3.1. Mean property variations

Since all simulated cases have both heated and cooled walls, the fluid is of high density near the cooled wall and of low density near the heated wall. Figures 4(a–c) show instantaneous values of density in cases A, B and C, respectively. Near the wall, the mixing between high-density fluid and low-density fluid is enhanced by turbulent motions. Furthermore, a clear interface of density change in the vicinity of the heated wall can be seen in case C, indicating a substantially large density gradient. The corresponding distributions of Prandtl number ( $\textit {Pr}$ ) are presented in figures 4(d–f). It is found that the maximum value of $\textit {Pr}$ , corresponding to the pseudo-boiling point (black contour line), is observed very close to the heated wall in all cases.

Figure 5 depicts the wall-normal profiles of normalised mean temperature and enthalpy in all cases. The pseudo-boiling point shifts towards the heated wall with increasing wall temperature difference, resulting in increasingly sharp gradients in temperature and enthalpy near the heated wall. This trend significantly impacts the density profile. Figure 6(a) illustrates the wall-normal distributions of the mean density. Significant mean density variation is observed near the heated wall in all cases. In particular, in case C with the largest temperature difference ( $\unicode{x0394} T = 150\ \text {K}$ ), the density near the heated wall ( $y\in [1.75, 2.0]$ ) decreases by approximately 5.8 times. Figure 6(b) shows the wall-normal profiles of the r.m.s. density fluctuation $\rho '_{\textit {rms}}$ relative to the local mean values of density $\bar {\rho }$ in all cases, where $\rho '_{\textit {rms}} = \sqrt {\overline {\rho ^2} - \bar {\rho }^2}$ . The r.m.s. fluctuations are much stronger near the heated wall than those near the cooled wall. This could be attributed to the fact that stronger mean property variations occur near the heated wall where the pseudo-critical point exists. In cases B and C with large mean density variation, $\rho '_{\textit {rms}}$ reaches 45 % to 55 % of the mean density near the heated wall. Such a level of density fluctuation renders Morkovin’s hypothesis invalid, i.e. the large amplitude of density fluctuation could exert a substantial influence on turbulence statistics. It determines the relative differences between Favre- and Reynolds-averaged statistics, resulting in discrepancies of approximately 12 % for the first-order moments, and approximately 30 % for the second-order moments (see Appendix A). To obtain deeper insight, the mechanisms underlying the interactions between density fluctuation and turbulence statistics are explored systematically in the following subsections. For demonstration, we will choose three typical turbulence statistics involved in the transport of momentum and energy in fully developed channel flows: TKE, turbulent shear stress and turbulent heat flux. They represent the salient characteristics of turbulent flows and heat transfer, and are also important for turbulence modelling in SCP fluid flows.

Figure 5.

Distributions of normalised mean (a) temperature, and (b) enthalpy. Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

Figure 6.

Distributions of (a) normalised mean density, and (b) r.m.s. fluctuation of density. Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

3.2. Effect of density fluctuation on TKE transport

The effect of density fluctuation on TKE transport is explored first. The TKE, defined as $k = \tfrac {1}{2} \overline {\rho }\, \widetilde {u_i'' u_i''}$ , can be decomposed into the mean density contribution $k_{-m}$ and the density fluctuation contribution $k_{-f}$ :

(3.1) \begin{equation} k = \tfrac {1}{2} \overline {\rho }\, \widetilde {u_i'' u_i''} = \underbrace {\tfrac {1}{2} \overline {\rho }\, \overline {u_i'' u_i''}}_{k_{-m}} + \underbrace {\tfrac {1}{2} \overline {(\rho ' u_i'' u_i'')}}_{k_{-f}}. \end{equation}

Figure 7(a) displays the TKE distributions along the wall-normal direction. In contrast to the trend of density fluctuation in figure 6(b), the magnitude of TKE near the cooled wall is much higher than that near the heated wall. In the cooled near-wall region ( $y \lt 0.2$ ), the TKE exhibits a rise with increasing density ratios, while the opposite trend occurs on the heated wall side. Figure 7(b) illustrates the distributions of the density fluctuation component, $k_{-f}$ , in TKE in all cases. Remarkably, the amplitude of the density fluctuation component ( $k_{-f}$ ) near the heated wall increases with the density ratio, and $k_{-f}$ tends to attenuate the TKE, as manifested by the negative sign of $k_{-f}$ troughs.

Figure 7.

Distributions of (a) TKE $k$ and (b) its fluctuations contribution $k_{-f}$ , as function s of $y$ . Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

Utilising DNS data, it becomes feasible to explicitly derive all terms in the TKE budget equation for a comprehensive evaluation of the density fluctuation effect on TKE. The equation for TKE is given by

(3.2) \begin{align} \frac {\partial k}{\partial t} + \frac {\partial (\tilde {u}_j k)}{\partial x_j} &= \underbrace {-\overline {\rho u_i'' u_j''} \frac {\partial \tilde {u}_i}{\partial x_j}}_{P_k} + \underbrace {\overline {p' \frac {\partial u_i''}{\partial x_i}}}_{\phi _k} + \underbrace {-\frac {\partial \overline {p' u_i''}}{\partial x_i}}_{\pi _k}\nonumber\\& \quad +\underbrace {-\overline {u_i''} \frac {\partial \overline {p}}{\partial x_i}}_{E_{k1}} + \underbrace {\overline {u_i''} \frac {\partial \overline {\tau _{ij}}}{\partial x_j}}_{E_{k2}} +\underbrace {\overline {-\tau _{ij}' \frac {\partial u_i''}{\partial x_j}}}_{\epsilon _k} + \underbrace {-\frac {1}{2} \frac {\partial \overline {\rho u_i'' u_i'' u_j''}}{\partial x_j}}_{T_k} + \underbrace {\frac {\partial \overline {\tau _{ij}' u_i''}}{\partial x_j}}_{V_k}, \end{align}

where $P_k$ is the turbulence production, $\phi _k$ and $\pi _k$ are the pressure–velocity-gradient correlation terms, $\epsilon _k$ is the turbulence dissipation, $T_k$ is the turbulence diffusion, $V_k$ is the viscous diffusion, and $E_{k1}$ and $E_{k2}$ are the additional terms induced by density fluctuation. Figure 8 presents the spatial distributions of the TKE budgets along the wall-normal direction. Here, $P_k$ augments near the cooled wall as the cooled wall temperature decreases from case A to case C, and the peak magnitude moves away from the wall. Adjacent to the heated wall, $P_k$ decreases as the heated wall temperature increases from case A to case C, and the peak moves towards the heated wall. Meanwhile, $T_k$ , $\epsilon _k$ and $V_k$ decrease to partially balance $P_k$ . This behaviour may be attributed to the significant decrease in viscosity on the heated wall side. Although previous studies explained this behaviour as the result of strong viscosity stratification phenomena on the heated wall side (Zonta, Marchioli & Soldati Reference Zonta, Marchioli and Soldati2012; Lee et al. Reference Lee, Jung, Sung and Zaki2013), we would like to elucidate it from the perspective of density fluctuation by decomposing $P_k$ into contributions from mean density and density fluctuation. Figure 9(a) illustrates the density fluctuation contribution, $P_{k-f}$ , to turbulence production. It is observed that $P_{k-f}$ tends to weaken $P_k$ near both cooled and heated walls. As the density ratio increases from case A to case C, the amplitude of $P_{k-f}$ decreases near the cooled wall but increases near the heated wall, which is consistent with the variational trend of density fluctuation in figure 6(b). It is thus deduced that density fluctuation is responsible for the reduction in $P_k$ . Stronger density fluctuation inhibits turbulence production more in the transcritical region near the heated wall. Moreover, as $P_k$ is closely related to the Reynolds shear stress, further analysis will be conducted subsequently for additional insights.

Figure 8.

TKE budgets along the wall-normal direction: (a) case A, (b) case B, and (c) case C.

Figure 9.

(a) Density fluctuation contribution $ P_{k-f}$ , and (b) the sum of additional terms induced by density fluctuation as a function of $ y/\delta$ . Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

The additional terms $E_{k1}$ and $E_{k2}$ in the TKE budgets also show interesting information. As shown in figures 8 and9(b), $E_{k1}$ and $E_{k2}$ are negligible near the cooled wall due to small density fluctuation. Adjacent to the heated wall, $E_{k1}$ and $E_{k2}$ become comparable to other terms, attributed to the intensification of density fluctuation induced $\overline {u_i''}$ (see Appendix A). Since $\overline {u_i''} = -\overline {\rho ' u_i'}/\overline {\rho }$ , density fluctuation can offset the reduction in turbulence production by additional mass flux.

As the turbulence production is dependent on the Reynolds shear stress (3.2), the Reynolds shear stress $\overline {\rho }\, \widetilde {u''v''}$ is examined to further explore the impact of the density fluctuation on turbulence quantity. Similar to TKE, $\overline {\rho }\, \widetilde {u''v''}$ is decomposed into the mean density contribution $R_m$ and fluctuation contribution $R_f$ :

(3.3) \begin{equation} \overline {\rho }\, \widetilde {u''v''} = \overline {\rho u''v''} = \underbrace {\overline {\rho }\, \overline {u''v''}}_{R_m} + \underbrace {\overline {\rho ' u''v''}}_{R_f}. \end{equation}

Figure 10 shows the Reynolds shear stress $\overline {\rho }\, \widetilde {u''v''}$ as well as its density fluctuation contribution $R_f$ . In SCP fluid flows, the strong variation of mean density renders asymmetric distributions of $\overline {\rho }\, \widetilde {u''v''}$ on two sides of the channel walls. The amplitude of the Reynolds shear stress is higher near the cooled wall. Furthermore, the impact of density fluctuation contribution to $\overline {\rho }\, \widetilde {u''v''}$ is significant in the near-wall regions ( $y=0{-}0.5$ on the cooled side, and $y=1.5{-}2$ on the heated side), with a particularly pronounced effect observed adjacent to the heated wall (figure 10 b). As shown in figure 11, these regions correspond mainly to the viscous wall region ( $y^*\lt 50$ ).

Figure 10.

(a) Reynolds shear stress $\overline {\rho }\, \widetilde {u''v''}$ , and (b) density fluctuation contribution $R_f$ . Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

Figure 11.

The correspondence between $y^*$ and $y$ : (a) cooled wall side, and (b) heated wall side.

Figure 12.

Quadrant analysis of Reynolds shear stress $\overline {\rho }\, \widetilde {u''v''}$ : (a) cooled side, and (b) heated side. Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

The specific mechanism underlying the effects of density fluctuation can be further scrutinised utilising the quadrant analysis (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972). Quadrant analysis provides insight into the impact of instantaneous fluctuations of individual elements on turbulence statistics. For the present study, this analysis can elucidate the positive or negative contributions of density fluctuation to Reynolds shear stress and turbulent heat flux.

We first employ the quadrant analysis to examine the density fluctuation effect on the Reynolds shear stress $\overline {\rho }\, \widetilde {u'' v''}$ . The plane of concern is categorised into four quadrants based on the signs of $u''$ and $v''$ , denoted as Q1 ( $u''\gt 0,\ v''\gt 0$ ), Q2 ( $u''\lt 0,\ v''\gt 0$ ), Q3 ( $u''\lt 0,\ v''\lt 0$ ) and Q4 ( $u''\gt 0,\ v''\lt 0$ ). Figure 12 presents the conditional means of the Reynolds shear stress $\overline {\rho }\, \widetilde {u'' v''}$ in each of these quadrants within the viscous wall region. In the cooled wall boundary layer, quadrants Q1 and Q3 correspond to ‘outward’ events (high-speed fluid from the sweep being reflected back towards the central region) and ‘inward’ events (low-speed fluid from the decelerated region being pushed back towards the wall), respectively, while quadrants Q2 and Q4 represent ‘ejection’ events (low-speed fluid moving away from the wall) and ‘sweep’ events (high-speed fluid moving towards the wall), respectively. As depicted in figure 12(a), the contributions from quadrants Q2 and Q4 are significant, and increase as the cooled wall temperature decreases. Given that the Reynolds shear stress is $\overline {\rho }\, \widetilde {u'' v''}\lt 0$ near the cooled wall (see figure 10 a), quadrant Q2 demonstrates a positive contribution from density fluctuation, manifested by positive density fluctuation with the physical event of the cold fluid moving away from the cooled wall ( $\rho ' \gt 0$ ). In quadrant Q4, where the hot fluid is cooled, negative density fluctuation ( $\rho '\lt 0$ ) leads to $\overline {\rho ' u'' v''}\gt 0$ , resulting in a negative contribution to the Reynolds shear stress. To provide a more intuitive description of the density fluctuation contribution, figure 13(a) illustrates $\overline {\rho ' u'' v''}$ near the cooled wall in the four quadrants. The dominance of Q2 and Q4 events aligns with figure 12(a). Furthermore, a noteworthy observation is that the absolute magnitude of the density fluctuation contribution in Q2 surpasses that in Q4, revealing that density fluctuation near the cooled wall predominantly contributes positively to Reynolds shear stress through ‘ejection’ motions. In proximity to the heated wall (figures 12 b, 13 b), where $\overline {\rho }\, \widetilde {u'' v''}$ is positive (see figure 10 a), the quadrants Q1 and Q3 now represent ‘sweep’ and ‘ejection’ events, respectively, while quadrants Q2 and Q4 denote ‘inward’ and ‘outward’ events, respectively. As shown in figure 12(b), the ‘sweep’ (Q1) and ‘ejection’ events (Q3) are prominent. Consequently, in quadrant Q1, the fluid adjacent to the hot wall is heated, accounting for $\rho ' \gt 0$ and a positive contribution from density fluctuation. In quadrant Q3, the opposite scenario unfolds. Figure 13(b) also illustrates $\overline {\rho ' u'' v''}$ near the heated wall in the four quadrants. The magnitude of the density fluctuation contribution in Q3 is considerably larger than that in Q1, suggesting that the density contribution on the heated wall side primarily poses a negative influence on the Reynolds shear stress through ‘ejection’ motions.

Figure 13.

Quadrant analysis of density fluctuation contribution $\overline {\rho ' u''v''}$ : (a) cooled side, and (b) heated side. Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

3.3. Effect of density fluctuation on turbulent heat flux

In this subsection, we further examine the turbulent heat flux to explore the mechanism of how large density fluctuation affects turbulent heat transfer. The turbulent heat flux is similarly decomposed into the form

(3.4) \begin{equation} \textit {THF} = \overline {\rho }\, \widetilde {v'' h''} = \overline {\rho v'' h''}= \underbrace {\overline {\rho }\,\overline {v'' h''}}_{\textit {THF}_{-m}} + \underbrace {\overline {\rho ' v'' h''}}_{\textit {THF}_{-f}}. \end{equation}

The turbulent heat flux and its density fluctuation contribution are shown along the wall-normal direction in figures 14(a) and 14(b), respectively. The turbulent heat flux near both cooled and heated walls increases with rising density ratio. Figure 14(b) reveals that the density fluctuation near the heated wall exerts a more pronounced influence, tending to weaken the turbulent heat flux. The underlying explanation will be provided using the budgets of turbulent heat flux. The transport equation for the turbulent heat flux is formulated as

(3.5) \begin{align} &\frac {\overline {\partial \rho u_i'' h''}}{\partial t}+ \frac {\partial \tilde {u}_j \overline {(\rho u_i'' h'')}}{\partial x_j} = \underbrace {-\left (\overline {\rho u_j'' h''} \frac {\partial \tilde {u}_i}{\partial x_j} + \overline {\rho u_i'' u_j''} \frac {\partial \tilde {h}}{\partial x_j}\right )}_{P_{e,i}} + \underbrace {-\frac {\partial \overline {\rho h'' u_i'' u_j''}}{\partial x_j}}_{T_{e,i}}+\underbrace {-\overline {h''\frac {\partial p'}{\partial x_i}} }_{\pi _{e,i}}\nonumber\\ & {}+ \underbrace {\frac {\partial \left (\overline {\tau _{ij}' h''} - \overline {q_j' u_i''}\right )}{\partial x_j}}_{V_{e,i}} + \underbrace {\left (\overline {q_j' \frac {\partial u_i''}{\partial x_j}}-\overline {\tau _{ij}' \frac {\partial h''}{\partial x_j}}\right )}_{\epsilon _{e,i}} +\underbrace {\left (\overline {h''} \frac {\partial \overline {\tau _{ij}}}{\partial x_j} - \overline {h''} \frac {\partial \overline {p}}{\partial x_i}\right )}_{E_{e,i1}} + \underbrace {-\overline {u_i''} \frac {\partial \overline {q_j}}{\partial x_j}}_{E_{e,i2}}. \end{align}

Here, the terms on the right-hand side in (3.5) are production $P_{e,i}$ , turbulence diffusion $T_{e,i}$ , enthalpy–pressure-gradient correlation $\pi _{e,i}$ , molecular diffusion $V_{e,i}$ , dissipation $\epsilon _{e,i}$ , and additional terms $E_{e,i1}$ , $E_{e,i2}$ that mainly stem from density fluctuation. Figure 15 plots the wall-normal turbulent heat flux budgets ( $i=2$ ) with respect to $y$ . Near the cooled wall, $P_{e,2}$ is primarily balanced by $\pi _{e,2}$ and $T_{e,2}$ . Yet the net increase in the production is still observed. All other terms near the cooled wall are relatively small, mainly due to the minor property variations. In the vicinity of the heated wall, the budget terms display substantial differences compared to the counterparts of the cooled wall. An increase in density ratio results in a pronounced increase in $\pi _{e,2}$ , which is due to the enhanced $h''$ induced by density fluctuation. This, in turn, damps $\overline {\rho }\, \widetilde {v'' h''}$ . However, the additional term $E_{e,21}$ , mainly originating from the density fluctuation, also experiences a significant increase, which helps to offset the decrease in the turbulent heat flux. Furthermore, it is noteworthy that the molecular diffusion $V_{e,2}$ exhibits a significant increase near the heated wall, attributed to fluctuations of thermal conductivity and isobaric specific capacity (not shown here). This increase further helps to balance the decrease in the turbulent heat flux, resulting in a net increase.

Figure 14.

(a) Turbulent heat flux $\overline {\rho }\, \widetilde {v'' h''}$ , and (b) fluctuating density contribution $\textit {THF}_{-f}$ . Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

A further examination of the density fluctuation effect on $\overline {\rho }\, \widetilde {v'' h''}$ is carried out using quadrant analysis. The turbulent heat flux $\overline {\rho }\, \widetilde {v'' h''}$ can be decomposed into four different quadrants based on the signs of $v''$ and $h''$ . In the proximity of the cooled wall, as illustrated in figure 16(a), quadrants Q1 ( $v''\gt 0$ , $h''\gt 0$ ) and Q3 ( $v''\lt 0$ , $h''\lt 0$ ) signify ‘hot ejection’ events (hot fluid moving away from the wall) and ‘cold sweep’ events (cold fluid moving towards the wall), respectively. Conversely, quadrants Q2 ( $v''\gt 0$ , $h''\lt 0$ ) and Q4 ( $v''\lt 0$ , $h''\gt 0$ ) represent ‘cold ejection’ events (cold fluid moving from the wall) and ‘hot sweep’ events (hot fluid moving towards the wall), respectively, and they emerge as the dominant contributions to $\overline {\rho }\, \widetilde {v'' h''}$ near the cooled wall. In quadrant Q2, the cold fluid undergoes heating while moving away from the wall, along with an overall positive density fluctuation $\rho '\gt 0$ . This leads to $\overline {\rho ' v'' h''} \lt 0$ , indicating a positive contribution of density fluctuation to $\overline {\rho }\, \widetilde {v'' h''}$ . In contrast, the density fluctuation has a negative contribution in quadrant Q4. Figure 17(a) depicts $\overline {\rho ' v'' h''}$ near the cooled wall in the four quadrants. Note that the density fluctuation contributions in the Q2 and Q4 quadrants are almost compensatory, rendering a relatively small effect of density fluctuation on the cooled wall side, which is consistent with the observation in figure 14(b).

Figure 15.

Turbulent heat flux $\overline {\rho }\, \widetilde {v'' h''}$ budgets along the wall-normal direction. Solid line indicates case A, dashed line indicates case B, dotted line indicates case C.

Figure 16.

Quadrant analysis of wall-normal turbulent heat flux $\overline {\rho }\, \widetilde {v''h''}$ : (a) cooled side, and (b) heated side. Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

Proximity to the heated wall introduces a reversal observation, with quadrants Q1 and Q3 now representing ‘hot sweep’ and ‘cold ejection’ events, respectively. Meanwhile, quadrants Q2 and Q4 showcase ‘cold sweep’ and ‘hot ejection’ events, respectively. Figure 16(b) shows that Q2 and Q4 continue to dominate near the heated wall. In quadrant Q2, the cold fluid experiences heating, leading to $\rho ' \gt 0$ and $\overline {\rho ' v'' h''} \lt 0$ . Thus a positive contribution from density fluctuation is formed. Conversely, in quadrant Q4, the hot fluid away from the heated wall undergoes cooling, with $\rho ' \lt 0$ and $\overline {\rho ' v'' h''} \gt 0$ , generating a negative contribution of density fluctuation. Figure 17(b) shows the distribution of $\overline {\rho ' v'' h''}$ in the four quadrants near the heated wall. The dominance of Q2 and Q4 events aligns with the result in figure 16(b). Notably, the absolute magnitude of the density fluctuation contribution in quadrant Q4 surpasses that in quadrant Q2, revealing that density fluctuation predominantly contributes negatively to $\overline {\rho }\, \widetilde {v'' h''}$ through ‘hot ejection’ motions near the heated wall.

Table 3.

The skin friction coefficient and Nusselt number at the cooled and heated walls, respectively.

Figure 17.

Quadrant analysis of density fluctuation contribution $\overline {\rho ' v''h''}$ : (a) cooled side, and (b) heated side. Black solid line indicates case A, red dashed line indicates case B, blue dotted line indicates case C.

3.4. Effect of density fluctuation on skin friction coefficient and Nusselt number

Since the effects of density on turbulence and heat transfer, as observed, ultimately manifest in the skin friction coefficient and Nusselt number, we have computed these quantities in table 3. The equations used for these calculations are

(3.6) \begin{align} C_{f_{wc}} = \frac {\tau _{wc}^\dagger }{\frac {1}{2} \rho _{b}^\dagger (u_b^\dagger )^2}, \quad C_{f_{wh}} = \frac {\tau _{wh}^\dagger }{\frac {1}{2} \rho _{b}^\dagger (u_b^\dagger )^2},\end{align}

(3.7) \begin{align} Nu_{wc} = \frac {2 q_{w}^\dagger \delta ^\dagger }{\lambda _{b}^\dagger | T_{wc}^\dagger - T_{b}^\dagger |}, \quad Nu_{wh} = \frac {2 q_{w}^\dagger \delta ^\dagger }{\lambda _{b}^\dagger | T_{wh}^\dagger - T_{b}^\dagger |},\end{align}

where the subscript ‘b’ refers to the bulk values. Here, $ \tau _w^\dagger$ and $ q_w^\dagger$ represent the wall shear stress and total wall heat flux, respectively.

Near the cooled wall, where density fluctuations are minimal, the skin friction coefficients $ C_{f_{wc}}$ are higher than those of $ C_{f_{wh}}$ on the heated wall side, where density fluctuations are more pronounced in all cases. In addition, $ C_{f_{wh}}$ continues to decrease with increasing density ratio. This suggests that significant density fluctuations could have a negative effect on the skin friction coefficient. For the Nusselt number, it is observed that $ Nu_{wc}$ is larger than $ Nu_{wh}$ , which shows that augmented density fluctuations could attenuate the heat transfer at the heated wall. The shear stress and heat flux balances can further facilitate understanding the effect of density fluctuations on skin friction coefficient and Nusselt number. For the shear stress, we derived the stress balance from the mean momentum equation

(3.8) \begin{equation} \overline {\mu\, {{\rm d}u}/{{\rm d}y}} - \overline {\rho }\, \widetilde {u''v''} = \overline {\tau }_{{total}}, \end{equation}

where on the left-hand side, the first term represents viscous shear stress, and the second term represents turbulent shear stress; $ \overline {\tau }_{{total}}$ represents the total shear stress. The wall-normal distribution of viscous shear stress term is shown in figure 18(a). We observe that the viscous shear stress increases with density ratio at the cold wall, whereas it decreases at the hot wall. In most regions, the viscous shear stress is relatively small compared to the turbulent shear stress (see figure 10 a). As a result, the total shear stress is primarily governed by turbulent shear stress. This finding is consistent with previous work (Nemati et al. Reference Nemati, Patel, Boersma and Pecnik2016b ), which quantified the individual laminar contribution (induced by viscous shear stress) and turbulent contribution (induced by turbulent shear stress) to the total skin friction using the Fukagata–Iwamoto–Kasagi identity (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002). Their results demonstrated that the turbulent contribution is dominant. As shown in § 3.2, our findings reveal that density fluctuations enhance turbulent shear stress at the cold wall but weaken it at the hot wall, which aligns with the skin friction coefficient in table 3. It is reasonable to conclude that density fluctuations affect the total shear stress in a similar manner.

Figure 18.

Distributions of (a) viscous shear stress $\overline {\mu\, {{\rm d}u}/{{\rm d}y}}$ , and (b) molecular heat flux $\overline {\alpha\, {{\rm d}h}/{{\rm d}y}}$ along the wall-normal direction.

A similar analysis is performed on the mean energy equation, yielding the following total heat flux balance:

(3.9) \begin{equation} \overline {\alpha\, {{\rm d}h}/{{\rm d}y}} - \overline {\rho }\, \widetilde {v''h''} = \overline {q}_{{total}}, \end{equation}

where on the left-hand side, the first term represents molecular heat flux, and the second term turbulent heat flux; $ \overline {q}_{{total}}$ represents the total heat flux. The normal distributions of molecular heat fluxes are shown in figure 18(b). It is found that the molecular heat flux increases with density ratio at the cold wall, whereas it decreases at the hot wall. Furthermore, in most regions, the molecular heat flux is small, and turbulent heat flux (see figure 14 a) dominates the total heat flux, in line with previous findings (Nemati et al. Reference Nemati, Patel, Boersma and Pecnik2016b ). Consequently, it is suggested that the effects of density fluctuations on turbulent heat flux observed in § 3.3 also extend to the total heat flux. That is, the large density fluctuation near the heated wall could exert a negative contribution to the total heat flux (consistent with $ Nu_{wh}$ in table 3) by ‘hot ejection’ motions.

3.5. Instantaneous flow characteristics

Figure 19.

Instantaneous cross-sectional views ( $x-z$ plane) of SCP channel flow in case C: (a) cooled wall side at $ y^* = 25$ with black contour lines of $ \rho ' = -0.1$ , and blue contour lines of $ \rho ' = 0.1$ ; (b) heated side at $ y^* = 16$ with black contour lines of $ \rho ' = -0.7$ , and blue contour lines of $ \rho ' = 0.4$ .

Figure 20.

Iso-surfaces of near-wall (a) low-speed ( $ u''/u_\infty = -0.15$ ) and (b) high-speed ( $ u''/u_\infty = 0.15$ ) structures for case C.

Figure 19 provides snapshots of the streamwise velocity fluctuation $ u''$ , wall-normal velocity fluctuation $ v''$ , Reynolds shear stress fluctuation term $ \rho ' u'' v''$ , and turbulent heat flux term $ \rho ' v'' h''$ at $ y^* = 25$ (cooled side) and $ y^* = 16$ (heated side) on the $ x$ – $ z$ plane in case C, where $ u''$ and $ v''$ are normalised using the maximum value of the streamwise velocity, $ u_\infty$ . Contour lines represent density fluctuation at prescribed values. In figure 19(a), it can be observed that contour lines at $ \rho ' = 0.1$ correspond to the upward low-speed region ( $ u'' \lt 0$ , $ v'' \gt 0$ ), while contour lines at $ \rho ' = -0.1$ are associated with the downward high-speed region ( $ u'' \gt 0$ , $ v'' \lt 0$ ). These regions respectively signify the prevalence of ‘ejection’ and ‘sweep’ events near the cooled wall, where density fluctuation generates additional negative or positive Reynolds shear stress and turbulent heat flux. This observation aligns with the quadrant analysis results. Near the heated wall, as shown in figure 19(b), density fluctuation is stronger, where contour lines at $ \rho ' = 0.4$ coincide with the downward high-speed region ( $ u'' \gt 0$ , $ v'' \gt 0$ ), and $ \rho ' = -0.7$ is highly correlated with the upward low-speed region ( $ u'' \lt 0$ , $ v'' \lt 0$ ). These also correspond to ‘sweep’ and ‘ejection’ events near the heated wall, where significant additional Reynolds shear stress and turbulent heat flux are generated.

Figures 20(a) and 20(b) present iso-surface plots of high-speed and low-speed structures, coloured by density fluctuation. It is revealed that on the cooled wall side, the velocity structures with small density fluctuation lift and gradually form horseshoe shapes (red circle), which induces substantial Reynolds shear stress (Robinson Reference Robinson1991). On the heated wall side, the structures with large density fluctuation become sparse and elongated (orange circle). Thus the Reynolds stress is suppressed.