end{gather}$$

(4.7)$$\begin{gather}C_f^U=-\frac{1}{q_{\infty} y_b} \int_0^{y_b}\left(y_b-y\right)\langle\rho\rangle \frac{\partial \tilde{u} }{\partial t} {{\rm d}y}, \end{gather}$$

where $q_\infty =\rho _{\infty }u_\infty ^2/2$ is the dynamic pressure and $y_b$ denotes the upper bound of integrals. According to the physical meanings of the integrands, each term can be interpreted as: $C_f^M$ is the mean-convection term; $C_f^T$ is the turbulent-convection term; $C_f^P$ is the pressure-gradient term; $C_f^B$ is the boundary-layer term and $C_f^U$ is the unsteady term.

In the same way, based on the conditionally averaged internal energy equation

(4.8)\begin{align} \left\langle\rho\right\rangle\frac{\partial \tilde{T}}{\partial t}+ \left\langle\rho\right\rangle\tilde{u}_j\frac{\partial \tilde{T} }{\partial x_j}+ \frac{\partial \left\langle\rho\right\rangle \widetilde{ T^{\prime \prime} u_j^{\prime \prime}}}{\partial x_j}=-\frac{1}{c_v}\left\langle \,p \frac{\partial u_j}{\partial x_j}\right\rangle+\frac{\sqrt{\gamma}}{c_v} \frac{ M}{ {\textit{Re}}}\left\langle \tau_{i j} \frac{\partial u_i}{\partial x_j}\right\rangle+ \frac{1}{c_v \alpha} \frac{\partial}{\partial x_j}\left\langle\mu \frac{\partial T}{\partial x_j}\right\rangle\!, \end{align}

the conditionally averaged heat-transfer coefficient can be decomposed as

(4.9)\begin{equation} \left\langle C_h\right\rangle=C_h^M+C_h^T+C_h^P+C_h^V+C_h^B+C_h^U, \end{equation}

specifically,

(4.10)$$\begin{gather} C_h^M=-\frac{c_v}{\beta y_b} \int_0^{y_b}\left(y_b-y\right) \langle\rho\rangle \tilde{u}_j \frac{\partial \tilde{T} }{\partial x_j} {{\rm d}y}, \end{gather}$$

(4.11)$$\begin{gather}C_h^T=-\frac{c_v}{\beta y_b} \int_0^{y_b}\left(y_b-y\right)\left(\frac{\partial\langle\rho\rangle \widetilde{ T^{\prime \prime} u^{\prime \prime}} }{\partial x}+\frac{\partial\langle\rho\rangle \widetilde{ T^{\prime \prime} w^{\prime \prime}} }{\partial z}\right)+\langle\rho\rangle \widetilde{ T^{\prime \prime} v^{\prime \prime}} {{\rm d}y}, \end{gather}$$

(4.12)$$\begin{gather}C_h^P=-\frac{1}{\beta y_b} \int_0^{y_b}\left(y_b-y\right) \left\langle \,p \frac{\partial u_j}{\partial x_j}\right\rangle {{\rm d}y}, \end{gather}$$

(4.13)$$\begin{gather}C_h^V=\frac{1}{\beta y_b} \frac{\sqrt{\gamma} M}{{\textit{Re}}} \int_0^{y_b}\left(y_b-y\right)\left\langle\tau_{i j} \frac{\partial u_i}{\partial x_j}\right\rangle {{\rm d}y}, \end{gather}$$

(4.14)$$\begin{gather}C_h^B=\frac{c_v}{\beta y_b} \frac{1}{c_v \alpha} \int_0^{y_b}\left(y_b-y\right)\left(\frac{\partial}{\partial x}\left\langle\mu \frac{\partial T}{\partial x} \right\rangle +\frac{\partial}{\partial z}\left\langle\mu \frac{\partial T}{\partial z} \right\rangle \right)+\left\langle\mu\frac{\partial T}{\partial y}\right\rangle {{\rm d}y}, \end{gather}$$

(4.15)$$\begin{gather}C_h^U=-\frac{c_v}{\beta y_b} \int_0^{y_b}\left(y_b-y\right)\langle\rho\rangle \frac{\partial \tilde{T} }{\partial t} {{\rm d}y}, \end{gather}$$

where the parameter $\beta$ is defined as $\beta =\rho _{\infty } u_{\infty } c_p(T_r-T_w)$. Similarly, each term can be interpreted according to the physical meanings of the integrands as: $C_h^M$ is the mean-convection term; $C_h^T$ is the turbulent-convection term; $C_h^P$ is the pressure-dilatation term; $C_h^V$ is the viscous-dissipation term; $C_h^B$ is the boundary-layer term and $C_h^U$ is the unsteady term. The detailed derivations of the conditional skin-friction and heat-transfer decompositions are given in Appendix B. Note that compared with the conventional skin-friction and heat-transfer decomposition based on spanwise homogeneous and mean flow (Wenzel et al. Reference Wenzel, Gibis and Kloker2022; Xu et al. Reference Xu, Wang and Chen2022), the main differences are that some spanwise-gradient terms and unsteady terms are still retained in the present decompositions. Particularly, the accuracy of the conditional skin-friction and heat-transfer decompositions has been verified in Appendix C.

4.3. Conditional analysis on extreme events of wall shear stress

In this subsection, the above-introduced volumetric conditionally averaged fluctuation fields, in combination with the proposed decomposition identities, are used to analyse the coherent structures and their corresponding momentum and energy transport mechanisms in extreme events. Figure 9 shows the conditional analysis results on extreme $\tau _w^\prime$ events for M2T05. In figure 9(a), a pair of energetic counter-rotating quasi-streamwise vortices is identified by the isosurface of $Q$-criterion coloured by the streamwise vorticity $\omega _x$. These vortices induce strong sweep motions that facilitate high-speed flow from the outer layer to the inner layer. Consequently, a high-speed streak shown by the isosurface of $u^+=2.5$ coloured by the wall-normal velocity $v$ is centred between the counter-rotating vortices near the wall. This isosurface evolves from thin to wide and to thin again along the streamwise direction, which indicates the local streamwise velocity gradient changes from positive to negative, as shown in the $x$–$y$ plane shifted from $\Delta z^+=0$. In the view of quadrant analysis (Wallace Reference Wallace2016), these structures associated with EP events of $\tau _w^\prime$ induce a strong Q4 event. This flow topology is consistent with those observed in the incompressible TBL (Pan & Kwon Reference Pan and Kwon2018), pipe flow (Guerrero et al. Reference Guerrero, Lambert and Chin2020) and compressible TBL (Tong et al. Reference Tong, Dong, Lai, Yuan and Li2022). These identified coherent structures are also consistent with the conceptual model proposed by Guerrero et al. (Reference Guerrero, Lambert and Chin2020). Due to the orientation of rotation, it can be deduced that the counter-rotating vortex pair is composed of the legs of two individual spanwise-arranged hairpin-like vortices, which are widely present in wall-bounded turbulent flows.

Figure 9. Conditional analysis on extreme events of $\tau _w^\prime$ for M2T05: (a,b) conditionally averaged fluctuation fields, the isosurfaces of high-/low-speed streaks are flooded by $v$, the vortices displayed by the isosurfaces of $Q$-criterion are flooded by $\omega _x$. An $x$–$y$ plane at $\Delta z^+=0$ and a $y$–$z$ plane at $\Delta x^+=0$ are shifted aside for clearer observation. Components of conditionally averaged skin-friction coefficients normalized by $C_f$: (c,e) $C_f^i$; (d,f) $C_f^i(y_b)$ as a function of the upper bound of integrals. Panels (a,c,d) and (b,e,f) are for the EP and EN events, respectively.

The components of conditionally decomposed skin-friction coefficients normalized by $C_f$ are displayed in figure 9(c,e). First, the present sample numbers of extreme events used for decompositions are sufficient to achieve statistical convergence, which has been proved in Appendix D. For a detailed comparison, the components of the mean-convection term $C_f^M$ and the turbulent-convection term $C_f^T$ in three directions are shown. The legend of lines in figure 9(d,f) can refer to figure 9(c,e) in which the colour correspondences are the same. As indicated in figure 9(d), $C_f^i(y_b)$ are independent on the integral upper bounds when $y^+_b>800$, within which the choice of $y_b$ does not affect the integral results. The dominating terms are the streamwise mean-convection term $C_f^{Mx}$, wall-normal mean-convection term $C_f^{My}$, spanwise turbulent-convection term $C_f^{Tz}$ and unsteady term $C_f^{U}$, whose magnitudes reach several times that of $C_f$. Due to the streamwise evolving of the high-speed streak, the near-wall streamwise velocity gradient has a large positive value, producing the coefficient $C_f^{Mx}$ of a large negative integral value. Moreover, it is worth noting that the profile of $C_f^{U}$ seems to be similar to that of $C_f^{Mx}$ but with an opposite sign. Provided with Taylor's frozen-flow model and assuming that the flow is streamwise dominated, it is possible to derive $\partial \tilde {u}/\partial t=-u_c\partial \tilde {u}/\partial x$ with $u_c$ being the convection velocity. If the convection velocity $u_c$ satisfies $u_c=\tilde {u}$, the profiles of $C_f^{Mx}$ and $C_f^{U}$ are expected to be opposite to each other. Actually, the convection velocity is found to be consistent with the mean velocity except for very near the wall (Zhao & He Reference Zhao and He2009; Zhang, Wan & Sun Reference Zhang, Wan and Sun2019). Eventually, the similar profiles of $C_f^{Mx}$ and $C_f^{U}$ could be achieved. The strong sweep motions (Q4) are the key reason to induce the EP events of $\tau _w^\prime$, which transfer amounts of momentum from the outer layer to the inner layer. The large-magnitude negative $\tilde {v}$ produces the coefficient $C_f^{My}$ of a large positive integral value. In addition, the location of EP events is between the spanwise arranged energetic vortex pairs, and the coherent vortex pair in a counter-rotating state can induce strong spanwise turbulent fluctuations. Thus, as a result of the large gradient of Reynolds stress $\widetilde {u^{\prime \prime }w^{\prime \prime }}$ in the spanwise, the coefficient $C_f^{Tz}$ also becomes an important component.

The coherent structures associated with the EN events of $\tau _w^\prime$ are shown in figure 9(b). A hairpin-like vortex following a pair of counter-rotating quasi-streamwise vortices is identified. The pair of counter-rotating vortices could also be regarded as two legs of a hairpin-like vortex. The rotating directions of the vortex pair are opposite to those in the EP events. An elongated low-speed streak is centred between the counter-rotating quasi-streamwise vortices, sustained by strong ejecting motions induced by these vortices. Therefore, the region of the streak which is shown by the isosurface of $u^+=-3$ is under a Q2 event. These identified coherent structures associated with the EN events also agree with the conceptual models proposed by Sheng et al. (Reference Sheng, Malkief and Katz2009) and Guerrero et al. (Reference Guerrero, Lambert and Chin2020). The extreme events tend to be located under the two continuously streamwise-arranged hairpin-like vortices. The dominating terms are partly the same as the terms of EP events: the streamwise mean-convection term $C_f^{Mx}$, wall-normal mean-convection term $C_f^{My}$, spanwise turbulent-convection term $C_f^{Tz}$ and unsteady term $C_f^{U}$. Notably, an additional dominating term is the pressure-gradient term $C_f^P$. Due to the opposite rotating direction and the difference between high- and low-speed streaks, the contributions of $C_f^{My}$ and $C_f^{Tz}$ for EN events are quantitatively opposite to their contributions for EP events. Eventually, the mechanism which contributes most to the EN events is the pressure-gradient term $C_f^P$. The two-part structures of coherent vortices are probably related to the convecting pressure wavepackets as shown in the shifted $x$–$y$ plane. The extreme events are located behind the wave valley, resulting in a large negative value of $C_f^P$.

As for coherent structures associated with the EP events of $\tau _w^\prime$ for M8T05 shown in figure 10(a), the structures are partly similar to the counterparts of M2T05. The energetic counter-rotating quasi-streamwise vortex pair and the high-speed streak centring between the vortex pair are also identified. The biggest difference is that a spanwise-extending bridge vortex located at the EP events connects the vortex pair. This vortex is related to dilatation motions, because it is induced by the upstream expansion and downstream compression motions shown by the isosurface of $\theta ^+=\pm 0.02$. These alternating negative and positive dilatational motions remind us of the presence of APNSs in M8T05 which tend to be embedded in streaks. Despite the slight increasing contribution of other terms, the dominating terms are the same as those in M2T05: the streamwise mean-convection term $C_f^{Mx}$, wall-normal mean-convection term $C_f^{My}$, spanwise turbulent-convection term $C_f^{Tz}$ and unsteady term $C_f^{U}$. The contributions of $C_f^{My}$ and $C_f^{Tz}$ are qualitatively in agreement with those in M2T05, because of the similar counter-rotating quasi-streamwise vortex pair. Due to the strong sweep motion at EP events, the momentum is transferred from the outer layer to the inner layer, and $C_f^{My}$ contributes most to extreme $\tau _w^\prime$. Here, $C_f^{Mx}$ and $C_f^{U}$ show opposite contributions compared to those in M2T05, resulting from the deceleration region located at the buffer and log layer, but the sum of their contributions is still small.

Figure 10. Conditional analysis on extreme events of $\tau _w^\prime$ for M8T05: (a,b) conditionally averaged fluctuation fields, the isosurfaces of dilatation $\theta ^+$ are flooded by itself, the isosurfaces of high-/low-speed streaks are flooded by $v$, the vortices displayed by the isosurfaces $Q$-criterion are flooded by $\omega _x$. An $x$–$y$ plane at $\Delta z^+=0$ and a $y$–$z$ plane at $\Delta x^+=0$ are shifted aside for clearer observation. Components of conditionally averaged skin-friction coefficients normalized by $C_f$: (c,e) $C_f^i$; (d,f) $C_f^i(y_b)$ as a function of upper bound of integrals. Panels (a,c,d) and (b,e,f) are for the EP and EN events, respectively.

The coherent structures associated with the EN events shown in figure 10(b) become more complex. A horseshoe-like vortex is located behind the EN event, and the two legs are also in a counter-rotating state. A brush-shaped low-speed elongated streak is above the EN events sustained by ejection motions, indicating that EN events of $\tau _w^\prime$ for M8T05 are also related to the Q2 event. A similar spanwise-extending vortex is located at the EN events, and it is also induced by strong dilatation motions. However, in contrast to the EP event, the vortex is caught by upstream compression and downstream expansion motions shown by the isosurface of $\theta ^+=\pm 0.04$. The dominating terms are only the streamwise mean-convection term $C_f^{Mx}$, wall-normal mean-convection term $C_f^{My}$ and unsteady term $C_f^{U}$. The magnitudes of these terms are relatively large, and $C_f^{U}$ even reaches nearly $20C_f$. Comparing the coherent structures in other extreme events, we find that there is no quasi-streamwise vortex pair near the EN event; thus, the contribution of the spanwise turbulent-convection term $C_f^{Tz}$ becomes much less important. On account of the acceleration in the buffer and log layers, the streamwise mean-convection term $C_f^{Mx}$ contributes to the EN events the most. This also suggests the correlation between the outer-layer large-scale structures and the inner-layer small-scale APNS. Meanwhile, the Q2 event induces momentum transport from the inner layer to the outer layer by wall-normal mean-convection $C_f^{My}$. These coherent structures make $C_f^{U}$ a dominating term to balance the contributions of $C_f^{Mx}$ and $C_f^{My}$.

Due to the co-existence of strong shear and dilatation motions, the coherent structures associated with extreme events of $\tau _w^\prime$ for M8T05 tend to be more complex than those for M2T05. To disentangle the coupled contributions of shear and dilatational motions, we perform the Helmholtz decomposition to cast the velocity $\boldsymbol {u}$ as the sum of a solenoidal part $\boldsymbol {u}_s$ and an irrotational part $\boldsymbol {u}_d$, which satisfy $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {u}_s=0$ and $\boldsymbol {\nabla } \times \boldsymbol {u}_d=0$, respectively. Based on the velocity field $\boldsymbol {u}$, these two parts can be obtained by solving Poisson equations of the vector potential $\boldsymbol {A}$ and scalar potential $\varphi$:

(4.16a,b)\begin{equation} \nabla^2 \boldsymbol{A}=-\boldsymbol{\nabla} \times \boldsymbol{u},\quad \nabla^2 \varphi=\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}. \end{equation}

Subsequently, the solenoidal and dilatational components can be calculated by

(4.17a,b)\begin{equation} \boldsymbol{u}_s=\boldsymbol{\nabla} \times \boldsymbol{A}, \quad \boldsymbol{u}_d=\boldsymbol{\nabla} \varphi. \end{equation}

Complemented with the boundary conditions (Hirasaki & Hellums Reference Hirasaki and Hellums1970)

(4.18a–c)\begin{equation} \frac{\partial \varphi}{\partial y}=0,\quad \frac{\partial A_y}{\partial y}=0,\quad A_x=A_z=0, \end{equation}

the vector potential $\boldsymbol {A}$ is shown to be unique. Figure 11(a) shows the $\langle u\rangle _{EP}$ field at the $\Delta z^+=0$ plane, with an elongated high-speed region locating near the wall. After the Helmholtz decomposition, the structures are clearly found to be composed of the high-speed streak exhibited in figure 11(c) by the solenoidal component $\langle u_s\rangle _{EP}$ and the APNSs exhibited in figure 11(e) by the dilatational component $\langle u_d\rangle _{EP}$. The high-speed streak denoted by $\langle u_s\rangle _{EP}$ is consistent with the original coherent structures observed in incompressible pipe flow (Guerrero et al. Reference Guerrero, Lambert and Chin2020) and supersonic TBL (Tong et al. Reference Tong, Dong, Lai, Yuan and Li2022) in the EP event of $\tau _w^\prime$; thus, the APNSs denoted by $\langle u_d\rangle _{EP}$ are additional structures caused by the compressible effects in high-Mach-number flow. Nonetheless, the magnitude of APNSs is an order of magnitude smaller than that of the high-speed streak. As for EN events, similarly and more apparently, the structures are also composed of the low-speed streak exhibited in figure 11(d) by $\langle u_s\rangle _{EN}$ and the APNSs exhibited in figure 11(f) by $\langle u_d\rangle _{EN}$. The difference is that the low-speed streak is thicker in the wall-normal direction, and the magnitude of the APNSs becomes larger, which is comparable to the magnitude of the streak. The phases of APNSs are reversed in EN and EP events, implying that EP or EN events are related to opposite dilatational motions.

Figure 11. Contours of the conditionally averaged fluctuation field of extreme $\tau _w^\prime$ events for M8T05 at $\Delta z^+=0$ plane: (a,b) the streamwise velocity $u$; (c,d) the solenoidal component $u_s$ and (d,f) the dilatational component $u_d$. Panels (a,c,e) and (b,d,f) are for EP and EN events, respectively.

To illustrate the ejection and sweep motions induced by these two structures, figure 12 shows contours of conditionally averaged wall-normal velocity at the $y^+=15$ plane. The strong sweep motion is observed near the location of extreme events in figure 12(a). The streak pattern in the $\langle v_s\rangle _{EP}$ field shown in figure 12(c) is more well shaped and extends longer than that in figure 12(a). Two ejecting regions simultaneously induced by the quasi-streamwise vortex pair flank the central sweeping region. The $\langle v_d\rangle _{EP}$ field in figure 12(e) shows the appearance of APNSs also with a relatively small magnitude. As for EN events, the structures in the $\langle v\rangle _{EP}$ field shown in figure 12(b) are split into several parts by weak and strong ejection motions along the $x$ direction, which consist of local ejection spots and elongated streaks. In the $\langle v_s\rangle _{EN}$ field shown in figure 12(d), the upstream ejection streak shrinks at $\Delta x^+\approx 15$ first, and then grows to be elongated. In the $\langle v_d\rangle _{EN}$ field shown in figure 12(f), the strong APNSs are identified, resulting in streamwise alternating sweeping and ejection motions. The spanwise and streamwise length scales of APNSs are consistent with the characteristic wavelength of the inner peak in figure 7(c). The APNSs also enhance ejection motions at the location of EN events. Moreover, there are phase differences in APNSs between the $u_d$ field and $v_d$ field. Similarly, Yu et al. (Reference Yu, Xu and Pirozzoli2019) performed Helmholtz decomposition to study compressibility effects on skin friction in turbulent channel flow. Through a modified FIK decomposition, they found a relatively small contribution of dilatational component $\boldsymbol {u}_d$ (approximately $4\,\%$) to the overall skin-friction coefficient $C_f$ in the cooled channel flow of a bulk Mach number $M_b=4.4$ with the presence of APNSs. However, the dilatational component $\boldsymbol {u}_d$ plays a much more important role in extreme $\tau _w^\prime$ events.

Figure 12. Contours of the conditionally averaged field based on extreme $\tau _w^\prime$ events for M8T05 at the $y^+=15$ plane: (a,b) the wall-normal velocity $v$; (c,d) the solenoidal component $v_s$; (e,f) the dilatational component $v_d$. Panels (a,c,e) and (b,d,f) are for EP and EN events, respectively.

4.4. Conditional analysis on extreme events of wall heat flux

Similarly, extreme $q_w^\prime$ events are analysed in the same framework. According to the joint p.d.f. in figure 5(b) where the $q_w^\prime$ is nearly linearly positively correlated with the $\tau _w^\prime$, the extreme events of $\tau _w^\prime$ and $q_w^\prime$ are mostly overlapped with each other for M2T05 referring to the overlapping percentages in table 5. Thus, the coherent structures associated with the EP events of $q_w^\prime$ shown in figure 13(a) are quite similar to those of $\tau _w^\prime$ shown in figure 9(a). A high-temperature elongated streak shown by the isosurface of $T^+=4$ centring between the counter-rotating quasi-streamwise vortex pair is subjected to sweep motions. The isosurface of this high-temperature streak evolves from thin to wide and to thin again along the streamwise direction, which indicates that the local streamwise temperature gradient changes from positive to negative as shown in the $x$–$y$ plane shifted from $\Delta z^+=0$. The dominating contributing terms are the streamwise mean-convection term $C_f^{Mx}$, the viscous-dissipation term $C_h^V$ and the unsteady term $C_f^{U}$. The counter-rotating vortex pair inducing sweep motions enhances the near-wall shear ${\partial u}/{\partial y}$, thus the EP event of $q_w^\prime$ for M2T05 is mainly the result of strong molecular viscous heating within the viscous sublayer $(y^+<5)$.

Figure 13. Conditional analysis on extreme events of $q_w^\prime$ for M2T05: (a,b) conditionally averaged fluctuation fields, the isosurfaces of high-/low-temperature streaks are flooded by $v$, the vortices displayed by the isosurfaces of $Q$-criterion are flooded by $\omega _x$. An $x$–$y$ plane at $\Delta z^+=0$ and a $y$–$z$ plane at $\Delta x^+=0$ are shifted aside for clearer observation. Components of conditionally averaged heat-transfer coefficients normalized by $C_h$: (c,e) $C_h^i$; (d,f) $C_h^i(y_b)$ as a function of the upper bound of integrals. Panels (a,c,d) and (b,e,f) are for the EP and EN events, respectively.

For the EN event of $q_w^\prime$, the coherent structures shown in figure 13(b) are also similar to those of $\tau _w^\prime$ shown in figure 9(b). A hairpin-like vortex following a pair of counter-rotating quasi-streamwise vortices is identified, and an elongated low-temperature streak shown by the isosurface of $T^+=-4$ centres between these vortices. Under these coherent structures, there is no very dominant term as the streamwise mean-convection term $C_f^{Mx}$, the wall-normal mean-convection term $C_f^{My}$, the spanwise turbulent-convection term $C_f^{Tz}$, the viscous-dissipation term $C_h^V$, the pressure-dilatation term $C_h^P$ and the unsteady term $C_f^{U}$ all have relatively considerable contributions. Compared with the dominating terms of EP events, the magnitudes of these terms of EN events are relatively small. The near-wall molecular viscous heating is highly mitigated. As is shown in the $x$–$y$ plane shifted from $\Delta z^+=0$, streamwise extended compression and expansion motions are located near the EN event, which could be related to the convecting wavepackets within the boundary layer; thus the most important transport mechanism of energy contributing to EN events becomes the pressure-dilatation work $C_h^P$.

The scenario in extreme events of $q_w^\prime$ for M8T05 is quite different. The coherent structures associated with extreme events shown in figure 14(a,b) are formed by local fine-scale spanwise-extending spotty structures near the wall, without the presence of the elongated velocity/temperature streaks and counter-rotating quasi-streamwise vortex pair observed in other extreme events. Intuitively, these dilational spanwise-extending spotty structures correspond to the APNSs directly observed in figure 6(d). The length scale of these structures is also consistent with the corresponding wavelength of the dominant spectral peak in figure 7(d). For the EP event, a fine-scale spanwise extending vortex with negative spanwise vorticity $(\omega _z<0)$ is induced by the upstream expansion and downstream compression shown by isosurfaces of $\theta ^+=\pm 0.07$, respectively. The EP events are favourable to occur behind the inflection points of APNSs where expansion transitions to compression. A vortex sheet with $(\omega _z>0)$ off the wall in the buffer layer is located downstream of the EP event. The high-temperature region of $T^+=2$ becomes a tiny isolated wall-mounted spot at the location of EP events. The dominant energy transport mechanisms seem to be much simpler than those in M2T05, the streamwise convection of the strong APNSs dominates due to their feature of convecting wavepackets. Meanwhile, the contribution of pressure-dilatation work $C_h^P$ is comparable to that of molecular viscous heating $C_h^V$. The mean-convection term $C_h^{Mx}$ and the unsteady term $C_h^{U}$ are overwhelming on other terms and comparable to each other. Their magnitudes are very large, and can even exceed $60C_h$.

Figure 14. Conditional analysis on extreme events of $q_w^\prime$ for M8T05: (a,b) conditionally averaged fluctuation fields, the isosurfaces of dilatation $\theta ^+$ and temperature $T^+$ are flooded by themselves, the vortices displayed by the $Q$-criterion are flooded by $\omega _z$. An $x$–$y$ plane at $\Delta z^+=0$ and a $y$–$z$ plane at $\Delta x^+=0$ are shifted aside for clearer observation. Components of conditionally averaged heat-transfer coefficients normalized by $C_h$: (c,e) $C_h^i$; (d,f) $C_h^i(y_b)$ as a function of the upper bound of integrals. Panels (a,c,d) and (b,e,f) are for the EP and EN events, respectively.

For the EN event, the coherent structures are also mainly associated with the APNSs. Compared with the EP event, the signs of the spanwise vorticity of the identified vortex and the dilatation are inverse, namely a fine-scale spanwise-extending near-wall vortex with positive spanwise vorticity $(\omega _z>0)$ is induced by the upstream compression and downstream expansion shown by isosurfaces of $\theta ^+\pm 0.07$. The EN events are favourable to occur behind the inflection points of APNSs where compression transitions to expansion. The low-temperature regions of $T^+=-3$ are shown to be two parts: one is also a tiny isolated wall-mounted spot at the location of EN events; another is a large-scale off-wall bubble. Similarly, the mean-convection term $C_h^{Mx}$ and the unsteady term $C_h^{U}$ are also overwhelming on other terms and comparable to each other. Compared with EP events, the growth of these two terms is slower persisting up to the wake, due to the low-temperature off-wall bubble. Meanwhile, the pressure-dilatation term $C_h^P$ is also contributive to EN events.

Since the two terms $C_h^{Mx}$ and $C_h^{U}$ are overwhelming and almost cancel each other in both EP and EN events, to more clearly compare with other mechanisms, $C_h^U$ and $C_h^{Mx}$ are combined and denoted by $C_h^{MU}=C_h^{Mx}+C_h^U$, which is a measure of comprehensive contributions by convecting structures. As shown in figure 15, the magnitudes of the combined components $C_h^{MU}$ become much smaller and comparable to other components. For EP events, the pressure-dilatation term $C_h^P$ and viscous-dissipation term $C_h^V$ play dominating roles, while $C_h^V$ is expected to be always positive; and for EN events, the dominating contribution of $C_h^P$ is highlighted. In this sense, the pressure-dilatation terms $C_h^P$, whose magnitudes grow rapidly near the wall $(y^+<10)$, have the greatest contribution to both EP and EN events. These findings are consistent with the observations of the near-wall strong-dilatation structures APNSs in the conditionally averaged field.

Figure 15. Recombined components of conditionally averaged heat-transfer coefficients for M8T05 normalized by $C_h$: (a,c) $C_h^i$; (b,d) $C_h^i(y_b)$ as a function of the upper bound of integrals. Panels (a,b) and (c,d) are for the EP and EN events, respectively.